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Abstract

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at “good enough” prime. In this paper, we initiate the study of this topic by imposing increasingly strong hypotheses on basic algebras. When the algebras satisfy the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. The examples include the Solomon descent algebras of finite Coxeter groups at large enough primes, nilCoxeter algebra, and certain finite semigroup algebras at an arbitrary prime.

1 Introduction

In the representation theory of finite groups, characteristic zero and “good enough” prime characteristic behave the same way. Namely in both cases, the group algebra is semisimple. If the fields are algebraically closed then the simple modules correspond and have the same dimensions. Here, the characteristic is good enough if it does not divide the group order.

One may wonder whether a similar situation holds for classes of algebras that are not semisimple in characteristic zero. A general setup might involve an algebra defined over a suitable ring of integers, where one can either pass to the field of fractions, or reduce modulo a maximal ideal, and then ask what it might mean for a maximal ideal to be good enough for the algebra.

There are various questions that arise here, and correspondingly there are various notions of “good enough”. For example, one might at least want the simple modules to correspond, and have the same dimensions. And then, how similar are the cohomology algebras? Of course, they cannot be isomorphic, as they have different characteristics. But at least we can compare their dimensions in each degree. To refine this, we could examine the product structure and secondary operations, define various constants in terms of these, and compare them. We can also compare the Loewy layers of their projective indecomposable modules, and other more subtle invariants of this sort.

Our purpose in this paper is to begin an investigation of this topic by examining the case where, over the field of fractions, we have a finite-dimensional basic algebra. We present a series of increasingly strong hypotheses on the algebra. It turns out that for the equality of dimensions of cohomology groups between simple modules, a fairly weak hypothesis suffices. The equality of multiplicities of simple modules in Loewy layers of projective indecomposables requires a much stronger hypothesis, and we give examples to show that the weaker hypotheses are not sufficient for this. Related questions are examined with different methods in [34, 35].

The examples of basic algebras we examine in detailed are the Solomon descent algebra of a finite Coxeter group [1–3, 7, 10–13, 16, 19, 24, 28, 29, 31, 33], the nilCoxeter algebra [8, 17, 22], and certain classes of finite semigroup algebras [6, 9, 25, 26, 30, 32]. The descent algebra was originally constructed by Solomon as a non-commutative analogue of the Mackey’s multiplication formula for permutation characters of the parabolic subgroups of the Coxeter group. But it can also be obtained as the opposite algebra of the fixed-point space of a certain face algebra of hyperplane arrangement in real space associated with the Coxeter group. The face semigroup algebra belongs to a broader class known as unital left regular band algebras, which, in turn, are part of the even larger class known as the |$\mathcal{R}$|-trivial monoid algebras. Notably, this largest class also includes the |$0$|-Hecke algebras.

We show that the strongest of our hypotheses holds for the descent algebra for sufficiently large primes and the nilCoxeter algebra for an arbitrary prime. In the case of the semigroup algebras that we investigate, we show that the weaker hypotheses hold under suitable conditions, but we leave open the question of when the stronger one does. While our results do not give an explicit description of the cohomology groups nor the multiplicities of simple modules in Loewy layers of projective indecomposables for these classes of algebras, they offer a general explanation why these notions behave the same way in characteristic zero and at “good enough” prime.

In preparation for this task, in the next section, we give a general discussion of various hypotheses (Hypotheses A–E) on a finitely generated algebra over a local principal ideal domain, and what they imply. The main results are summarised in Theorem 2.3. Sections 3 to 6 are devoted to the proof of the theorem. Then we apply the results to the cases of the descent algebra of a finite Coxeter group, the nilCoxeter algebra, and certain semigroup algebras and obtain Theorems 9.15, 11.2, 12.2, and 13.2. We begin the general discussion on the descent algebra in Section 7 and extend the result of the second author regarding the idempotents of the descent algebra to all types and all primes in Section 8. In Section 9, we show that the descent algebra satisfies our strongest of our hypotheses at large enough prime. The analysis is based on a certain characteristic free basis of the descent algebra and we further investigate that in Section 10. Another algebra satisfying our strongest hypothesis is the nilCoxeter algebra and that is discussed in Section 11. In Sections 12 and 13, we show that certain classes of semigroup algebras satisfy our weaker hypotheses.

2 The Hypotheses and Main Results

Throughout the paper, we shall use the following notation.

 

Notation 2.1.

  • (i)

    Let |${\mathcal{O}}$| be a local principal ideal domain with maximal ideal |$(\pi )$|⁠. Write |${\mathbb{K}}$| for the field of fractions of |${\mathcal{O}}$| and |${{\mathbb{F}}}$| for the residue field |${\mathcal{O}}/(\pi )$| of characteristic |$p$|⁠.

  • (ii)

    Let |$A$| be an |${\mathcal{O}}$|-torsion free |${\mathcal{O}}$|-algebra, and let |${\bar A}={{\mathbb{F}}}\otimes _{\mathcal{O}} A$|⁠, |${\hat A}={\mathbb{K}}\otimes _{\mathcal{O}} A$|⁠.

  • (iii)
    Let |$M$| and |$N$| be |${\mathcal{O}}$|-free |$A$|-modules, and set
    $$ \begin{align*} & \bar M={{\mathbb{F}}}\otimes_{\mathcal{O}} M,\ \bar N={{\mathbb{F}}}\otimes_{\mathcal{O}} N,\ \hat M={\mathbb{K}}\otimes_{\mathcal{O}} M,\ \hat N={\mathbb{K}}\otimes_{\mathcal{O}} N. \end{align*} $$
  • (iv)

    We write |$\textsf{Rad}(R)$| for the Jacobson radical of a ring |$R$|⁠, and we write |$J_{A}$| for |$A\cap \textsf{Rad}({\hat A})$|⁠. By convention, |$J_{A}^{0}=A$|⁠.

  • (v)

    We denote a general field |$k$| and assume that it has characteristic |$q$| (either |$q=0$| or |$q>0$|⁠).

 

Hypothesis A.

If |$M$| and |$N$| are finitely generated |${\mathcal{O}}$|-free |$A$|-modules with |$\hat N$| simple then |$\textsf{Ext}^{t}_{A}(M,N)$| is |${\mathcal{O}}$|-free for all |$t\geqslant 0$|⁠.

 

Hypothesis B.
Suppose that the algebra |$A$| has finite rank over |${\mathcal{O}}$|⁠, we have
$$ \begin{align*} & \textsf{Rad}(A)=\pi A+J_{A}, \end{align*} $$
and orthogonal idempotent decompositions of the identity in |${\bar A}$| lift to |$A$|⁠.

 

Hypothesis C.

We have |${\hat A}$| is basic and |$A$|⁠, |$A/J_{A}$| and |$J_{A}/J_{A}^{\,2}$| are all |${\mathcal{O}}$|-free.

 

Hypothesis D.

Let |$Q$| be a finite quiver, and let |${\mathcal{O}} Q$| be its quiver algebra. This is an |${\mathcal{O}}$|-free |${\mathcal{O}}$|-algebra with basis the directed paths in |$Q$|⁠. The paths of length zero are idempotent. The paths of length at least one form a two-sided ideal |$J_{Q}$| in |${\mathcal{O}} Q$|⁠. Let |$I$| be a two-sided ideal of |${\mathcal{O}} Q$| such that there exists |$n\geqslant 2$| with |$J_{Q}^{n}\leqslant I\leqslant J_{Q}^{2}$|⁠, and with the property that the quotient ring |$A={\mathcal{O}} Q/I$| is |${\mathcal{O}}$|-free of finite rank.

 

Notation 2.2.
If Hypothesis D holds, we write
$$ \begin{gather*} \hat J_{Q}={\mathbb{K}}\otimes_{\mathcal{O}} J_{Q},\qquad \bar J_{Q}={{\mathbb{F}}}\otimes_{\mathcal{O}} J_{Q},\\ \hat I={\mathbb{K}}\otimes_{\mathcal{O}} I\leqslant{\mathbb{K}} Q,\qquad\bar I={{\mathbb{F}}}\otimes_{\mathcal{O}} I \leqslant{{\mathbb{F}}} Q,\\ J_{A}=J_{Q}/I\leqslant A,\quad \hat J_{A}= {\mathbb{K}}\otimes_{\mathcal{O}} J_{A},\quad \bar J_{A}={{\mathbb{F}}}\otimes_{\mathcal{O}} J_{A}. \end{gather*} $$
We refer to the images in |$A$|⁠, |${\hat A,}$| and |${\bar A}$| of the paths of length zero in |${\mathcal{O}} Q$| as the vertex idempotents. Furthermore, there is a set |$S_{1},\dots ,S_{m}$| of |$A$|-modules that are |${\mathcal{O}}$|-free of rank one, corresponding to the vertex idempotents. For |$1\leqslant i\leqslant m$|⁠, we define |$\hat S_{i}={\mathbb{K}}\otimes _{\mathcal{O}} S_{i}$| as an |${\hat A}$|-module, and |$\bar S_{i}={{\mathbb{F}}}\otimes _{\mathcal{O}} S_{i}$| as an |${\bar A}$|-module. Then |$\hat S_{1},\dots ,\hat S_{m}$| is a complete set of simple |${\hat A}$|-modules, and |$\bar S_{1},\dots ,\bar S_{m}$| is a complete set of simple |${\bar A}$|-modules. We write |$P_{i}$| for the projective cover of |$S_{i}$|⁠, namely the image of the vertex idempotent on |$A$|⁠. Then |$\hat P_{i}={\mathbb{K}}\otimes _{\mathcal{O}} P_{i}$| is the projective cover of |$\hat S_{i}$| over |${\hat A}$|⁠, and |$\bar P_{i}={{\mathbb{F}}}\otimes _{\mathcal{O}} P_{i}$| is the projective cover of |$\bar S_{i}$| over |${\bar A}$|⁠.

 

Hypothesis E.

Hypothesis D holds, and for all |$n\geqslant 1$|⁠, |$A/J_{A}^{n}$| is |${\mathcal{O}}$|-free.

We investigate the implications of the above-mentioned hypotheses and study the Ext groups and radical layers of the projective indecomposable modules. In particular, we compare these under the change of field. The main result of this paper is given as follows.

 

Theorem 2.3.

  • (i)

    We have Hypothesis E  |$\Rightarrow $| Hypothesis D  |$\Leftrightarrow $| Hypothesis C  |$\Rightarrow $| Hypothesis B  |$\Rightarrow $| Hypothesis A.

  • (ii)
    Let |$t\geqslant 0$| and |$M,N$| be finitely generated |${\mathcal{O}}$|-free |$A$|-modules with |$\hat N$| simple. If Hypothesis A holds for |$A$| (so that |$\textsf{Ext}^{t}_{A}(M,N)$| is |${\mathcal{O}}$|-free), then we have
    $$ \begin{align*} {{\mathbb{F}}}\otimes_{\mathcal{O}}\textsf{Ext}^{t}_{A}(M,N) &\cong \textsf{Ext}^{t}_{\bar A}(\bar M,\bar N), \\{\mathbb{K}} \otimes_{\mathcal{O}} \textsf{Ext}^{t}_{A}(M,N) &\cong \textsf{Ext}^{t}_{\hat A}(\hat M,\hat N). \end{align*} $$
    Furthermore, |$\dim _{{\mathbb{F}}} \textsf{Ext}^{t}_{\bar A}(\bar M,\bar N)=\dim _{\mathbb{K}}\textsf{Ext}^{t}_{\hat A}(\hat M,\hat N)$|⁠.
  • (iii)
    Hypothesis E is equivalent to |$J_{A}^{n}/J_{A}^{n+1}$| are |${\mathcal{O}}$|-free for all |$n\geqslant 0$|⁠. In this case, the radical layer multiplicities of |$\hat P_{i}$| and |$\bar P_{i}$| are equal, that is, for all |$1\leqslant i,j\leqslant m$| and |$n\geqslant 0$|⁠,
    $$ \begin{align*} & [\textsf{Rad}^{n}(\hat P_{i})/\textsf{Rad}^{n+1}(\hat P_{i}):\hat S_{j}]= [\textsf{Rad}^{n}(\bar P_{i})/\textsf{Rad}^{n+1}(\bar P_{i}):\bar S_{j}].\end{align*} $$

The implication Hypothesis E  |$\Rightarrow $| Hypothesis D is clear. Hypothesis D  |$\Rightarrow $| Hypothesis B is proved in Corollary 3.2, Hypothesis D  |$\Leftrightarrow $| Hypothesis C is proved in 3.4, and Hypothesis B  |$\Rightarrow $| Hypothesis A is proved in Theorem 4.5. The consequence of Hypothesis A is proved in Theorem 5.1, and the consequence of Hypothesis E is proved in Theorem 6.3.

3 Quivers and Relations Over |${\mathcal{O}}$|

 

Theorem 3.1.

Suppose that |$A$|⁠, |${\hat A}$| and |${\bar A}$| satisfy Hypothesis D.

  • (i)

    We have |$\hat J_{A}=\textsf{Rad}(\hat A)$| and |$J_{A}=A\cap \hat J_{A}$|⁠.

  • (ii)

    |$J_{A}$| is a nilpotent ideal in |$A$|⁠, and |$A/J_{A}$| is a direct product of copies of |${\mathcal{O}}$| as an algebra, spanned by the vertex idempotents, which are primitive.

  • (iii)

    Idempotents in |${\bar A}$| lift to idempotents in |$A$|⁠.

  • (iv)

    We have |$\dim _{\mathbb{K}}{\hat A}=\dim _{{\mathbb{F}}}{\bar A}$|⁠.

  • (v)

    The ideal |$\hat I\leqslant{\hat A}$| satisfies |$\hat J_{Q}^{n}\leqslant \hat I\leqslant \hat J_{Q}^{2}$|⁠. The ideal |$\hat J_{A}$| is the radical of |${\hat A}$|⁠, and |${\hat A}/\hat J_{A}$| is a direct product of copies of |${\mathbb{K}}$| as an algebra, spanned by the vertex idempotents, which are primitive.

  • (vi)

    The ideal |$\bar I\leqslant{\bar A}$| satisfies |$\bar J_{Q}^{n}\leqslant \bar I\leqslant \bar J_{Q}^{2}$|⁠. The ideal |$\bar J_{A}$| is the radical of |${\bar A}$|⁠, and |${\bar A}/\bar J_{A}$| is a direct product of copies of |${{\mathbb{F}}}$| as an algebra, spanned by the vertex idempotents, which are primitive.

  • (vii)

    We have |$\textsf{Rad}(A)=\pi A+J_{A}=\pi A + (A\cap \textsf{Rad}({\hat A}))$|⁠.

  • (viii)

    Both |$A/J_{A}$| and |$J_{A}/J_{A}^{2}$| are |${\mathcal{O}}$|-free.

 

Proof.

(i) Let |$A_{0}$| be the |${\mathcal{O}}$|-subalgebra of |$A$| spanned by the vertex idempotents, and similarly |${\hat A}_{0}$| in |${\hat A}$|⁠. Then as an |${\mathcal{O}}$|-module we have |$A=A_{0}\oplus J_{A}$|⁠, |${\hat A}={\hat A}_{0}\oplus \hat J_{A}$|⁠. So |$J_{A}=A\cap \hat J_{A}$|⁠.

(ii) Since |$J_{Q}^{n}\leqslant I$|⁠, we have |$J_{A}^{n}=0$|⁠. The quotient |$A/J_{A}$| is isomorphic to |${\mathcal{O}} Q/J_{Q}$|⁠, which is a direct product of copies of |${\mathcal{O}}$| spanned by the vertex idempotents, which are primitive.

(iii) Every idempotent in |${\bar A}$| is conjugate to a sum of vertex idempotents, and therefore lifts to |$A$|⁠.

(iv) This follows from the fact that |$A$| is |${\mathcal{O}}$|-free.

(v) Using part (ii), we see that |$\hat J_{A}$| is nilpotent, and |${\hat A}/\hat J_{A}={\mathbb{K}}\otimes _{\mathcal{O}} (A/J_{A})$| is isomorphic to |${\mathbb{K}} Q/\hat J_{Q}$|⁠, which is a direct product of copies of |${\mathbb{K}}$|⁠, so |$\hat J_{A}=\textsf{Rad}({\hat A})$|⁠.

(vi) Again using part (ii), we see that |$\bar J_{A}$| is nilpotent, and |${\bar A}/\bar J_{A}={{\mathbb{F}}}\otimes _{\mathcal{O}}(A/J_{A})$| is isomorphic to |${{\mathbb{F}}} Q/\bar J_{Q}$|⁠, which is a direct product of copies of |${{\mathbb{F}}}$|⁠, so |$\bar J_{A}=\textsf{Rad}({\bar A})$|⁠.

(vii) If |${{\mathfrak{m}}}$| is a maximal left ideal of |$A$| then since |$1\not \in{{\mathfrak{m}}}$| we have |${{\mathfrak{m}}}\cap{\mathcal{O}}\leqslant (\pi )$|⁠, where |${\mathcal{O}}$| is regarded as embedded in |$A$| as multiples of the identity. Letting |$\bar{{\mathfrak{m}}}$| be the image of |${{\mathfrak{m}}}$| in |${\bar A}$|⁠, we therefore have |$\bar{{\mathfrak{m}}}\cap{{\mathbb{F}}}=0$|⁠. Thus, |${{\mathfrak{m}}}+(\pi )$| is a proper left ideal in |$A$|⁠, and so by maximality |$(\pi )\leqslant{{\mathfrak{m}}}$|⁠. It follows that |$(\pi )\leqslant \textsf{Rad}(A)$|⁠. Since |$J_{A}$| is nilpotent, we also have |$J_{A}\leqslant \textsf{Rad}(A)$|⁠. Finally, |$A/(J_{A}+(\pi ))\cong{\bar A}/\bar J_{A}$| is semisimple by part (vi), and so |$\textsf{Rad}(A)=J_{A}+(\pi )$|⁠.

(viii) Since |$I\leqslant J_{Q}^{2}$|⁠, we have |$J_{A}=J_{Q}/I$| and |$A/J_{A}\cong ({\mathcal{O}} Q/I)/(J_{Q}/I)\cong{\mathcal{O}} Q/J_{Q}$| and |$J_{A}/J_{A}^{2}\cong J_{Q}/J_{Q}^{2}$|⁠. Therefore, both |$A/J_{A}$| and |$J_{A}/J_{A}^{2}$| are |${\mathcal{O}}$|-free.

 

Corollary 3.2.

Hypothesis D implies Hypothesis B.

 

Proof.

This follows from parts (iii) and (vii) of Theorem 3.1.

 

Example 3.3.

Here are a couple of examples that do not satisfy Hypothesis D. The first is |$A={\mathcal{O}}[x]/(x^{2}-x^{3})$|⁠. In this case, |$Q$| has just one vertex, it has one loop corresponding to |$x$|⁠, but |$I=(x^{2}-x^{3})$| contains no power of |$J_{Q}=(x)\leqslant{\mathcal{O}} Q$|⁠. So |$J_{A}$| is not a nilpotent ideal in |$A$|⁠. The second example is |$A={\mathcal{O}}[x]/(\pi x^{2}, x^{3})$|⁠. Then |$x^{2}\ne 0$| but |$\pi x^{2}=0$| in |$A$|⁠, so |$A$| is not |${\mathcal{O}}$|-free. In this example we have |$\dim _{\mathbb{K}}{\hat A}=2$|⁠, |$\dim _{{\mathbb{F}}}{\bar A}=3$|⁠, so part (v) of Theorem 3.1 does not hold.

 

Theorem 3.4.

Hypothesis C is equivalent to Hypothesis D

 

Proof.

By Theorem 3.1 (viii), Hypothesis D implies Hypothesis C. Conversely, suppose that Hypothesis C holds. Since |${\mathcal{O}}$| is a PID and |$A$| is |${\mathcal{O}}$|-free, both |$J_{A}$| and |$J_{A}^{2}$| are |${\mathcal{O}}$|-free. Let |$B_{2}$| be an |${\mathcal{O}}$|-basis for |$J_{A}^{2}$|⁠. Given that |$J_{A}/J_{A}^{2}$| is |${\mathcal{O}}$|-free, the map |$J_{A}\to J_{A}/J_{A}^{2}$| splits. Let |$B_{1}$| be a lift of a basis for |$J_{A}/J_{A}^{2}$| so that |$B_{1}\cup B_{2}$| is a basis for |$J_{A}$|⁠. Similarly, together with the idempotent lifting, let |$B_{0}$| be a lift of a basis for |$A/J_{A}$| consisting of idempotents. So |$B=B_{0}\cup B_{1}\cup B_{2}$| is an |${\mathcal{O}}$|-basis for |$A$|⁠. As such, |$\hat B=\{1\otimes b:b\in B\}$| forms a basis for |${\hat A}$| and |$\{1\otimes b:b\in B_{0}\}$| is a complete set of primitive orthogonal idempotents for |${\hat A}$|⁠. Let |$Q$| be the Ext quiver of the basic algebra |${\hat A}$| where the vertices are labelled by |$B_{0}$|⁠. Define the algebra homomorphism |$\psi :{\mathcal{O}} Q\to A$| by |$\psi (e_{b})=1\otimes b$| for each |$b\in B_{0}$| and, for |$b,b^{\prime}\in B_{0}$|⁠, if |$(1\otimes b^{\prime})(J_{A}/J_{A}^{2})(1\otimes b)$| has a basis |$\{1\otimes (b^{\prime}xb):x\in S\}$| for some subset |$S\subseteq B_{1}$|⁠, mapping the arrows from |$b$| to |$b^{\prime}$| bijectively onto |$S$|⁠. In particular, |$\psi (J_{Q})\subseteq J_{A}$|⁠. By the construction, |$\psi $| is surjective with the kernel |$I$| such that |$I\leqslant J_{Q}^{2}$|⁠. Suppose that |$\textsf{Rad}^{n}({\hat A})=0$|⁠. We have |$\psi (J_{Q}^{n})\subseteq J_{A}^{n}=0$|⁠, that is, |$J_{Q}^{n}\leqslant I$|⁠. Thus, |${\hat A}$| satisfies Hypothesis D.

4 |${\mathcal{O}}$|-freeness of |$\textsf{Ext}$|

In this section, we investigate algebras satisfying Hypothesis B. In particular, we prove that they satisfy Hypothesis A.

 

Lemma 4.1.

If |$A$| satisfies Hypothesis B, then |$A$|⁠, |${\hat A,}$| and |${\bar A}$| are semiperfect. So finitely generated projective modules over these rings satisfy the Krull–Schmidt theorem, and furthermore, all finitely generated modules have minimal projective resolutions, unique up to non-unique isomorphism.

 

Proof.

Since |$\pi A \subseteq \textsf{Rad} (A)$|⁠, |$A/\textsf{Rad} (A)$| is a finite dimensional semisimple |${{\mathbb{F}}}$|-algebra and idempotents lift to |$A$|⁠, it follows that |$A$| is semiperfect. See Lam [23, Chapter 8], especially Definition 23.1 and Propositions 23.5, 24.10, and 24.12.

 

Remark 4.2.

Recall that a surjection from a projective module |$P\to M$| is a projective cover if and only if the kernel is contained in |$\textsf{Rad}(P)$|⁠.

 

Lemma 4.3.

Let |$A$| satisfy Hypothesis B and |$M$| be a finitely generated |${\mathcal{O}}$|-free |$A$|-module with projective cover |$P\xrightarrow{f} M$|⁠. Then the kernel of |$P \xrightarrow{f} M$| is contained in |$P\cap \textsf{Rad}(\hat P)$|⁠, and |$\hat P \to \hat M$| is a projective cover of the |${\hat A}$|-module |$\hat M={\mathbb{K}}\otimes _{\mathcal{O}} M$|⁠.

 

Proof.
If |$x\in \textsf{Ker}(f)$| then by Remark 4.2, |$x \in \textsf{Rad}(P)$|⁠, which by Hypothesis B is equal to |$\pi P+(P\cap \textsf{Rad}(\hat P))$|⁠. If there is such an |$x$| which is not in |$P\cap \textsf{Rad}(\hat P)$|⁠, choose one of the form |$x=\pi ^{n} z+y$| with |$y\in \textsf{Rad}(\hat P)$|⁠, |$z\in P$|⁠, |$z\not \in \textsf{Rad}(P)$| and |$n>0$|⁠. Since |$f(x)=0$| we have
$$ \begin{align*} & \pi^{n} f(z)=-f(y)\in \pi^{n} M\cap \textsf{Rad}(\hat M)= f(\pi^{n} P\cap \textsf{Rad}(\hat P)), \end{align*} $$
and we can write |$f(y)=\pi ^{n} f(y^{\prime})=f(\pi ^{n} y^{\prime})$| with |$y^{\prime} \in P\cap \textsf{Rad}(\hat P)\subseteq \textsf{Rad}(P)$|⁠. Thus,
$$ \begin{align*} & \pi^{n} f(z+y^{\prime})=f(\pi^{n}z)+f(\pi^{n} y^{\prime})=f(\pi^{n}z+y)=f(x)=0 \end{align*} $$
and so |$f(z+y^{\prime})=0$|⁠. But |$z+y^{\prime}\not \in \textsf{Rad}(P)$|⁠, contradicting Remark 4.2.

 

Lemma 4.4.
If |$A$| satisfies Hypothesis B, |$M$| is a finitely generated |${\mathcal{O}}$|-free |$A$|-module, and
$$ \begin{align*} & \dots \to P_{1}\to P_{0}\to M \to 0 \end{align*} $$
is a minimal projective resolution of |$M$| over |$A$|⁠, then after tensoring with |${\mathbb{K}}$|⁠, the sequence
$$ \begin{align*} & \dots \to \hat P_{1}\to \hat P_{0} \to \hat M \to 0 \end{align*} $$
is a minimal projective resolution of |$\hat M$| over |${\hat A}$|⁠.

 

Proof.

The tensored sequence is certainly a projective resolution. It follows from Lemma 4.3 that it is minimal.

 

Theorem 4.5.

If |$A$| satisfies Hypothesis B, and |$M$| and |$N$| are finitely generated and |${\mathcal{O}}$|-free, with |$\hat N$| simple, then for all |$t\geqslant 0$|⁠, |$\textsf{Ext}^{t}_{A}(M,N)$| is |${\mathcal{O}}$|-free.

 

Proof.
Consider a minimal projective resolution of |$M$|  
$$ \begin{align*} & \cdots \to P_{1} \to P_{0} \to M \to 0. \end{align*} $$
By Lemma 4.4, this remains a minimal resolution after tensoring with |${\mathbb{K}}$|⁠. So if |$\hat N$| is simple, the differential in the complex |$\textsf{Hom}_{\hat A}(\hat P_{*},\hat N)$| is zero. Since |$\textsf{Hom}_{A}(P_{*},N)$| is contained in |$\textsf{Hom}_{\mathcal{O}}(P_{*},N)$|⁠, it is |${\mathcal{O}}$|-free and |$\textsf{Hom}_{\hat A}(\hat P_{*},\hat N)={\mathbb{K}}\otimes _{\mathcal{O}}\textsf{Hom}_{A}(P_{*},N)$|⁠. So the differential on |$\textsf{Hom}_{A}(P_{*},N)$| is also zero, and |$\textsf{Ext}^{*}_{A}(M,N)$| is |${\mathcal{O}}$|-free.

 

Example 4.6.
The algebra |$A={\mathcal{O}}[X]/(X^{2})$| satisfies Hypothesis B. Let |$M=A/(X)$| as an |$A$|-module, and let |$N$| be the ideal |$\textsf{Rad}(A)=(\pi ,X)$| as an |$A$|-module. The minimal resolution of |$M$| is given by
$$ \begin{align*} & \cdots \to A \xrightarrow{X}A \xrightarrow{X}A\to M \to 0. \end{align*} $$
Ignoring the augmentation and taking homomorphisms to |$N$|⁠, we get
$$ \begin{align*} & N \xrightarrow{X} N \xrightarrow{X} N \to \cdots \end{align*} $$
so |$\textsf{Hom}_{A}(M,N)\cong{\mathcal{O}}$|⁠, and |$\textsf{Ext}^{t}_{A}(M,N)=(X)/(\pi X) \cong{{\mathbb{F}}}$| for all |$t>0$|⁠. This shows why |$\hat N$| has to be simple in the proof of Theorem 4.5.

5 Comparing |$\textsf{Ext}$| Over |${\mathcal{O}}$|⁠, |${{\mathbb{F}}}$| and |${\mathbb{K}}$|

Let |${\mathcal{O}}$|⁠, |${\mathbb{K,}}$| and |${{\mathbb{F}}}$| be as in the introduction. For this section, we let |$A$| be an |${\mathcal{O}}$|-torsion free |${\mathcal{O}}$|-algebra. Notice that |$A$| is not necessarily finitely generated. But we may continue to assume this if the reader wishes to. Let |${\bar A}={{\mathbb{F}}}\otimes _{\mathcal{O}} A$| and |${\hat A}={\mathbb{K}}\otimes _{\mathcal{O}} A$|⁠. Then |${\hat A}$| is flat as an |$A$|-module.

 

Theorem 5.1.

Let |$M$| and |$N$| be |${\mathcal{O}}$|-torsion free |$A$|-modules.

  • (i)
    Suppose that for |$t\geqslant 0$|⁠, |$\textsf{Ext}^{t}_{A}(M,N)$| is |${\mathcal{O}}$|-torsion free. Then for |$t\geqslant 0$|⁠, we have an isomorphism
    $$ \begin{align*} & {{\mathbb{F}}}\otimes_{\mathcal{O}}\textsf{Ext}^{t}_{A}(M,N) \cong \textsf{Ext}^{t}_{\bar A}(\bar M,\bar N). \end{align*} $$
  • (ii)
    If |$M$| is finitely presented, then for |$t\geqslant 0$|⁠, we have an isomorphism
    $$ \begin{align*} & {\mathbb{K}} \otimes_{\mathcal{O}} \textsf{Ext}^{t}_{A}(M,N) \cong \textsf{Ext}^{t}_{\hat A}(\hat M,\hat N). \end{align*} $$

 

Proof.
(i) Tensoring the short exact sequence
$$ \begin{align*} & 0 \to A \xrightarrow{\pi} A\to{\bar A} \to 0. \end{align*} $$
with |$M$|⁠, we get an exact sequence
$$ \begin{align*} & \cdots \to \textsf{Tor}^{A}_{1}(A,M) \to \textsf{Tor}^{A}_{1}({\bar A},M) \to M \xrightarrow{\pi} M \to \bar M \to 0. \end{align*} $$
We have |$\textsf{Tor}_{s}^{A}(A,M)=0$| for |$s\geqslant 1$|⁠, so |$\textsf{Tor}^{A}_{s}({\bar A},M)=0$| for |$s\geqslant 2$|⁠. Since |$M$| is |${\mathcal{O}}$|-torsion free and multiplication with |$\pi $| is injective, it also follows that |$\textsf{Tor}^{A}_{1}({\bar A},M)=0$|⁠. By Cartan and Eilenberg [15, VI, Proposition 4.1.3], we get an isomorphism
(2)
Next, since |$N$| is |${\mathcal{O}}$|-torsion free, we have a short exact sequence
$$ \begin{align*} & 0 \to N \xrightarrow{\pi} N \to \bar N \to 0. \end{align*} $$
This gives us a long exact sequence
$$ \begin{align*} & \cdots \to \textsf{Ext}^{t}_{A}(M,N) \xrightarrow{\pi} \textsf{Ext}^{t}_{A}(M,N) \to \textsf{Ext}^{t}_{A}(M,\bar N)\to \cdots \end{align*} $$
By the hypothesis, |$\textsf{Ext}^{t}_{A}(M,N)$| is |${\mathcal{O}}$|-torsion free, so multiplication by |$\pi $| is injective. The sequence therefore breaks up into short exact sequences, which shows that
(3)
Combining (2) with (3) proves part (i).
(ii) Consider the exact sequence
$$ \begin{align*} &\cdots\to P_{1}\to P_{0}\to M\to 0\end{align*} $$
with |$P_{t}$|’s projective |$A$|-modules. Since |${\mathbb{K}}$| is flat over |${\mathcal{O}}$|⁠, tensoring with |${\mathbb{K}}$| over |${\mathcal{O}}$|⁠, we have the exact sequence |$\cdots \to \hat P_{1}\to \hat P_{0}\to \hat M\to 0$| where |$\hat P_{t}$|’s are projectives. Since |${\mathbb{K}}\otimes _{\mathcal{O}} \textsf{Hom}_{A}(P_{t},N)\cong \textsf{Hom}_{{\hat A}}(\hat P_{t},\hat N)$|⁠, we get the isomorphism |${\mathbb{K}} \otimes _{\mathcal{O}} \textsf{Ext}^{t}_{A}(M,N) \cong \textsf{Ext}^{t}_{\hat A}(\hat M,\hat N)$|⁠.

6 |${\mathcal{O}}$|-freeness of |$J^{n}/J^{n+1}$|

In this section, we investigate Hypothesis D. We begin with a preliminary lemma.

 

Lemma 6.1.
Let |$X$| be a finitely generated free |${\mathcal{O}}$|-module and let |$Y$| be a submodule. Let |$\hat X={\mathbb{K}}\otimes _{\mathcal{O}} X$|⁠, |$\hat Y={\mathbb{K}}\otimes _{\mathcal{O}} Y$|⁠, |$\bar X={{\mathbb{F}}}\otimes _{\mathcal{O}} X$|⁠, and |$\bar Y$| be the image of |${{\mathbb{F}}}\otimes _{\mathcal{O}} Y\to{{\mathbb{F}}}\otimes _{\mathcal{O}} X$|⁠. Then we have
$$ \begin{align*} & \dim_{{\mathbb{F}}}\bar Y \leqslant \dim_{\mathbb{K}} \hat Y \end{align*} $$
with equality if and only if |$X/Y$| is |${\mathcal{O}}$|-free.

 

Proof.

We have |$\hat X/\hat Y\cong{\mathbb{K}}\otimes _{\mathcal{O}}(X/Y)$|⁠, and

Since |$Y$| is |${\mathcal{O}}$|-free, we have
$$ \begin{align*} \dim_{{\mathbb{F}}} \bar Y&=\dim_{{\mathbb{F}}}{{\mathbb{F}}}\otimes_{\mathcal{O}} Y - \dim_{{\mathbb{F}}}\textsf{Tor}_{1}^{\mathcal{O}}({{\mathbb{F}}},X/Y)\\ &\leqslant \dim_{{\mathbb{F}}}{{\mathbb{F}}}\otimes_{\mathcal{O}} Y=\textsf{rank}_{\mathcal{O}} Y=\dim_{\mathbb{K}}\hat Y \end{align*} $$
with equality if and only if |$\textsf{Tor}_{1}^{\mathcal{O}}({{\mathbb{F}}},X/Y)=0$|⁠. Since |$X/Y$| is finitely generated, this is equivalent to the condition that it is |${\mathcal{O}}$|-free.

The following lemma is useful in verifying Hypothesis E.

 

Lemma 6.2.

Let |$J$| be a nilpotent ideal of |$A$| and suppose that |$A$| is |${\mathcal{O}}$|-free of finite rank. Then the following are equivalent:

  • (i)

    For all |$n\geqslant 1$|⁠, |$A/J^{n}$| is |${\mathcal{O}}$|-free.

  • (ii)

    For all |$n\geqslant 0$|⁠, |$J^{n}/J^{n+1}$| is |${\mathcal{O}}$|-free.

  • (iii)

    We can find an |${\mathcal{O}}$|-basis |${{\mathcal{B}}}$| of |$A$| and a descending chain of subsets |${{\mathcal{B}}}\supseteq{{\mathcal{B}}}_{1}\supseteq{{\mathcal{B}}}_{2}\supseteq \cdots $| such that each |${{\mathcal{B}}}_{i}$| is an |${\mathcal{O}}$|-basis for |$J^{i}$|⁠.

Furthermore, if |$J=J_{A}=A\cap \textsf{Rad}({\hat A})$|⁠, then the following are equivalent to statements (i)–(iii) above.

  • (iv)

    For all |$n\geqslant 1$|⁠, we have |$\dim _{{\mathbb{F}}}\textsf{Rad}^{n}({\bar A})\geqslant \dim _{\mathbb{K}}\textsf{Rad}^{n}({\hat A})$|⁠.

  • (v)

    For all |$n\geqslant 1$|⁠, we have |$\dim _{{\mathbb{F}}}\textsf{Rad}^{n}({\bar A})=\dim _{\mathbb{K}}\textsf{Rad}^{n}({\hat A})$|⁠.

 

Proof.

(i) |$\Leftrightarrow $| (ii): If |$A/J^{n+1}$| is |${\mathcal{O}}$|-free then so is the submodule |$J^{n}/J^{n+1}$|⁠. Conversely, |$A/J$| is |${\mathcal{O}}$|-free with basis the vertex idempotents. So if each |$J^{n}/J^{n+1}$| is |${\mathcal{O}}$|-free then |$A/J^{n}$| has a finite filtration with |${\mathcal{O}}$|-free subquotients, so it is |${\mathcal{O}}$|-free.

(ii) |$\Rightarrow $| (iii): Since |$A/J$| is |${\mathcal{O}}$|-free, we can choose an |${\mathcal{O}}$|-basis. Then |$A\to A/J$| is |${\mathcal{O}}$|-split, so we let |${{\mathcal{B}}}\setminus{{\mathcal{B}}}_{1}$| be the lift of this basis using the splitting. Suppose by induction on |$n\geqslant 1$| that we have chosen a free |${\mathcal{O}}$|-basis for |$A/J^{n}$| and used a splitting of |$A\to A/J^{n}$| to lift to |${{\mathcal{B}}}\setminus{{\mathcal{B}}}_{n}$|⁠. Then |$J^{n}$| is |${\mathcal{O}}$|-free, and |$J^{n}/J^{n+1}$| is |${\mathcal{O}}$|-free, so |$J^{n}\to J^{n}/J^{n+1}$| is |${\mathcal{O}}$|-split, and we apply a splitting to a free basis of |$J^{n}/J^{n+1}$| to get |${{\mathcal{B}}}_{n}\setminus{{\mathcal{B}}}_{n+1}$|⁠. This gives us a set |${{\mathcal{B}}}\setminus{{\mathcal{B}}}_{n+1}$| whose image in |$A/J^{n+1}$| is a free |${\mathcal{O}}$|-basis. When |$n$| is sufficiently large, |$J^{n}=0$| and we are done.

(iii) |$\Rightarrow $| (ii): Given such subsets |${{\mathcal{B}}}\supseteq{{\mathcal{B}}}_{1}\supseteq{{\mathcal{B}}}_{2}\supseteq \cdots $|⁠, the image of |${{\mathcal{B}}}_{n}\setminus{{\mathcal{B}}}_{n+1}$| in |$J^{n}/J^{n+1}$| is a free |${\mathcal{O}}$|-basis.

Suppose further that |$J=J_{A}$|⁠.

(iv) |$\Leftrightarrow $| (v) |$\Leftrightarrow $| (i): We have |$\textsf{Rad}({\hat A})=\hat J$| and |$\textsf{Rad}({\bar A})=\bar J$|⁠. So applying Lemma 6.1 with |$X=A$| and |$Y=J^{n}$|⁠, we have |$\dim _{{\mathbb{F}}}\textsf{Rad}^{n}({\bar A})\leqslant \dim _{\mathbb{K}}\textsf{Rad}^{n}({\hat A})$|⁠, with equality if and only if |$A/J^{n}$| is |${\mathcal{O}}$|-free.

 

Theorem 6.3.
If Hypothesis E holds then the radical layer multiplicities of |$\hat P_{i}$| and |$\bar P_{i}$| are equal, that is, for all |$1\leqslant i,j\leqslant m$| and |$n\geqslant 0$|⁠,
$$ \begin{align*} & [\textsf{Rad}^{n}(\hat P_{i})/\textsf{Rad}^{n+1}(\hat P_{i}):\hat S_{j}]= [\textsf{Rad}^{n}(\bar P_{i})/\textsf{Rad}^{n+1}(\bar P_{i}):\bar S_{j}]. \end{align*} $$

 

Proof.
Let |$J=J_{A}$|⁠. Since |$A$| satisfies Hypothesis D, |$J$| is nilpotent by Theorem 3.1(ii). By Lemma 6.2, we have an |${\mathcal{O}}$|-basis |$\Omega $| for the |$A$|-module |$J^{n}/J^{n+1}$|⁠. Let |$\hat e_{i},\hat e_{j}\in{\hat A}$| and |$\bar e_{i},\bar e_{j}\in{\hat A}$| be the corresponding vertex idempotents. Furthermore, by assumption, we have
$$ \begin{align*} \hat s_{ij}:=&[\textsf{Rad}^{n}(\hat P_{i})/\textsf{Rad}^{n+1}(\hat P_{i}):\hat S_{j}]=\dim_{\mathbb{K}}\hat e_{j}({\mathbb{K}}\otimes_{\mathcal{O}}(J^{n}/J^{n+1}))\hat e_{i}=\textrm{span}_{\mathbb{K}}\{\hat e_{j}\hat b\hat e_{i}:b\in\Omega\},\\ \bar s_{ij}:=&[\textsf{Rad}^{n}(\bar P_{i})/\textsf{Rad}^{n+1}(\bar P_{i}):\bar S_{j}]=\dim_{{\mathbb{F}}}\bar e_{j}({{\mathbb{F}}}\otimes_{\mathcal{O}}(J^{n}/J^{n+1}))\bar e_{i}=\textrm{span}_{{\mathbb{F}}}\{\bar e_{j}\bar b\bar e_{i}:b\in\Omega\}. \end{align*} $$
Since |$\hat e_{j}\hat b\hat e_{i}\in A$|⁠, we get |$\hat s_{ij}\geqslant \bar s_{ij}$|⁠. Summing over all |$i,j$|⁠, we have
$$ \begin{align*} &\sum_{i,j}\hat s_{ij}=\dim_{\mathbb{K}} ({\mathbb{K}}\otimes_{\mathcal{O}}(J^{n}/J^{n+1}))=\dim_{{\mathbb{F}}} ({{\mathbb{F}}}\otimes_{\mathcal{O}}(J^{n}/J^{n+1}))=\sum_{i,j}\bar s_{ij}\end{align*} $$
and get |$\hat s_{ij}= \bar s_{ij}$|⁠.

 

Example 6.4.

Let |$Q$| be the quiver

and |$A={\mathcal{O}} Q/I$| with |$I=(edc-\pi ba)$|⁠. Let |$J=J_{A}$|⁠. The algebra |$A$| satisfies Hypothesis D, but not Hypothesis E, because |$ba$| is in |$J^{2}$| but not in |$J^{3}$| while |$\pi ba$| is in |$J^{3}$|⁠. So as an |${\mathcal{O}}$|-module, |$J^{2}/J^{3}\cong{{\mathbb{F}}}\oplus{\mathcal{O}}\oplus{\mathcal{O}}$|⁠, spanned by |$ba$|⁠, |$dc$| and |$ed$|⁠. Thus, |$A/J^{3}\cong{{\mathbb{F}}}\oplus{\mathcal{O}}^{\oplus 12}$| is not |${\mathcal{O}}$|-free.

The projective covers of the module |$1$| over |${\hat A}$| and |${\bar A}$| are given by the following diagrams, respectively:

In particular, the radical lengths are different.

7 The Descent Algebras

In this section, we review some elementary properties for (finite) Coxeter groups and their descent algebras for the discussions in Sections 8 to 10. For the general knowledge about Coxeter groups, we refer the reader to [20, 21]. We will investigate the algebra over a general field |$k$| of characteristic |$q$| (see Hypothesis 2.1).

Irreducible finite Coxeter groups can be classified in terms of the Coxeter–Dynkin diagrams. Let |$(W,S)$| be a (finite irreducible) Coxeter system and we call |$W$| a Coxeter group. A Coxeter element of |$W$| is a product of all |$s\in S$| in any given order. It is known that any two Coxeter elements of |$W$| are conjugate in |$W$| by a cyclic shift. As such, by abuse, we may speak of the Coxeter element of |$W$|⁠. For each subset |$J\subseteq S$|⁠, we write |$W_{J}$| for the parabolic subgroup of |$W$| generated by |$J$|⁠. The pair |$(W_{J},J)$| is again a Coxeter system. We write |$c_{J}$| for the Coxeter element of |$W_{J}$|⁠.

For two subsets |$J,K$| of |$S$|⁠, we write |$J\subseteq _{W} K$| (respectively, |$J=_{W}K$|⁠) if there exists |$w\in W$| such that |${}^{w}J\subseteq K$| (respectively, |${}^{w}J=K$|⁠). In the case when |$J=_{W}K$|⁠, we say that |$J$| and |$K$| are conjugate in |$W$|⁠. In the case when |$J\subseteq _{W} K$| but |$J\neq _{W}K$|⁠, we write |$J\subsetneq _{W}K$|⁠. The following theorem is well known.

 

Theorem 7.1.

Let |$(W,S)$| be a Coxeter system and |$J,K$| be subsets of |$S$|⁠. The following statements are equivalent:

  • (i)

    |$W_{J}$| and |$W_{K}$| are conjugate in |$W$|⁠;

  • (ii)

    |$J=_{W}K$|⁠; and

  • (iii)

    |$c_{J}$| and |$c_{K}$| are conjugate in |$W$|⁠.

Let |$\ell :W\to{{\mathbb{Z}}}_{\geqslant 0}$| be the length function for |$(W,S)$|⁠. For each |$J\subseteq S$|⁠, let |$X_{J}$| be the distinguished set of left coset representatives of |$W_{J}$| in |$W$| consisting of minimal length representatives, that is,

$$ \begin{align*} &X_{J}=\{w\in W: \ell(ws)>\ell(w)\ \textrm{for all}\ s\in J\}.\end{align*} $$

The set |$X_{J}^{-1}=\{w^{-1}:w\in X_{J}\}$| is thus the distinguished set of minimal length right coset representatives of |$W_{J}$| in |$W$|⁠. For another subset |$K\subseteq S$|⁠, |$X_{JK}:=X_{J}^{-1}\cap X_{K}$| is therefore a set of double coset representatives of |$(W_{J},W_{K})$| in |$W$|⁠. We let

$$ \begin{align*} &x_{J}=\sum_{w\in X_{J}}w\in{{\mathbb{Z}}} W.\end{align*} $$

Solomon [33] showed that, over |${{\mathbb{Z}}}$|⁠, for |$J,K\subseteq S$|⁠,

$$ \begin{align*} &x_{J}x_{K}=\sum_{L\subseteq S}a_{JKL}x_{L},\end{align*} $$

where |$a_{JKL}$| is the number of elements |$w\in X_{JK}$| such that |$L=w^{-1}Jw\cap K$|⁠. Thus, the |${{\mathbb{Z}}}$|-span of the set |$\{x_{J}:J\subseteq S\}$| is an |${{\mathbb{Z}}}$|-subalgebra of |${{\mathbb{Z}}} W$| and it is called the Solomon descent algebra of |$W$| (over |${{\mathbb{Z}}}$|⁠). We denote it |$\mathscr{D}_{{\mathbb{Z}}}$|⁠. Furthermore, |$\mathscr{D}_{{\mathbb{Z}}}$| is |${{\mathbb{Z}}}$|-free of rank |$2^{|S|}$|⁠. Since the structure constants |$a_{JKL}$| are integers, for any integral domain |$R$|⁠, considering |$a_{JKL}\cdot 1_{R}$|⁠, we obtain the descent algebra |$\mathscr{D}_{R}:=R\otimes _{{\mathbb{Z}}}\mathscr{D}_{{\mathbb{Z}}}$| which is considered as a subalgebra of |$RW$|⁠. The set |$\{x_{J}:J\subseteq S\}$| forms an |$k$|-basis for |$\mathscr{D}_{k}$|⁠. Let |$Y_{J}$| and |$Y^\circ _{J}$| be the subspaces of |$\mathscr{D}_{k}$| spanned by |$\{x_{K}:K\subseteq _{W}J\}$| and |$\{x_{K}:K\subsetneq _{W} J\},$| respectively. We have the following.

 

Lemma 7.2.

Let |$J\subseteq S$|⁠. The subspaces |$Y_{J}$| and |$Y^\circ _{J}$| are two-sided ideals of |$\mathscr{D}_{k}$|⁠. Furthermore, |$Y^\circ _{J}\subsetneq Y_{J}$|⁠.

In [33], Solomon also determined the radical |$\textsf{Rad}(\mathscr{D}_{\mathbb{Q}})$|⁠. Together with the work by Atkinson–van Willigenburg [2] (for the type |$\mathbb{A}$| case) and Atkinson–Pfeiffer–van Willigenburg [3] (for the general case), the radical of |$\mathscr{D}_{k}$| is known. We now describe their results.

For each |$J\subseteq S$|⁠, let |$\varphi _{J}$| be the permutation character of the induced module from the trivial module for |$W_{J}$| to |$W$|⁠. Notice that |$\varphi _{J}$| takes integer values on the elements of |$W$|⁠. Let |$\mathscr{C}_{{\mathbb{Z}}}$| be the |${{\mathbb{Z}}}$|-span of the set |$\{\varphi _{J}:J\subseteq S\}$|⁠. We have the well-known Mackey formula

$$ \begin{align*} &\varphi_{J}\varphi_{K}=\sum_{L\subseteq S}a_{JKL}\varphi_{L,}\end{align*} $$

where |$a_{JKL}$|’s are precisely the integers we have obtained earlier. Therefore, we obtain a homomorphism of |${{\mathbb{Z}}}$|-algebras

$$ \begin{align*} &\theta:\mathscr{D}_{{\mathbb{Z}}}\to \mathscr{C}_{{\mathbb{Z}}}\end{align*} $$

given by |$\theta (x_{J})=\varphi _{J}$|⁠. Let |$\mathscr{C}_{k}$| be the |$k$|-span of the set |$\{\varphi ^{k}_{J}:J\subseteq S\}$| where |$\varphi ^{k}_{J}(x)=\varphi _{J}(x)\cdot 1_{k}$| for each |$x\in W$|⁠. As such, tensoring with |$k$|⁠, the map |$\theta $| reduces to an |$k$|-algebra homomorphism |$\theta _{k}:\mathscr{D}_{k}\to \mathscr{C}_{k}$| where |$\theta _{k}(x_{J})=\varphi ^{k}_{J}$|⁠.

 

Theorem 7.3
([33, Theorem 3], [3, Theorem 3]).

The radical |$\textsf{Rad}(\mathscr{D}_{k})$| is spanned by elements |$x_{J} - x_{K}$| such that |$J =_{W} K$| and together with elements |$x_{L}$| such that |$q\mid [\textrm{N}_{W}(W_{L}):W_{L}]$|⁠. Furthermore, |$\textsf{Rad}(\mathscr{D}_{k})=\textsf{Ker}\theta _{k}$|⁠. In particular, |$\mathscr{D}_{k}$| is a basic algebra.

Suppose now that |$n>0$| and let |${\textsf{J}}(n)$| be the set consisting of pairs |$(\underline{J},\underline{K})$| of sequences |$\underline{J}=(J_{1},\ldots ,J_{n})$| and |$\underline{K}=(K_{1},\ldots ,K_{n})$| of subsets |$S$| such that |$J_{i}=_{W} K_{i}$| for each |$i\in [1,n]$|⁠. In this case, we write |$\underline{J}=_{W}\underline{K}$|⁠. For |$(\underline{J},\underline{K})\in{\textsf{J}}(n)$|⁠, denote

$$ \begin{align*} &x_{\underline{J},\underline{K}}=\prod_{i=1}^{n}(x_{J_{i}}-x_{K_{i}})\in \textsf{Rad}^{n}(\mathscr{D}_{k}).\end{align*} $$

For example, if |$n=1$| and |$J=_{W}K$|⁠, we have |$x_{(J),(K)}=x_{J}-x_{K}\in \textsf{Rad}(\mathscr{D}_{k})$|⁠. By Theorem 7.3, if |$q\nmid |W|$|⁠, we have

$$ \begin{align*} &\textsf{Rad}^{n}(\mathscr{D}_{k})=\textrm{span}_{k}\{x_{\underline{J},\underline{K}}:(\underline{J},\underline{K})\in{\textsf{J}}(n)\}.\end{align*} $$

Therefore, we have the following lemma.

 

Lemma 7.4.

Let |$(W,S)$| be a Coxeter group, |$n$| be a positive integer and suppose that |$q\nmid |W|$|⁠. There is a subset |$T\subseteq{\textsf{J}}(n)$| such that |$\{x_{\underline{J},\underline{K}}:(\underline{J},\underline{K})\in T\}$| forms a basis for |$\textsf{Rad}^{n}(\mathscr{D}_{k})$|⁠.

Consider the action of |$W$| on the subsets of |$S$|⁠. The equivalence classes are called the Coxeter classes of |$W$|⁠. Let |${\mathcal{R}}$| be a complete set of representatives of the Coxeter classes and we also call the elements in |${\mathcal{R}}$| Coxeter classes. We fix a total order |$\leq $| for |${\mathcal{R}}$| such that, if |$J\subseteq _{W} K$|⁠, we have |$J\leq K$|⁠. For |$J,K\subseteq S$|⁠, we define the mark and the parabolic table of marks of |$W$| as |$\beta _{JK}=|\textrm{Fix}_{W_{K}}(W/W_{J})|$| and |$M=(\beta _{JK})_{J,K\in{\mathcal{R}}}$|⁠, respectively.

 

Lemma 7.5
([20, Proposition 2.4.4] and [3, Lemma 1]).
Let |$(W,S)$| be a Coxeter system and |$J,K$| be subsets of |$S$|⁠. We have
$$ \begin{align*} &\beta_{JK}=\varphi_{J}(c_{K})=a_{JKK}=[\textrm{N}_{W}(W_{J}):W_{J}]\cdot |\{W^\omega_{J}:\omega\in W,\ W_{K}\subseteq W_{J}^\omega\}|.\end{align*} $$
Furthermore, we have the following:
  • (i)

    |$\beta _{JJ}=[\textrm{N}_{W}(W_{J}):W_{J}]$| and |$\beta _{JJ}\mid \beta _{JK}$|⁠.

  • (ii)

    If |$\beta _{JK}\neq 0$|⁠, then |$K\subseteq _{W} J$|⁠. In particular, the matrix |$M$| is lower triangular.

Let |${\mathcal{R}}_{q}$| be the subset of |${\mathcal{R}}$| consisting of |$J\in{\mathcal{R}}$| such that |$q\nmid \beta _{JJ}=[\textrm{N}_{W}(W_{J}):W_{J}]$| and |$M_{k}$| be the entry-wise modulo |$q$| of the principal submatrix of |$M$| labelled by |${\mathcal{R}}_{q}$|⁠, that is, for |$J,K\in{\mathcal{R}}_{q}$|⁠,

$$ \begin{align*} &(M_{k})_{J,K}=\beta_{JK}\cdot 1_{k}=\varphi^{k}_{J}(c_{K}).\end{align*} $$

Notice that |$M_{k}$| is also lower triangular and |${\mathcal{R}}_{0}={\mathcal{R}}$|⁠. We call the elements in |${\mathcal{R}}_{q}$| the |$q$|-Coxeter classes. For example, in the |$\mathbb{A}_{n-1}$| case, |${\mathcal{R}}$| and |${\mathcal{R}}_{q}$| are labelled by partitions and |$q$|-regular partitions of |$n,$| respectively.

 

Lemma 7.6.

Let |$J\in{\mathcal{R}}_{q}$| and |$I\in{\mathcal{R}}$| such that |$\varphi ^{k}_{K}(c_{I})=\varphi ^{k}_{K}(c_{J})$| for all |$K\subseteq S$|⁠. Then |$I\subseteq _{W} J$|⁠.

 

Proof.

Since |$\varphi ^{k}_{J}(c_{I})=\varphi ^{k}_{J}(c_{J})\neq 0$|⁠, by Lemma 7.5, we have |$I\subseteq _{W} J$|⁠.

Furthermore, we have the following lemma.

 

Lemma 7.7
([3, Lemmas 2, 3]).

The matrix |$M_{k}$| is lower triangular of rank |$|{\mathcal{R}}_{q}|$|⁠. Furthermore, the complete set of non-isomorphic simple |$\mathscr{D}_{k}$|-modules is labelled by |${\mathcal{R}}_{q}$| and defined by the columns of |$M_{k}$| in the sense that, for each |$J\in{\mathcal{R}}_{q}$|⁠, the simple |$\mathscr{D}_{k}$|-module |$S_{J,k}$| is one-dimensional such that |$x\in \mathscr{D}_{k}$| acts via the multiplication by |$\theta _{k}(x)(c_{J})\in k$|⁠.

For each |$J\subseteq S$|⁠, let |$\textrm{char}_{J,k}$| denote the characteristic function on the Coxeter class labelled by |$J$| over |$k$|⁠, that is, for each |$K\in{\mathcal{R}}_{q}$|⁠,

$$ \begin{align*} &\textrm{char}_{J,k}(c_{K})=\left \{\begin{array}{ll} 1&\textrm{if}\ K=J,\\ 0&\text{otherwise.}\end{array}\right.\end{align*} $$

By Lemma 7.7, |$\{\textrm{char}_{J,k}:J\in{\mathcal{R}}_{q}\}$| forms a basis and complete set of primitive orthogonal idempotents for |$\textrm{im}\hspace{.08cm} \theta _{k}$|⁠.

We now investigate the simple modules for the descent algebras further. Let |$J\in{\mathcal{R}}$| and |$S_{J,{{\mathbb{Z}}}}$| be the free |${{\mathbb{Z}}}$|-module of rank 1 and let |$x_{K}\in \mathscr{D}_{{\mathbb{Z}}}$| act via the multiplication by |$\theta (x_{K})(c_{J})=\varphi _{K}(c_{J})$|⁠. Notice that |$k\otimes _{{\mathbb{Z}}} S_{J,{{\mathbb{Z}}}}\cong k\otimes _{{\mathbb{Z}}} S_{J^{\prime},{{\mathbb{Z}}}}$| as |$\mathscr{D}_{k}$|-modules if and only if |$\varphi ^{k}_{K}(c_{J})=\varphi ^{k}_{K}(c_{J^{\prime}})$| for all |$K\subseteq S$|⁠. Furthermore, if |$J\in{\mathcal{R}}_{q}$|⁠, then |$S_{J,k}\cong k\otimes _{{\mathbb{Z}}} S_{J,{{\mathbb{Z}}}}$|⁠.

Let |$A$| be a finite-dimensional |$k$|-algebra. Recall that we have a contravariant exact functor

$$ \begin{align*} &\textrm{D}(-)=\textsf{Hom}_{k}(-,k):\textsf{mod}-A\ \to A\ -\textsf{mod}\end{align*} $$

from the category of right |$A$|-modules to the category of left |$A$|-modules. More precisely, for a right |$A$|-module |$M$| and |$\xi \in \textsf{Hom}_{k}(M,k)$|⁠, we have |$(a\xi )(m)=\xi (ma)$| for all |$a\in A$| and |$m\in M$|⁠. Furthermore, |$M$| is projective if and only if |$\textrm{D}(M)$| is injective.

Let |$P_{J,k}$| and |$I_{J,k}$| be the projective cover and injective hull of the simple module |$S_{J,k}$| respectively. We have the following proposition.

 

Proposition 7.8.

Let |$J\in{\mathcal{R}}_{q}$| and |$e\in \mathscr{D}_{k}$| be a primitive idempotent such that |$\theta _{k}(e)=\textrm{char}_{J,k}$|⁠. Then |$P_{J,k}\cong \mathscr{D}_{k} e$| and |$I_{J,k}\cong \textrm{D}(e\mathscr{D}_{k})$|⁠.

 

Proof.
Let |$P=\mathscr{D}_{k}e$| and |$K\subseteq S$|⁠. Since
$$ \begin{align*} &\theta_{k}((x_{K}-\varphi_{K}^{k}(c_{J}))e)=(\varphi^{k}_{K}-\varphi^{k}_{K}(c_{J}))\textrm{char}_{J,k}=0,\end{align*} $$
we have |$(x_{K}-\varphi ^{k}_{K}(c_{J}))e\in \textsf{Ker}\theta _{k}=\textsf{Rad}(\mathscr{D}_{k})$|⁠. As such, |$x_{K}$| acts by multiplication by |$\varphi _{K}^{k}(c_{J})$| on |$Q:=P/\textsf{Rad}(P)$| and hence |$Q$| is isomorphic to |$S_{J,k}$|⁠. So |$P\cong P_{J,k}$|⁠.
Now consider the right |$\mathscr{D}_{k}$|-module |$P^{\prime}:=e\mathscr{D}_{k}$|⁠. Similar as in the previous paragraph, |$P^{\prime}$| is the projective cover of the right simple |$A$|-module |$T:=S^{\prime}_{J,k}$| where |$T$| is one-dimensional spanned by |$\{v\}$| and |$v\cdot x_{K}=\varphi ^{k}_{K}(c_{J})v$| for each |$K\subseteq S$|⁠. Therefore, |$\textrm{D}(P^{\prime})$| is the injective hull of |$\textrm{D}(T)=\textsf{Hom}_{k}(T,k)$|⁠. We only need to check that |$\textrm{D}(T)\cong S_{J,k}$|⁠. Let |$\{\xi \}$| be the dual basis of |$\textrm{D}(T)$| with respect to |$\{v\}$|⁠. We have, for any |$K\subseteq S$|⁠,
$$ \begin{align*} &(x_{K}\xi)(v)=\xi(v\cdot x_{K})=\xi(\varphi^{k}_{K}(c_{J})v)=\varphi^{k}_{K}(c_{J}).\end{align*} $$
Therefore, |$x_{K}\xi =\varphi ^{k}_{K}(c_{J})\xi $| and |$\textrm{D}(T)\cong S_{J,k}$| as desired.

8 Idempotents for the Descent Algebras

In this section, we generalise the construction of the idempotents for the descent algebras in the type |$\mathbb{A}$| case (see [24]) to arbitrary Coxeter groups. Since the construction is similar, we omit some proofs and computational details but keep the notations as close as possible as in [24].

As before, throughout this section, |$k$| is a field of characteristic |$q$| (⁠|$q=0$| or |$q>0$|⁠). For each |$J\in{\mathcal{R}}_{q}$|⁠, let

$$ \begin{align*} &\gamma_{J}=\beta_{JJ}\cdot 1_{k}=\varphi^{k}_{J}(c_{J}).\end{align*} $$

By definition, |$\gamma _{J}\neq 0$|⁠. Recall the total order |$\leq $| we have fixed earlier for the set |${\mathcal{R}}$| (and hence on the subset |${\mathcal{R}}_{q}$|⁠). Let |$M_{k}^{-1}=(b_{JK})_{J,K\in{\mathcal{R}}_{q}}$| be the inverse matrix of |$M_{k}=(\varphi ^{k}_{J}(c_{K}))_{J,K\in{\mathcal{R}}_{q}}$|⁠. For each |$J\in{\mathcal{R}}_{q}$|⁠, define

(4)

Furthermore, recall the two-sided ideals |$Y^\circ _{J}$| and |$Y_{J}$| of |$\mathscr{D}_{k}$| as in Lemma 7.2.

The proof of the following lemma is similar to that of [24, Lemma 3.2] and we shall leave it to the readers.

 

Lemma 8.2.

Let |$(W,S)$| be a Coxeter system and let |$J\in{\mathcal{R}}_{q}$|⁠.

  • (i)

    We have |$\theta _{k}(f_{J})=\textrm{char}_{J,k}$|⁠.

  • (ii)
    For any |${\mathcal{R}}_{q}\ni K\not \subseteq _{W} J$|⁠, we have |$b_{JK}=0$|⁠. In particular,
    $$ \begin{align*} &\textstyle f_{J}=\frac{1}{\gamma_{J}}x_{J}+\sum_{{\mathcal{R}}_{q}\ni K\subsetneq_{W} J} b_{JK}x_{K}\in Y_{J}.\end{align*} $$
  • (iii)

    For any positive integer |$r$|⁠, we have |$(f_{J})^{r}=\frac{1}{\gamma _{J}}x_{J}+\epsilon _{J,r}$| for some |$\epsilon _{J,r}\in Y^\circ _{J}$|⁠.

  • (iv)

    We have |$\sum _{J\in{\mathcal{R}}_{q}}f_{J}=1$|⁠.

As |$\theta _{k}(f_{J})=\textrm{char}_{J,k}$| and |$\textsf{Ker}\theta _{k}=\textsf{Rad}(\mathscr{D}_{k})$|⁠, we identify |$\mathscr{D}_{k}/\textsf{Rad}(\mathscr{D}_{k})$| with |$\mathscr{C}_{k}$| via the |$k$|-algebra epimorphism |$\theta _{k}$| and can lift and orthogonalise the set

$$ \begin{align*} &\{f_{J}+\textsf{Rad}(\mathscr{D}_{k}):J\in{\mathcal{R}}_{q}\}\end{align*} $$

to obtain a complete set of orthogonal primitive idempotents for |$\mathscr{D}_{k}$|⁠. Let |$m=|{\mathcal{R}}_{q}|$|⁠. We denote

$$ \begin{align*} &{\mathcal{R}}_{q}=\{J_{i}:i \in [1,m]\},\end{align*} $$

where |$J_{i}<J_{i+1}$| for each |$i\in [1,m-1]$| and define |$f_{i}=f_{J_{i}}$|⁠, |$f_{\geqslant i}=\sum _{j\geqslant i} f_{j}$|⁠, |$\textrm{char}_{i}=\textrm{char}_{J_{i},k}$|⁠, |$\textrm{char}_{\geqslant i}=\sum _{j\geqslant i}\textrm{char}_{i}$|⁠, |$\gamma _{i}=\gamma _{J_{i}}$|⁠, |$Y_{\leqslant i}=\sum _{k\in [1,i]} Y_{J_{k}}$| and |$Y^\circ _{\leqslant i}=Y^\circ _{J_{i}}+\sum _{k\in [1,i-1]} Y_{J_{k}}$| one for each |$i\in [1,m]$|⁠. Notice that |$J_{m}=S$|⁠.

By Lemmas 7.2 and 7.5, we have the following result.

 

Lemma 8.3.

Let |$i\in [1,m]$|⁠. We have

  • (i)

    |$Y^\circ _{\leqslant i}$| and |$Y_{\leqslant i}$| are two-sided ideals of |$\mathscr{D}_{k}$|⁠;

  • (i)

    |$x_{J_{i}}^{r}=\gamma _{i}^{r-1}x_{J_{i}}(\textrm{mod}\ Y^\circ _{\leqslant i})$| for any positive integers |$r$|⁠; and

  • (iii)

    |$Y^\circ _{\leqslant 1}\subsetneq Y_{\leqslant 1}\subsetneq Y^\circ _{\leqslant 2}\subsetneq Y_{\leqslant 2}\cdots \subsetneq Y^\circ _{\leqslant m}\subsetneq Y_{\leqslant m}=\mathscr{D}_{k}$|⁠.

Since we are dealing with the characteristic zero case (⁠|$q=0$|⁠) as well, we need to replace the idempotent-lifting procedure in [24, Definition 3.5] with the |$(3a^{2}-2a^{3})$|-construction (see the proof of [5, Theorem 1.7.3]) and provide proofs for both Lemma 8.6 and Theorem 8.7. For this, we need to introduce some extra notation as follows.

 

Proposition 8.4.
Let |$N$| be a nilpotent ideal of a finite-dimensional |$k$|-algebra |$A$| and |$f\in A$| such that |$f+N\in A/N$| is an idempotent. Let |$a_{1}=f$| and, inductively, for |$i\geqslant 1$|⁠, let
$$ \begin{align*} &a_{i+1}=3a_{i}^{2}-2a_{i}^{3}.\end{align*} $$
Then there exists an integer |$r$| such that |$a_{r}$| is an idempotent of |$A$| and thus |$a_{r^{\prime}}=a_{r}$| for any |$r^{\prime}\geqslant r$|⁠. We denote |$a_\infty =a_{r}$|⁠. Furthermore, |$a_{i}$| commutes with |$a_{1}$| for all |$i\geqslant 1$| and  
$$ \begin{align*} &(a_{1}a_\infty)_\infty=a_\infty.\end{align*} $$

 

Proof.
The first assertion is well known. The fact that |$a_{i}$| commutes with |$a_{1}$| can be proved easily using induction on |$i$|⁠. Notice that |$(a_{1}a_\infty )+N$| is again an idempotent. To show that |$(a_{1}a_\infty )_\infty =a_\infty $|⁠, let |$b_{1}=a_{1}a_\infty $| and, inductively, for all |$i\geqslant 1$|⁠, let |$b_{i+1}=3b_{i}^{2}-2b_{i}^{3}$|⁠. We check that |$b_{i}=a_{i}a_\infty $| for all |$i\geqslant 1$| using induction. Since |$a_\infty $| is an idempotent, we have
$$ \begin{align*} b_{i+1}&=3b_{i}^{2}-2b_{i}^{3}=3(a_{i}a_\infty)^{2}-2(a_{i}a_\infty)^{3}=3a_{i}^{2}a_\infty^{2}-2a_{i}^{3}a_\infty^{3}=3a_{i}^{2}a_\infty-2a_{i}^{3}a_\infty=a_{i+1}a_\infty. \end{align*} $$
As such, |$b_\infty =a_\infty ^{2}=a_\infty $|⁠.

In the case of the descent algebra, notice that we have |$\theta _{k}(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})=\textrm{char}_{\geqslant i}$|⁠. This allows us to define the following.

 

Definition 8.5.
In |$\mathscr{D}_{k}$|⁠, let |$f^{\prime}_{1}=1$| and, inductively, for |$i\in [2,m]$|⁠, we define |$f_{i}^{\prime}=(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty $| and |$f_{m+1}^{\prime}=0$|⁠. For each |$i\in [1,m]$|⁠, define
$$ \begin{align*} &e_{J_{i}}=f_{i}^{\prime}-f_{i+1}^{\prime}.\end{align*} $$

The following lemma is similar to [24, Lemma 3.6].

 

Lemma 8.6.

  • (i)

    For each |$i\in [1,m]$|⁠, the element |$f_{i}^{\prime}$| is an idempotent such that |$\theta _{k}(f_{i}^{\prime})=\textrm{char}_{\geqslant i}$| and for |$i>1$|⁠, we have |$f_{i}^{\prime}\in f_{i-1}^{\prime}\mathscr{D}_{k} f_{i-1}^{\prime}$|⁠.

  • (ii)
    For |$i\in [2,m+1]$|⁠,
    $$ \begin{align*} &\textstyle f_{i}^{\prime}=1-\left (\frac{1}{\gamma_{i-1}}x_{J_{i-1}}+\epsilon_{i-1}\right )\end{align*} $$
    for some |$\epsilon _{i-1}\in Y^\circ _{\leqslant i-1}$|⁠.

 

Proof.
We first prove part (i) by using induction on |$i$|⁠. It is clearly true when |$i=1$|⁠. Suppose that |$i>1$| and assume that the statement holds true for |$i-1$|⁠. Let |$a_{1}=f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime}$| and |$a_{r+1}=3a_{r}^{2}-2a_{r}^{3}$| for |$r\geqslant 1$|⁠. Let |$A=f_{i-1}^{\prime}\mathscr{D}_{k} f_{i-1}^{\prime}$|⁠. Notice that |$a_{1}\in A$| and, by Lemma 8.2,
$$ \begin{align*} &\theta_{k}(a_{1})=\theta_{k}(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})=\textrm{char}_{\geqslant i-1}\textrm{char}_{\geqslant i}\textrm{char}_{\geqslant i-1}=\textrm{char}_{\geqslant i}.\end{align*} $$
Suppose that |$\theta _{k}(a_{r})=\textrm{char}_{\geqslant i}$| and |$a_{r}\in A$| for some positive integer |$r$|⁠. We have
$$ \begin{align*} &\theta_{k}(a_{r+1})=\theta_{k}(3a_{r}^{2}-2a_{r}^{3})=3\ \textrm{char}_{\geqslant i}^{2}-2\ \textrm{char}_{\geqslant i}^{3}=\textrm{char}_{\geqslant i}.\end{align*} $$
As such, |$f_{i}^{\prime}=a_\infty =(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty $| is an idempotent of |$\mathscr{D}_{k}$| such that |$\theta _{k}(f_{i}^{\prime})=\textrm{char}_{\geqslant i}$|⁠. Furthermore, as |$f_{i-1}^{\prime}$| is an idempotent and |$a_{r}\in A$|⁠,
$$ \begin{align*} &a_{r+1}=3a_{r}^{2}-2a_{r}^{3}=f_{i-1}^{\prime}(3a_{r}^{2}-2a_{r}^{3})f_{i-1}^{\prime}\in A.\end{align*} $$
Therefore, |$f_{i}^{\prime}=a_\infty \in A$|⁠.
We again argue by induction on |$i$| to prove part (ii). Suppose that |$i=2$|⁠. Let |$a_{1}=f_{\geqslant 2}=1-f_{1}\equiv 1-\frac{1}{\gamma _{1}}x_{J_{1}}(\textrm{mod}\ Y^\circ _{\leqslant 1})$| and |$a_{r+1}=3a_{r}^{2}-2a_{r}^{3}$| as before. Suppose that |$a_{r}\equiv 1-\frac{1}{\gamma _{1}}x_{J_{1}}(\textrm{mod}\ Y^\circ _{\leqslant 1})$| for some |$r\geqslant 1$|⁠. By Lemma 8.3, we have
$$ \begin{align*} a_{r+1}&\equiv{\textstyle 3(1-\frac{1}{\gamma_{1}}x_{J_{1}})^{2}-2(1-\frac{1}{\gamma_{1}}x_{J_{1}})^{3}}(\textrm{mod}\ Y^\circ_{\leqslant 1})\\ &= {\textstyle3(1-\frac{2}{\gamma_{1}}x_{J_{1}}+\frac{1}{\gamma_{1}}x_{J_{1}})-2(1-\frac{3}{\gamma_{1}}x_{J_{1}}+\frac{3}{\gamma_{1}}x_{J_{1}}-\frac{1}{\gamma_{1}}x_{J_{1}})}\\ &={\textstyle1-\frac{1}{\gamma_{1}}x_{J_{1}}}. \end{align*} $$
As such, |$f_{2}^{\prime}=a_\infty $| has the desired form as in the statement. Suppose now that |$f_{i}^{\prime}=1-(\frac{1}{\gamma _{i-1}}x_{J_{i-1}}+\epsilon _{i-1})$| for some |$\epsilon _{i-1}\in Y^\circ _{\leqslant i-1}$| and |$\epsilon _{i-1}^{\prime}=\frac{1}{\gamma _{i-1}}x_{J_{i-1}}+\epsilon _{i-1}\in Y_{\leqslant i-1}$|⁠. Again, let |$a_{1}=f_{i}^{\prime}f_{\geqslant{i+1}}f_{i}^{\prime}$| and |$a_{r+1}=3a_{r}^{2}-2a_{r}^{3}$| for |$r\geqslant 1$|⁠. By Lemmas 8.2 and 8.3, let |$f_{i}=\frac{1}{\gamma _{i}}x_{J_{i}}+z_{i}$| where |$z_{i}\in Y^\circ _{\leqslant i}$|⁠, we have
$$ \begin{align*} a_{1}&=(1-\epsilon_{i-1}^{\prime})(1-f_{1}-\cdots-f_{i})(1-\epsilon_{i-1}^{\prime})\\ &={\textstyle(1-\epsilon_{i-1}^{\prime})(1-f_{1}-\cdots-f_{i-1}-(\frac{1}{\gamma_{i}}x_{J_{i}}+z_{i}))(1-\epsilon_{i-1}^{\prime})}\\ &\equiv{\textstyle1-\frac{1}{\gamma_{i}}x_{J_{i}}(\textrm{mod}\ Y^\circ_{\leqslant i})}. \end{align*} $$
Suppose now that |$a_{r}\equiv 1-\frac{1}{\gamma _{i}}x_{J_{i}}(\textrm{mod}\ Y^\circ _{\leqslant i})$| for some |$r\geqslant 1$|⁠. Same as the previous calculation, we have
$$ \begin{align*} a_{r+1}&\equiv{\textstyle3(1-\frac{1}{\gamma_{i}}x_{J_{i}})^{2}-2(1-\frac{1}{\gamma_{i}}x_{J_{i}})^{3}(\textrm{mod}\ Y^\circ_{\leqslant i})}\\ &={\textstyle1-\frac{1}{\gamma_{i}}x_{J_{i}}}. \end{align*} $$
As such, |$f_{i+1}^{\prime}=a_\infty $| has the desired form. The proof is now complete.

We are now ready to prove the main result in this section.

 

Theorem 8.7.
The set |$\{e_{J}:J\in{\mathcal{R}}_{q}\}$| is a complete set of orthogonal primitive idempotents of |$\mathscr{D}_{k}$| such that |$\theta _{k}(e_{J})=\textrm{char}_{J,k}$| and |$\sum _{J\in{\mathcal{R}}_{q}}e_{J}=1$|⁠. Furthermore,
$$ \begin{align*} &e_{J}=\textstyle\frac{1}{\gamma_{J}}x_{J}+\epsilon_{J},\end{align*} $$
where |$\epsilon _{J}$| is a linear combination of some |$x_{K}$| such that |$K\subsetneq _{W} J$|⁠, that is, |$\epsilon _{J}\in Y^\circ _{J}$| with coefficients belonging to the prime subfield of |$k$|⁠.

 

Proof.
Lemma 8.6 and [24, Proposition 3.3(ii)] give us the first sentence of our theorem. We now prove inductively on |$i$| that |$e_{J}$| has the desired form. When |$i=1$|⁠, since |$Y^\circ _{\leqslant 1}=Y^\circ _{J_{1}}$|⁠, we have
$$ \begin{align*} &\textstyle e_{J_{1}}=f_{1}^{\prime}-f_{2}^{\prime}\equiv 1-(1-\frac{1}{\gamma_{1}}x_{J_{1}})=\frac{1}{\gamma_{1}}x_{J_{1}}(\textrm{mod}\ Y^\circ_{J_{1}}).\end{align*} $$
For |$i\in [2,m]$|⁠, as |$Y^\circ _{\leqslant i-1}\subseteq Y_{\leqslant i-1}\subseteq Y^\circ _{\leqslant i}$|⁠, by Lemma 8.6(ii), we have
$$ \begin{align*} &\textstyle e_{J_{i}}=-\left (\frac{1}{\gamma_{i-1}}x_{J_{i-1}}+\epsilon_{i-1}\right )+\left (\frac{1}{\gamma_{i}}x_{J_{i}}+\epsilon_{i}\right )\equiv \frac{1}{\gamma_{i}}x_{J_{i}}(\textrm{mod}\ Y^\circ_{\leqslant i}).\end{align*} $$
To complete the proof, we are left to check that |$e_{J_{i}}\in Y_{J_{i}}$| so that |$e_{J_{i}}-\frac{1}{\gamma _{i}}x_{J_{i}}\in Y^\circ _{\leqslant i}\cap Y_{J_{i}}=Y^\circ _{J_{i}}$|⁠. When |$i=1$|⁠, |$e_{J_{1}}=\frac{1}{\gamma _{1}}x_{J_{1}}+\epsilon _{1}\in Y_{\leqslant 1}=Y_{J_{1}}$|⁠. Suppose now that |$i\in [2,m]$|⁠. We have
$$ \begin{align*} f_{i+1}^{\prime}&=(f_{i}^{\prime}f_{\geqslant i+1}f_{i}^{\prime})_\infty=(f_{i}^{\prime}(f_{\geqslant i}-f_{i})f_{i}^{\prime})_\infty\equiv (f_{i}^{\prime}f_{\geqslant i}f_{i}^{\prime})_\infty (\textrm{mod}\ Y_{J_{i}})\\ &=((f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty f_{\geqslant i}(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty)_\infty\\ &=((f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime}(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty)_\infty\\ &=(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime}(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty)_\infty\\ &=(f_{i-1}^{\prime}f_{\geqslant i}f_{i-1}^{\prime})_\infty=f_{i}^{\prime}, \end{align*} $$
where the equivalence is obtained using |$f_{i}^{\prime}f_{i}f_{i}^{\prime}\in Y_{J_{i}}$| and the construction in Proposition 8.4, the third equality follows from Lemma 8.6(i) and the fourth and fifth equalities follow from Proposition 8.4. As such, we get |$e_{J_{i}}=f_{i+1}^{\prime}- f_{i}^{\prime}\equiv 0(\textrm{mod}\ Y_{J_{i}})$|⁠.

The last assertion regarding the coefficients follows from our construction throughout this section. The proof is now complete.

Recall the simple module |$S_{J,k}$| and the functor |$\textrm{D}(-)=\textsf{Hom}_{k}(-,k)$| in Section 7. We have the following corollary.

 

Corollary 8.8.

Let |$J\in{\mathcal{R}}_{q}$|⁠. Then |$P_{J,k}\cong \mathscr{D}_{k} e_{J}$| and |$I_{J,k}\cong \textrm{D}(e_{J}\mathscr{D}_{k})$|⁠.

 

Proof.

Use Theorem 8.7 and Proposition 7.8.

9 Sufficient Conditions for the Descent Algebras

Recall that |${\mathcal{O}}$| is a local principal ideal domain with maximal ideal |$(\pi )$|⁠, |${\mathbb{K}}$| is the field of fractions of |${\mathcal{O}}$| and |${{\mathbb{F}}}={\mathcal{O}}/(\pi )$|⁠. Let |$p$| be the characteristic of |${{\mathbb{F}}}$|⁠. In this section, we show that the descent algebras |$\mathscr{D}_{\mathcal{O}}$| of |$W$| over |${\mathcal{O}}$| satisfy Hypothesis B if |$p\nmid |W|$| and Hypothesis E if |$p\nmid n_{W}$| (see Definition 9.9).

Similar as before, we write |${\mathscr{D}}=\mathscr{D}_{\mathcal{O}}$|⁠, |${\hat{\mathscr{D}}}={\mathbb{K}}\otimes _{\mathcal{O}} \mathscr{D}_{\mathcal{O}}$|⁠, |${\bar{\mathscr{D}}}={{\mathbb{F}}}\otimes _{\mathcal{O}} \mathscr{D}_{\mathcal{O}}$| and, for any |${\mathscr{D}}$|-module |$M$| that is |${\mathcal{O}}$|-free, we write |$\hat M={\mathbb{K}}\otimes _{\mathcal{O}} M$| and |$\bar M={{\mathbb{F}}}\otimes _{\mathcal{O}} M$|⁠. For each |$z\in{\mathscr{D}}$|⁠, the coefficients of the |$x_{J}$|’s in |$z$| belongs in |${\mathcal{O}}$| and we write |$\bar z$| for the element in |${\bar{\mathscr{D}}}$| by reducing the coefficients modulo |$\pi $|⁠. Furthermore, since we will be using the elements |$f_{i}^{\prime}$|⁠, |$f_{J}$| and |$e_{J}$| in Section 8 over different fields |${\mathbb{K}}$| and |${{\mathbb{F}}}$|⁠, we use |$f_{i,{\mathbb{K}}}^{\prime}$| and so on to emphasise the field which we work over.

9.1 Verifying Hypothesis B

In this subsection, we verify that the descent algebras satisfy Hypothesis B if |$p\nmid |W|$|⁠. For the next lemma, we note that if |$s\in{\mathcal{O}}\backslash (\pi )$|⁠, then |$s^{-1}\in{\mathcal{O}}$|⁠. As such, an element in |${\mathbb{K}}$| belongs in |${\mathcal{O}}$| if it can be written in the form |$rs^{-1}$| where |$r,s\in{\mathcal{O}}$| and |$s\not \in (\pi )$|⁠.

 

Lemma 9.1.

Suppose that |$p\nmid |W|$|⁠. We have that |${\mathcal{R}}={\mathcal{R}}_{p}={\mathcal{R}}_{0}$| and the entries in |$M_{\mathbb{K}}^{-1}$| belongs in |${\mathcal{O}}$|⁠. Moreover, the matrix |$M_{{{\mathbb{F}}}}^{-1}$| is obtained from |$M_{\mathbb{K}}^{-1}$| by taking modulo |$\pi $| on its entries. In particular, for each |$J\in{\mathcal{R}}$|⁠, we have |$f_{J,{\mathbb{K}}}\in{\mathscr{D}}$| and |$\bar f_{J,{\mathbb{K}}}=f_{J,{{\mathbb{F}}}}$|⁠.

 

Proof.

Since |$p\nmid |W|$|⁠, we have |$\beta _{JJ}=[\textrm{N}_{W}(W_{J}):W_{J}]\not \in (\pi )$| for all |$J\in{\mathcal{R}}$| and hence |${\mathcal{R}}={\mathcal{R}}_{p}={\mathcal{R}}_{0}$|⁠. Notice that |$\beta _{JK}\in{\mathcal{O}}$| for each |$J,K\in{\mathcal{R}}$| and |$M_{{\mathbb{F}}}=(\overline \beta _{JK})$|⁠. The adjugate matrix of |$M_{\mathbb{K}}$| have entries belonging in |${\mathcal{O}}$| and the adjugate matrix of |$M_{{\mathbb{F}}}$| is obtained from that of |$M$| modulo |$\pi $|⁠. Furthermore, |$\textsf{det}(M_{\mathbb{K}})=\prod _{J\in{\mathcal{R}}}\beta _{JJ}\not \in (\pi )$| and hence |$\textsf{det}(M_{\mathbb{K}})^{-1}\in{\mathcal{O}}$|⁠. Furthermore, |$\textsf{det}(M_{{\mathbb{F}}})=\overline{\textsf{det}(M)}\neq 0$|⁠. As such, the entries of |$M_{\mathbb{K}}^{-1}$| belong in |${\mathcal{O}}$| and they give the entries for |$M_{{{\mathbb{F}}}}^{-1}$| upon modulo |$\pi $|⁠.

The proof of the following lemma requires the construction presented in Section 8 which we shall not go into the details.

 

Lemma 9.2.

Suppose that |$p\nmid |W|$|⁠. For each |$J\in{\mathcal{R}}$|⁠, we have |$e_{J,{\mathbb{K}}}\in{\mathscr{D}}$| and |$\bar e_{J,{\mathbb{K}}}=e_{J,{{\mathbb{F}}}}$|⁠. In other words, the complete set of primitive orthogonal idempotents |$\{e_{J,{{\mathbb{F}}}}:J\in{\mathcal{R}}\}$| of |${\bar{\mathscr{D}}}$| are liftable to |${\mathscr{D}}$|⁠.

 

Proof.

For each |$J\in{\mathcal{R}}$|⁠, by Lemma 9.1, we have |$f_{J,{\mathbb{K}}}\in{\mathscr{D}}$| and |$\bar f_{J,{\mathbb{K}}}=f_{J,{{\mathbb{F}}}}$|⁠. Since the structure constants |$a_{IKL}$|’s belong in |${\mathcal{O}}$| and the construction of the idempotents are obtained by iterating the process |$a_{r+1}=3a_{r}^{2}-2a_{r}^{3}$|⁠, and it stabilises after finite number of steps, we have that |$f_{i,{\mathbb{K}}}^{\prime}\in{\mathscr{D}}$| and hence |$e_{J,{\mathbb{K}}}\in{\mathscr{D}}$|⁠. The construction “commutes” with the action of taking modulo |$\pi $|⁠. As such, |$\bar e_{J,{\mathbb{K}}}=e_{J,{{\mathbb{F}}}}$|⁠.

 

Corollary 9.3.
Suppose that |$p\nmid |W|$|⁠. We have
$$ \begin{align*} &{\mathscr{D}}=\textrm{span}_{{\mathcal{O}}}\{e_{J,{\mathbb{K}}}:J\in{\mathcal{R}}\}\oplus (\textsf{Rad}({\hat{\mathscr{D}}})\cap{\mathscr{D}})\end{align*} $$
and |$\textsf{Rad}({\mathscr{D}})=\pi{\mathscr{D}}+(\textsf{Rad}({\hat{\mathscr{D}}})\cap{\mathscr{D}})$|⁠.

 

Proof.
By Lemma 9.2, |$e_{J,{\mathbb{K}}}\in{\mathscr{D}}$| for any |$J\in{\mathcal{R}}$|⁠. Since |$e_{J,{\mathbb{K}}}$| are idempotents, the sum in the right-hand side of the display equation in the statement is clearly a direct sum. Let |$U:=\textrm{span}_{{\mathcal{O}}}\{e_{J,{\mathbb{K}}}:J\subseteq S\}\oplus (\textsf{Rad}({\hat{\mathscr{D}}})\cap{\mathscr{D}})$|⁠. Clearly, |$U\subseteq{\mathscr{D}}$|⁠. For the reverse inclusion, we argue by induction on the total order |$(\leq , {\mathcal{R}})$|⁠. When |$J=\emptyset \in{\mathcal{R}}$|⁠, we have |$e_{\emptyset ,{\mathbb{K}}}=\frac{1}{\gamma _\emptyset }x_\emptyset $| where |$\frac{1}{\gamma _\emptyset }\in{\mathcal{O}}$|⁠. Thus, |$x_\emptyset \in U$|⁠. For any |$J\in{\mathcal{R}}$|⁠, by induction, we have
$$ \begin{align*} &x_{J}=\gamma_{J}e_{J,{\mathbb{K}}}+\gamma_{J}({\mathcal{O}}\ \text{-linear combination of}\ x_{K}\ \textrm{where}\ K\subsetneq_{W} J)\in U.\end{align*} $$
For any |$J^{\prime}\subseteq S$| such that |$J^{\prime}=_{W}J$|⁠, we have |$x_{J^{\prime}}=x_{J}+(x_{J^{\prime}}-x_{J})\in U$| as |$x_{J^{\prime}}-x_{J}\in \textsf{Rad}({\hat{\mathscr{D}}})\cap{\mathscr{D}}$|⁠. Therefore, |${\mathscr{D}}=U$|⁠. The expression for |$\textsf{Rad}({\mathscr{D}})$| now follows from the decomposition.

 

Theorem 9.4.

Suppose that |$p\nmid |W|$|⁠. The descent algebra |${\mathscr{D}}$| satisfies Hypothesis B. In particular, Theorem 2.3(ii) applies for |${\mathscr{D}}$|⁠.

 

Proof.

The first statement follows by applying both Lemma 9.2 and Corollary 9.3. Therefore, |${\mathscr{D}}$| satisfies Hypothesis A using Theorem 2.3(i). Hence, Theorem 2.3(ii) applies for |${\mathscr{D}}$|⁠.

As a special case of Theorem 9.4, we have the following corollary.

 

Corollary 9.5.

Let |$W$| be a finite Coxeter group and suppose that |$p\nmid |W|$|⁠. Then the Ext quivers of |${\hat{\mathscr{D}}}$| and |${\bar{\mathscr{D}}}$| are identical.

We remark that Schocker [31] computed the Ext quivers for the descent algebras of type |$\mathbb{A}$| over any field of characteristic zero. Saliola [29] extended the computation to both types |$\mathbb{A}$| and |$\mathbb{B}$| over |$k$| given that |$p\nmid |W|$| and particularly showed that the Ext quivers of |${\hat{\mathscr{D}}}$| and |${\bar{\mathscr{D}}}$| are identical. Our corollary asserts that such phenomenon holds for arbitrary Coxeter group. In particular, Corollary 9.5 and Schocker’s result give an alternative proof to Saliola’s result in the type |$\mathbb{A}$| case.

9.2 Verifying Hypothesis E

Recall the set |${\textsf{J}}(n)$| we have introduced in Section 7. For each |$J\in{\mathcal{R}}$|⁠, we fix |$J_{1},\ldots ,J_{n_{J}}$| the distinct subsets of |$S$| such that |$J_{i}=_{W}J$|⁠. Let

(6)

Notice that, for any |$1\leqslant i<j\leqslant n_{J}$|⁠, we have |$x_{J_{i}}-x_{J_{j}}=\sum ^{j-1}_{k=i}(x_{J_{k}}-x_{J_{k+1}})$|⁠. Notice also that |${\mathcal{B}}$| forms a basis for |$\textsf{Rad}({\hat{\mathscr{D}}})$| (respectively, for |$\textsf{Rad}({\bar{\mathscr{D}}})$| when |$p\nmid |W|$|⁠) (cf. Lemma 7.4).

 

Definition 9.7.

We fix a total order for |${\textsf{J}}(n)$| (so that |$\Omega $| inherits that of |${\textsf{J}}(1)$|⁠). Define the matrix |$[{\textsf{J}}(n)]$| where the rows and columns of |$[{\textsf{J}}(n)]$| are labelled by |${\textsf{J}}(n)$| and |$\Omega ,$| respectively, and the corresponding entry is the coefficient of |$x_{L}-x_{L^{\prime}}$| in |$x_{\underline{J},\underline{K}}$| as elements in |${\mathscr{D}}$| where |$(\underline{J},\underline{K})\in{\textsf{J}}(n)$| and |$(L,L^{\prime})\in \Omega $|⁠. Let |$[{\textsf{J}}(n)]_{\mathbb{K}}={\mathbb{K}}\otimes _{\mathcal{O}} [{\textsf{J}}(n)]$| and |$[{\textsf{J}}(n)]_{{\mathbb{F}}}={{\mathbb{F}}}\otimes _{\mathcal{O}} [{\textsf{J}}(n)]$|⁠.

 

Lemma 9.8.

Let |$(W,S)$| be a Coxeter system and |$n$| be a positive integer.

  • (i)

    The entries of |$[{\textsf{J}}(n)]$| belong in |${{\mathbb{Z}}}$|⁠.

  • (ii)

    Suppose further that |$p\nmid |W|$|⁠. Then |$\textsf{rank}([{\textsf{J}}(n)]_{\mathbb{K}})=\dim _{\mathbb{K}} \textsf{Rad}^{n}({\hat{\mathscr{D}}})$| and |$\textsf{rank}([{\textsf{J}}(n)]_{{\mathbb{F}}})=\dim _{{\mathbb{F}}} \textsf{Rad}^{n}({\bar{\mathscr{D}}})$|⁠.

 

Proof.

Part (ii) follows from Lemma 7.4. We now prove part (i). For each |$(\underline{J},\underline{K})\in{\textsf{J}}(n)$| and |$L\subseteq S$|⁠, since the structure constants of the descent algebra belong in |${{\mathbb{Z}}}$|⁠, we have |$x_{\underline{J},\underline{K}}\in \mathscr{D}_{{\mathbb{Z}}}$|⁠. Also, since |$x_{\underline{J},\underline{K}}\in \textsf{Rad}(\mathscr{D}_{\mathbb{Q}})=\textrm{span}_{\mathbb{Q}} {\mathcal{B}}$|⁠, the sum of all the coefficients in terms of the basis |$\{x_{J}:J\subseteq S\}$| is |$0$| and hence the entries in |$[{\textsf{J}}(n)]$| belong in |${{\mathbb{Z}}}$|⁠.

 

Definition 9.9.
Let |$(W,S)$| be a Coxeter system and |$s=\dim _{\mathbb{K}} \textsf{Rad}^{n}({\hat{\mathscr{D}}})\neq 0$| (or equivalently, |$[{\textsf{J}}(n)]$| is not a zero matrix). Let |$d_{W,n}$| be the greatest common divisor of the determinants of all possible |$(s\times s)$|-submatrices of |$[{\textsf{J}}(n)]$|⁠. Furthermore, let |$n_{W,n}=\textsf{lcm}(d_{W,n},|W|)$| and |$n_{W}$| be the least common multiple of the |$n_{W,n}$|’s such that |$d_{W,n}\neq 0$|⁠, that is,
$$ \begin{align*} &n_{W}=\textsf{lcm}\{n_{W,n}:n\leqslant \ell\},\end{align*} $$
where |$\ell $| is the radical length of |${\hat{\mathscr{D}}}$| (i.e., the largest non-negative integer |$\ell $| such that |$\textsf{Rad}^\ell ({\hat{\mathscr{D}}})\neq 0$|⁠).

Notice that, if |$\Omega \neq \emptyset $|⁠, then |$[{\textsf{J}}(1)]$| contains an identity submatrix and hence |$d_{W,1}=1$|⁠. At the moment, the number |$d_{W,n}$| seemed to depend on the choice of |$\Omega $| (and hence the basis |${\mathcal{B}}$|⁠). In Section 10, we will show that it is not the case if we replace |$\Omega $| by another basis |$\Omega ^{\prime}$| within |${\textsf{J}}(1)$|⁠.

Before proceeding to the next lemma, we provide some examples.

 

Example 9.10.

Consider the Coxeter group |$(W,S)$| of the dihedral type |$\mathbb{I}_{n}$| where |$S=\{s_{1},s_{2}\}$| so that |$W$| is the dihedral group of order |$2n$|⁠. It is well known that |$\{s_{1}\}=_{W}\{s_{2}\}$| if and only if |$n$| is odd. Therefore, |$\Omega =\emptyset $| unless |$n$| is odd. In the case when |$n$| is odd, |$B=\{x_{\{s_{1}\}}-x_{\{s_{2}\}}\}$| and |$[{\textsf{J}}(r)]$| is zero for |$r\geqslant 2$|⁠. Therefore, for each |$n$|⁠, we have |$n_{W}=2n$|⁠.

 

Example 9.11.
Consider the Coxeter group of |$(W,S)$| type |$\mathbb{A}_{3}$| where |$S=\{s_{1},s_{2},s_{3}\}$|⁠. We have
$$ \begin{align*} &\Omega=\{x_{\{s_{1}\}}-x_{\{s_{2}\}},x_{\{s_{2}\}}-x_{\{s_{3}\}},x_{\{s_{1},s_{2}\}}-x_{\{s_{2},s_{3}\}}\}.\end{align*} $$
The matrix |$[{\textsf{J}}(r)]$| is zero for |$r\geqslant 3$|⁠. For |$r=2$|⁠, |$\textsf{Rad}^{2}({\hat{\mathscr{D}}})$| is one-dimensional. Since
$$ \begin{align*} &(x_{\{s_{1},s_{2}\}}-x_{\{s_{2},s_{3}\}})^{2}=x_{\{s_{1}\}}-2x_{\{s_{2}\}}+x_{\{s_{3}\}},\end{align*} $$
the corresponding row of |$[{\textsf{J}}(2)]$| is
$$\begin{pmatrix} 1&-1&0 \end{pmatrix}$$
⁠. Therefore, |$d_{W,2}=1$|⁠. Thus, |$n_{W}=24$|⁠.

 

Example 9.12.
By Magma [14] computation, for type |$\mathbb{E}_{6}$| or |$\mathbb{E}_{7}$| and for each |$n\in \{2,3,4\}$|⁠, we found a submatrix of |$[{\textsf{J}}(n)]$| with the prime factors of whose determinant are only |$2,3,5$|⁠. For example, in the case of |$\mathbb{E}_{7}$|⁠, one of the determinants is the following 53-digit number:
$$ \begin{align*} &17617366159747974325631727124547934643940229120000000=2^{105}3^{33}5^{7}.\end{align*} $$

 

Question 9.13.

Let |$(W,S)$| be a Coxeter system and |$p$| be a prime number. By definition, |$p\nmid n_{W}$| implies |$p\nmid |W|$|⁠. Is the converse true?

 

Lemma 9.14.

Let |$(W,S)$| be a Coxeter system, |$n$| a positive integer and suppose that |$p\nmid n_{W,n}$|⁠. We have |${\mathcal{R}}_{p}={\mathcal{R}}$| and there exists a subset |$B(n)\subseteq{\textsf{J}}(n)$| such that the sets |$\{x_{\underline{J},\underline{K}}:(\underline{J},\underline{K})\in B(n)\}\subseteq{\hat{\mathscr{D}}}$| and |$\{\bar x_{\underline{J},\underline{K}}:(\underline{J},\underline{K})\in B(n)\}\subseteq{\bar{\mathscr{D}}}$| form bases for |$\textsf{Rad}^{n}({\hat{\mathscr{D}}})$| and |$\textsf{Rad}^{n}({\bar{\mathscr{D}}}),$| respectively. In particular, |$\dim _{\mathbb{K}} \textsf{Rad}^{n}({\hat{\mathscr{D}}})=\dim _{{\mathbb{F}}} \textsf{Rad}^{n}({\bar{\mathscr{D}}})$|⁠.

 

Proof.
Since |$p\nmid |W|$|⁠, we have |${\mathcal{R}}_{p}={\mathcal{R}}$|⁠. By Lemma 7.4, there is a subset |$B(n)$| of |${\textsf{J}}(n)$| such that |$T:=\{\bar x_{\underline{J},\underline{K}}:(\underline{J},\underline{K})\in B(n)\}$| forms a basis for |$\textsf{Rad}^{n}({\bar{\mathscr{D}}})$|⁠. Let |$[B(n)]_{{\mathbb{F}}}$| be the submatrix of |$[{\textsf{J}}(n)]_{{\mathbb{F}}}$| labelled by the rows |$B(n)$|⁠. Let |$s=|B(n)|$|⁠. There is a |$(s\times s)$|-submatrix of |$[B(n)]_{{\mathbb{F}}}$| with non-zero determinant. Therefore, the corresponding submatrix of |$[B(n)]$| has also non-zero determinant. As such, by Lemma 9.8(ii), we get
$$ \begin{align*} &\dim_{\mathbb{K}} \textsf{Rad}^{n}({\hat{\mathscr{D}}})=\textsf{rank}([{\textsf{J}}(n)]_{\mathbb{K}})\geqslant \textsf{rank}([B(n)]_{\mathbb{K}})=s=\textsf{rank}([{\textsf{J}}(n)]_{{{\mathbb{F}}}})=\dim_{{\mathbb{F}}} \textsf{Rad}^{n}({\bar{\mathscr{D}}}).\end{align*} $$

Conversely, let |$s^{\prime}=\dim _{\mathbb{K}} \textsf{Rad}^{n}({\hat{\mathscr{D}}})=\textsf{rank}([{\textsf{J}}(n)]_{\mathbb{K}})$|⁠. By the assumption, |$p\nmid d_{W,n}$|⁠, there exists a submatrix |$D$| of |$[{\textsf{J}}(n)]_{\mathbb{K}}$| such that |$\textsf{det}(D)\not \equiv 0(\textrm{mod}\ \pi )$|⁠. Since |$D$| has entries belonging in |${\mathcal{O}}$|⁠, taking modulo |$\pi $|⁠, we obtain the corresponding submatrix |$D_{{\mathbb{F}}}$| of |$[{\textsf{J}}(n)]_{{\mathbb{F}}}$|⁠. As such, |$\textsf{det}(D_{{\mathbb{F}}})\neq 0$|⁠. We get |$\dim _{{\mathbb{F}}} \textsf{Rad}^{n}({\bar{\mathscr{D}}})\geqslant s^{\prime}$|⁠. Since |$s=s^{\prime}$|⁠, |$B(n)$| is a desired subset.

 

Theorem 9.15.

Let |$W$| be a finite Coxeter group and suppose that |$p\nmid n_{W}$|⁠. Then |${\mathscr{D}}$| satisfies Hypothesis E. In particular, Theorem 2.3(ii) and (iii) apply for |${\mathscr{D}}$|⁠.

 

Proof.

Clearly, |${\mathscr{D}}$| is |${\mathcal{O}}$|-free of finite rank. By Lemma 9.14, we have |$\dim _{\mathbb{K}} \textsf{Rad}^{n}({\hat{\mathscr{D}}})=\dim _{{\mathbb{F}}} \textsf{Rad}^{n}({\bar{\mathscr{D}}})$| for all |$n\geqslant 0$|⁠. Using Lemma 6.2 ((v) |$\Rightarrow $| (i)), we obtain that |${\mathscr{D}}$| satisfies our strongest hypothesis Hypothesis E. As such, by Theorem 2.3(i), the descent algebra satisfies the remaining hypotheses in Section 1. In particular, Theorem 2.3(ii) and (iii) apply for |${\mathscr{D}}$|⁠.

10 Independence of Choice of Basis for |$n_{W}$|

In Definition 9.9, we defined the numbers |$d_{W,n}$| and |$n_{W}$| based on the choice of the basis |$\Omega $| within |${\textsf{J}}(1)$| (see Equation 6). In this section, we address the question whether these numbers are independent of the choice |$\Omega $|⁠. The answer is positive and it is given in Corollary 10.4. We shall begin with the discussion on totally unimodular matrices.

A matrix |$A$| over |${{\mathbb{Z}}}$| is said to have the consecutive-ones property if |$A$| is a 0-1 matrix (that is, its entries can only take values 0 or 1) and there is a rearrangement of the columns of |$A$| so that, in each row, the ones appear consecutively. It is known that such a matrix is totally unimodular, that is, every square non-singular submatrix of |$A$| has determinant |$\pm 1$| (see [18]). It follows that we have the following lemma.

 

Lemma 10.1.

Suppose that |$A$| is a square non-singular matrix with the consecutive-ones property. Then every entry of |$A^{-1}$| can only take values |$0$|⁠, |$1$| or |$-1$|⁠.

 

Proof.

This follows from the totally unimodularity that the cofactor matrix of |$A$| admits entries with values |$0$|⁠, |$1$| or |$-1$|⁠. Furthermore, |$\textsf{det}(A)=\pm 1$|⁠. Therefore, we get the desired property for |$A^{-1}$|⁠.

We say that a matrix |$A$| over |${{\mathbb{Z}}}$| has the signed consecutive-ones property if there exists a diagonal matrix |$D$| with diagonal entries |$\pm 1$| and a matrix |$B$| with consecutive-ones property such that |$A=DB$|⁠, that is, every row of |$A$| has the consecutive-ones or consecutive-minus-ones property. Since |$A^{-1}=B^{-1}D$|⁠, we get the following.

 

Corollary 10.2.

Suppose that |$A$| is a square non-singular matrix with the signed consecutive-ones property. Then every entry of |$A^{-1}$| can only take values |$0$|⁠, |$1$| or |$-1$|⁠.

In view of Definition 9.9, for each non-zero matrix |$M$| over |${{\mathbb{Z}}}$| of size |$(m\times n)$| and |$s$| be a positive integer not more than the rank of |$M$|⁠, we write |$\textsf{gcd}_{s}(M)$| for the greatest common divisor of the determinants of all the |$(s\times s)$|-submatrices of |$M$|⁠.

 

Lemma 10.3.

Let |$M$| be a non-zero |$(m\times n)$|-matrix over |${{\mathbb{Z}}}$|⁠, |$s\leqslant \textsf{rank}(M)$|⁠, |$c_{1},\ldots ,c_{t}$| be distinct numbers in |$[1,n]$| for some positive integer |$t$| and |$(b_{ij})$| be an |$(t\times t)$|-matrix over |${{\mathbb{Z}}}$|⁠. Suppose that |$M^{\prime}$| is the matrix obtained by replacing, for each |$j\in [1,t]$| and |$i\in [1,m]$|⁠, the |$(i,c_{j})$|-entry of |$M$| by |$\sum _{k=1}^{t}M_{i,c_{k}}b_{kj}$|⁠, that is, |$M^{\prime}=MA$| where |$A$| is obtained from the identity matrix of size |$n$| by replacing its |$(c_{i},c_{j})$|-entry by |$b_{ij}$|⁠. Then |$\textsf{gcd}_{s}(M)\mid \textsf{gcd}_{s}(M^{\prime})$|⁠.

 

Proof.
Let |$d=\textsf{gcd}_{s}(M)$| and, for subsets |$I\subseteq [1,m]$| and |$J\subseteq [1,n]$| of size |$s$|⁠, we write |$M_{IJ}$| for the corresponding |$(s\times s)$|-submatrix of |$M$|⁠. Similarly, for |$M^{\prime}$|⁠. Furthermore, let
$$ \begin{align*} &\Gamma=\{(I,J):I\subseteq [1,m],\ J\subseteq [1,n],\ |I|=s=|J|\}.\end{align*} $$
Consider |$(I,J)\in \Gamma $| and let |$J\cap \{c_{j}:j \in [1,t]\}=\{j_{1},\ldots ,j_\ell \}$|⁠. By multilinear and alternating properties of determinant, we have
$$ \begin{align*} \textsf{det}(M^{\prime}_{IJ})&=\sum_{c_{j}\in J}\sum_{(k_{1},\ldots,k_\ell)}b_{k_{1}j_{1}}\cdots b_{k_\ell j_\ell}\textsf{det}(M_{I}(k_{1},\ldots,k_\ell)), \end{align*} $$
where the second sum is taken over all |$(k_{1},\ldots ,k_\ell )\in [1,t]^\ell $| and |$M_{I}(k_{1},\ldots ,k_\ell )$| is the matrix obtained from |$M_{IJ}$| by replacing the columns correspond to |$c_{j_{1}},\ldots ,c_{j_\ell }$| by the columns correspond to |$c_{k_{1}},\ldots ,c_{k_\ell }$| in the submatrix |$M_{I,[1,n]}$|⁠. Notice that
$$ \begin{align*} &\textsf{det}(M_{I}(k_{1},\ldots,k_\ell))=\left \{\begin{array}{ll} 0 &\textrm{if}\ k_{r}=k_{r^{\prime}}\ \textrm{for some}\ r\neq r^{\prime},\\ \textsf{det}(M_{I,K})& \text{otherwise,}\end{array}\right.\end{align*} $$
where |$K$| is obtained from |$J$| by removing |$c_{j_{1}},\ldots ,c_{j_\ell }$| and then adding the distinct |$c_{k_{1}},\ldots ,c_{k_\ell }$|⁠. As such, |$d\mid \textsf{det}(M^{\prime}_{IJ})$| and hence |$d\mid \textsf{gcd}_{s}(M^{\prime})$|⁠.

We obtain the following corollary addressing the independence of |$n_{W}$| of the choice of basis within |${\textsf{J}}(1)$|⁠.

 

Corollary 10.4.

For each positive integer |$n$|⁠, the number |$d_{W,n}$| is independent of the basis within |${\textsf{J}}(1)$| we have chosen in Definition 9.9. In particular, the same holds true for |$n_{W}$|⁠.

 

Proof.
By Lemma 9.8(i), we obtain that |$M:=[{\textsf{J}}(n)]$| has integer entries. Let |$\Omega ^{\prime}$| be another subset of |${\textsf{J}}(1)$| such that |${\mathcal{B}}^{\prime}$| forms another basis of |$\textsf{Rad}(\mathscr{D}_{\mathbb{Q}})$| and let |$M^{\prime}:=[{\textsf{J}}^{\prime}(n)]$| be the matrix with respect to |${\mathcal{B}}^{\prime}$| which also has integer entries. For each |$J\subseteq S$|⁠, recall from Section 9, |$J_{1},\ldots ,J_{n_{J}}$| are the distinct subsets of |$S$| such that |$J_{i}=_{W} J$| for each |$i\in [1,n_{J}]$|⁠. For each |$x_{J_{i},J_{j}}\in T$|⁠, we have
$$ \begin{align*} &x_{J_{i},J_{j}}=\left \{\begin{array}{ll}x_{J_{i},J_{i+1}}+\cdots+x_{J_{j-1},J_{j}}&\textrm{if}\ i<j,\\ -(x_{J_{i},J_{i+1}}+\cdots+x_{J_{j-1},J_{j}})&\textrm{if}\ i>j.\end{array}\right.\end{align*} $$
Therefore, the transition matrix |$A$| from |${\mathcal{B}}$| to |${\mathcal{B}}^{\prime}$| has the signed consecutive-ones property. By Corollary 10.2, |$A^{-1}$| has entries in |$\{0,1,-1\}$|⁠. Notice that we can rearrange the columns of |$M$| and |$M^{\prime}$| so that |$M=M^{\prime}A$|⁠. Let |$s$| be the rank of |${\mathbb{Q}}\otimes _{{\mathbb{Z}}} M$| (or |${\mathbb{Q}}\otimes _{{\mathbb{Z}}} M^{\prime}$|⁠). Applying Lemma 10.3 twice, we have
$$ \begin{align*} &{\textsf{gcd}}_{s}(M)\mid{\textsf{gcd}}_{s}(M^{\prime})\mid{\textsf{gcd}}_{s}(M),\end{align*} $$
which forces the equality |${\textsf{gcd}}_{s}(M)= {\textsf{gcd}}_{s}(M^{\prime})$|⁠. Therefore, |$d_{W,n}$| does not depend on the choice of basis within |${\textsf{J}}(1)$|⁠.

11 nilCoxeter Algebras Satisfy Hypothesis E

In this section, we show that the nilCoxeter algebra satisfies our strongest hypothesis. The algebra originates in [8, Theorem 3.4]. Subsequently, for example, it has been used in the study of the Schubert polynomials in [17] and studied for its categorification aspect in [22].

Fix a finite Coxeter system |$(W,S)$|⁠. For any integral domain |$R$|⁠, we define the nilCoxeter algebra |$\mathscr{N}_{R}(W)$| as the |$R$|-algebra generated by |$\{x_{s}:s\in S\}$| subject to the relations |$x_{s}^{2}=0$| and

$$ \begin{align*} &\underbrace{x_{s}x_{s^{\prime}}x_{s}\cdots}_{m\ \textrm{factors}}=\underbrace{x_{s^{\prime}}x_{s}x_{s^{\prime}}\cdots}_{m\ \textrm{factors}}\end{align*} $$

for distinct |$s,s^{\prime}\in S$| and where |$m$| is the order of |$ss^{\prime}$|⁠. As a consequence, |$\mathscr{N}_{R}(W)$| has an |$R$|-basis |$\{x_{w}:w\in W\}$| such that

$$ \begin{align*} &x_{u}x_{v}=\left \{\begin{array}{ll} x_{uv}&\textrm{if}\ \ell(uv)=\ell(u)+\ell(v),\\ 0&\text{otherwise.}\end{array}\right.\end{align*} $$

Since |$x_{w}$| is nilpotent for any |$1\neq w\in W$|⁠, it belongs in |$\textsf{Rad}(\mathscr{N}_{k}(W))$|⁠. As such, |$\mathscr{N}_{k}(W)$| is basic with a single simple module. In general, the |$n$|th radical layer |$\textsf{Rad}^{n}(\mathscr{N}_{k}(W))$| has a basis |$\{x_{w}:\ell (w)\geq n\}$|⁠.

 

Lemma 11.1.
Let |$A=\mathscr{N}_{k}(W)$| and |$T$| be the unique simple |$A$|-module. The projective cover |$P_{T}$| of |$T$| is the regular |$A$|-module and
$$ \begin{align*} [\textsf{Rad}^{n}(P_{T})/\textsf{Rad}^{n+1}(P_{T}):T]&=\dim_{k} \textsf{Rad}^{n}(A)/\textsf{Rad}^{n+1}(A)=|\{w\in W:\ell(w)=n\}|. \end{align*} $$

Similar as before, let |${\mathscr{N}}:=\mathscr{N}_{\mathcal{O}}(W)$|⁠, |${\hat{\mathscr{N}}}:=\mathscr{N}_{\mathbb{K}}(W)$|⁠, and |${\bar{\mathscr{N}}}:=\mathscr{N}_{{\mathbb{F}}}(W)$|⁠.

 

Theorem 11.2.

Let |$W$| be a Coxeter group. The algebra |${\mathscr{N}}$| satisfies Hypothesis E.

 

Proof.

By Lemma 11.1, we get |$\dim _{{\mathbb{F}}} \textsf{Rad}^{n}({\bar{\mathscr{N}}})/\textsf{Rad}^{n+1}({\bar{\mathscr{N}}})=\dim _{\mathbb{K}} \textsf{Rad}^{n}({\hat{\mathscr{N}}})/\textsf{Rad}^{n+1}({\hat{\mathscr{N}}})$|⁠. Using Lemma 6.2 ((v) |$\Rightarrow $| (i)), we obtain that |${\mathscr{N}}$| satisfies our strongest hypothesis Hypothesis E.

Let |$T$| be the unique simple |$\mathscr{N}_{k}(W)$|-module. Theorems 11.2 and 2.3 imply that both |$\dim _{k}\textsf{Ext}^{i}_{\mathscr{N}_{k}(W)}(T,T)$| and hence the Hilbert–Poincaré series for the Ext algebra |$\textsf{Ext}^{*}_{\mathscr{N}_{k}(W)}(T,T)$| are independent of the field |$k$|⁠. In the recent article [4], the first author studied the type |$\mathbb{A}$| case. He computed the Hilbert–Poincaré series and presented a set of generators and relations for the Ext algebra. We refer the readers to the article for the full statement.

 

Theorem 11.3
([4, Theorem 1.1]).
Suppose that |$n\geqslant 2$|⁠. The Ext algebra |$\textsf{Ext}^{*}_{\mathscr{N}_{{\mathbb{Z}}}({\mathfrak{S}}_{n})}({{\mathbb{Z}}},{{\mathbb{Z}}})$| has the Hilbert–Poincaré series
$$ \begin{align*} &\sum_{i=0}^\infty t^{i}\left (\textsf{rank}_{{\mathbb{Z}}}\textsf{Ext}^{i}_{\mathscr{N}_{{\mathbb{Z}}}({\mathfrak{S}}_{n})}({{\mathbb{Z}}},{{\mathbb{Z}}})\right )=\frac{1}{(1-t)^{n-1}}.\end{align*} $$

12 |${\mathcal{R}}$|-trivial Monoid Algebras Satisfy Hypothesis B

A (finite) monoid |$M$| is |${\mathcal{R}}$|-trivial if the subsets |$\sigma M$| are all distinct when |$\sigma $| runs through all elements of |$M$|⁠, that is, if |$\sigma M=\tau M$| then |$\sigma =\tau $|⁠. As shown by Schocker [32], this class of algebras includes the unital left regular band algebras (see Section 13) and |$0$|-Hecke algebras. In [6], Berg–Bergeron–Bhargava–Saliola proved that the notion of |${\mathcal{R}}$|-trivial monoid is equivalent to weakly ordered monoid and gave a recursive formula to construct a complete set of primitive orthogonal idempotents of |${\mathbb{C}} M$|⁠. Subsequently, for a general ring |$R$| (with known complete system of primitive orthogonal idempotents, for example, |${{\mathbb{Z}}}$| or a field), a recursive formula has also been given by Nijholt–Rink–Schwenker [26] for |$R M$| by using the observation that |${{\mathbb{Z}}} M$| is isomorphic to a subalgebra |$\textsf{U}_{M}^{{\mathbb{Z}}}$| of the upper triangular matrices |$\textsf{U}({{\mathbb{Z}}},n)$| where |$n=|M|$|⁠. In particular, we have the following theorem.

 

Theorem 12.1
([26]).
There are |$m$| disjoint nonempty subsets |$T_{1},\ldots ,T_{m}$| of |$[1,n]$| such that |$m$| is the number of simple modules for |${{\mathbb{Z}}} M$| and a complete set of primitive orthogonal idempotents |$\{E_{1},\ldots ,E_{m}\}$| of |$\textsf{U}_{M}^{{\mathbb{Z}}}$| such that, for each |$i\in [1,m]$| and |$j\in [1,n]$|⁠,
$$ \begin{align*} &(E_{i})_{jj}=\left \{\begin{array}{ll} 1&\textrm{if}\ j\in T_{i},\\ 0&\textrm{if}\ j\not\in T_{i},\end{array}\right.\end{align*} $$
Furthermore, for arbitrary ring |$R$|⁠, we have |$R M\cong R\otimes _{{\mathbb{Z}}} \textsf{U}^{{\mathbb{Z}}}_{M}=\textsf{U}^{R}_{M}$| and, over a field |$k$|⁠, |$\{\bar E_{1},\ldots ,\bar E_{m}\}$| is a complete set of primitive orthogonal idempotents of |$\textsf{U}^{k}_{M}$|⁠.

Again, let |${\mathcal{O}}$| be a local principal ideal domain with maximal ideal |$(\pi )$|⁠, |${\mathbb{K}}$| be the field of fractions of |${\mathcal{O}}$| and |${{\mathbb{F}}}={\mathcal{O}}/(\pi )$|⁠.

 

Theorem 12.2.

The |${\mathcal{R}}$|-trivial monoid algebras satisfy Hypothesis B. In particular, Theorem 2.3(ii) applies for this class of algebras.

 

Proof.
We identify |${\mathcal{O}} M$| with |$\textsf{U}^{\mathcal{O}}_{M}$| under the isomorphism in Theorem 12.1. The algebra |${\mathcal{O}} M$| has finite |${\mathcal{O}}$|-rank as |$M$| is finite. By Theorem 12.1, the primitive orthogonal idempotents of |$\textsf{U}^{{\mathbb{F}}}_{M}$| are obtained from those in |$\textsf{U}^{{\mathbb{Z}}}_{M}$| (and hence |$\textsf{U}^{\mathcal{O}}_{M}$|⁠) by taking entry-wise modulo |$(\pi )$|⁠. In particular, they are liftable to |$\textsf{U}^{\mathcal{O}}_{M}$|⁠. We are left to show that
$$ \begin{align*} &\textsf{Rad}(\textsf{U}^{\mathcal{O}}_{M})=\pi \textsf{U}^{\mathcal{O}}_{M}+(\textsf{Rad}(\textsf{U}^{\mathbb{K}}_{M})\cap \textsf{U}^{\mathcal{O}}_{M}).\end{align*} $$
Let |$\textsf{T}$| and |$\textsf{U}^{+}$| be the diagonal and strictly upper triangular matrices in |$\textsf{U}({\mathcal{O}},n),$| respectively. Clearly, |$\textsf{Rad}(\textsf{U}^{\mathcal{O}}_{M})$| belongs in |$\pi \textsf{T}\oplus \textsf{U}^{+}$|⁠. Let |$C\in \textsf{Rad}(\textsf{U}^{\mathcal{O}}_{M})$|⁠. By Theorem 12.1, there exist |$c_{1},\ldots ,c_{m}\in (\pi )$| such that |$C-\sum _{i=1}^{m}c_{i} E_{i}\in \textsf{U}^{+}$|⁠. Since |$\textsf{Rad}(\textsf{U}^{\mathbb{K}}_{M})=\textsf{U}^{\mathbb{K}}_{M}\cap \textsf{U}^{+}$| and |$\textsf{U}^{\mathbb{K}}_{M}$| is obtained from |$\textsf{U}^{\mathcal{O}}_{M}$| by extension of scalar entry-wise, we have |$C-\sum _{i=1}^{m}c_{i} E_{i}\in \textsf{Rad}(\textsf{U}^{\mathbb{K}}_{M})\cap \textsf{U}^{\mathcal{O}}_{M}$|⁠. Thus, |${\mathcal{O}} M$| satisfies Hypothesis B. By Theorem 2.3(i), |${\mathcal{O}} M$| satisfies Hypothesis A and hence Theorem 2.3(ii) applies for |${\mathcal{O}} M$|⁠.

 

Example 12.3.

Let |$(W,S)$| be a Coxeter system as in Section 7 and |$S=\{s_{i}:i\in I\}$|⁠. The |$0$|-Hecke algebra |$\mathscr{H}_{R}$| over an arbitrary commutative ring |$R$| with 1 is the |$R$|-algebra with identity generated by |$\{T_{i}:i\in I\}$| subject to the relations

  • (i)

    |$T_{i}^{2}=-T_{i}$| for all |$i\in I$|⁠, and

  • (ii)
    for |$i,j\in I$| with |$i\neq j$|⁠, if |$n_{ij}$| is the order of |$s_{i}s_{j}$|⁠, we have
    $$ \begin{align*} &\underbrace{T_{i}T_{j}T_{i}\cdots}_{n_{ij}\ \textrm{factors}}=\underbrace{T_{j}T_{i}T_{j}\cdots}_{n_{ij}\ \textrm{factors}}.\end{align*} $$

Notice that |$\mathscr{H}_{\mathbb{K}}\cong{\mathbb{K}}\otimes _{\mathcal{O}}\mathscr{H}_{\mathcal{O}}$| and |$\mathscr{H}_{{\mathbb{F}}}\cong{{\mathbb{F}}}\otimes _{\mathcal{O}} \mathscr{H}_{\mathcal{O}}$|⁠. Norton [27] showed that the |$0$|-Hecke algebras are basic over arbitrary field and their simple modules are parametrised by the subsets of |$S$|⁠. For any |$J\subseteq S$|⁠, let |$S_{J}$| be the |${\mathcal{O}}$|-free module of rank |$1$| defined by the character
$$ \begin{align*} &\lambda_{J}(T_{i})=\left \{\begin{array}{ll} 0&\textrm{if}\ s_{i}\in J,\\ -1&\textrm{if}\ s_{i}\not\in J.\end{array}\right.\end{align*} $$
As such, |$\hat S_{J}:={\mathbb{K}}\otimes _{\mathcal{O}} S_{J}$| and |$\bar S_{J}:={{\mathbb{F}}}\otimes _{\mathcal{O}} S_{J}$| are the simple modules for |$\mathscr{H}_{\mathbb{K}}$| and |$\mathscr{H}_{{\mathbb{F}}}$|⁠, respectively. Following Schocker [32], the |$0$|-Hecke algebras are |${\mathcal{R}}$|-trivial monoid algebras. As such, by Theorem 12.2, for |$I,J\subseteq S$| and all |$t\geqslant 0$|⁠, |$\textsf{Ext}^{t}_{\mathscr{H}_{\mathcal{O}}}(S_{I},S_{J})$| is |${\mathcal{O}}$|-free and furthermore,
$$ \begin{align*} \textsf{Ext}^{t}_{\mathscr{H}_{{\mathbb{F}}}}(\bar S_{I},\bar S_{J})&\cong{{\mathbb{F}}}\otimes_{\mathcal{O}}\textsf{Ext}^{t}_{\mathscr{H}_{\mathcal{O}}}(S_{I},S_{J}),\\ \textsf{Ext}^{t}_{\mathscr{H}_{\mathbb{K}}}(\hat S_{I},\hat S_{J})&\cong{\mathbb{K}}\otimes_{\mathcal{O}}\textsf{Ext}^{t}_{\mathscr{H}_{\mathcal{O}}}(S_{I},S_{J}). \end{align*} $$
In particular, |$\dim _{{\mathbb{F}}} \textsf{Ext}^{t}_{\mathscr{H}_{{\mathbb{F}}}}(\bar S_{I},\bar S_{J})=\dim _{\mathbb{K}} \textsf{Ext}^{t}_{\mathscr{H}_{\mathbb{K}}}(\hat S_{I},\hat S_{J})$|⁠.

13 Connected CW Left Regular Bands Satisfy Hypothesis D

In this section, we consider the class of algebras |$k B$| where |$B$| is a left regular band, that is, |$B$| is a semigroup satisfying the identities |$x^{2}=x$| and |$xyx=xy$| for all |$x,y\in B$|⁠. Since |$B$| may not have an identity, the algebra |$k B$| does not necessarily admit a unit. However, following Margolis–Saliola–Steinberg [25], to guarantee a unit, it suffices to assume that |$B$| is a connected CW left regular band; we shall not need the full definition here and refer the reader to their paper. In the case |$B$| is a monoid, it is also |${\mathcal{R}}$|-trivial (see Section 12). This class of algebras includes the face algebras of hyperplane arrangements in real spaces. In the special case when the hyperplane arrangement is given by the reflection arrangement of a (finite) Coxeter group |$W$|⁠, the descent algebra of |$W$| is isomorphic to the opposite of the fixed point space of the face algebra under the action of |$W$| as shown by Bidigare [9].

Throughout this section, we assume that |$B$| is a connected CW left regular band. Let |$\Lambda (B)=\{Bb:b\in B\}$| be the support semilattice of |$B$| and

$$ \begin{align*} &\sigma:B\to \Lambda(B)\end{align*} $$

be the support map, defined as |$\sigma (a)=Ba$|⁠, which is a semigroup homomorphism. Recall that, given a graded (finite) poset |$(\leq ,P)$|⁠, the closed interval |$[X,Z]$| is the subset |$\{Y:X\leq Y\leq Z\}$| of |$P$|⁠. Similarly, we denote |$(X,Z)$| for the open interval. The rank (or length) |$\ell ([X,Z])$| of |$[X,Z]$| is the length of a longest chain starting at |$X$| and ending at |$Y$|⁠.

 

Theorem 13.1
([25, Theorem 1.1(4,5)]).
Let |$Q$| be the Ext quiver of |$k B$|⁠. The support semilattice |$\Lambda (B)$| is a graded poset and |$Q$| is the Hasse diagram of |$\Lambda (B)$|⁠. Furthermore, |$Q$| is acyclic, |$k B$| is basic and |$k B\cong k Q/I$| where |$I$| is the ideal generated by, for any two vertices |$X,Z$| in |$Q$| such that |$\ell ([X,Z])=2$|⁠, the relations
$$ \begin{align*} &\sum_{Y\in (X,Z)} (X\to Y\to Z).\end{align*} $$

Same as before, let |${\mathcal{O}}$| be a local principal ideal domain with maximal ideal |$(\pi )$|⁠, |${\mathbb{K}}$| be the field of fractions of |${\mathcal{O}}$| and |${{\mathbb{F}}}={\mathcal{O}}/(\pi )$|⁠.

 

Theorem 13.2.

Let |$B$| be a connected CW left regular band. The algebra |${{\mathcal{O}} B}$| satisfies Hypothesis D. In particular, Theorem 2.3(ii) applies for |${{\mathcal{O}} B}$|⁠.

 

Proof.
Let |$Q$| be the Ext quiver of |${{{\mathbb{F}}} B}$| (or |${{\mathbb{K}} B}$|⁠). Consider the ideal |$I$| of |${\mathcal{O}} Q$| generated by
$$ \begin{align*} &\sum_{Y\in (X,Z)} (X\to Y\to Z)\in{\mathcal{O}} Q\end{align*} $$
and let |$A={\mathcal{O}} Q/I\cong{{\mathcal{O}} B}$|⁠. By Theorem 13.1, |${{{\mathbb{F}}} B}\cong{{\mathbb{F}}}\otimes _{\mathcal{O}} A$|⁠, |${{\mathbb{K}} B}\cong{\mathbb{K}}\otimes _{\mathcal{O}} A$| and |$J_{Q}^{n}=0$| for some positive integer |$n\geqslant 2$| as |$Q$| is acyclic. Clearly, |$I\leqslant J_{Q}^{2}$|⁠. Thus, |$A$| satisfies Hypothesis D and hence Hypothesis B by Theorem 2.3(i). Therefore, Theorem 2.3(ii) applies for |$A$|⁠.

At this point, it is not clear whether the algebra |${{\mathcal{O}} B}$| satisfies our strongest hypothesis Hypothesis E. We consider an example.

 

Example 13.3.

Consider the face algebra |${\mathcal{O}}{\mathcal{F}}$| with the hyperplane arrangement defined by the hyperplanes |$H_{1},\ldots ,H_{n}$| in |${\mathbb{R}}^{d}$|⁠. Suppose further that there exists a subspace |$X$| such that |$H_{i}\cap H_{j}=X$| for all |$1\leqslant i\neq j\leqslant n$|⁠. By [30, Proposition 8.5], the Ext quiver |$Q$| of |${{\mathbb{F}}}{\mathcal{F}}$| (or |${\mathbb{K}}{\mathcal{F}}$|⁠) is given by the diagram below.

The only relation generating the ideal |$I$| is |$\sum ^{n}_{i=1}\beta _{i}\alpha _{i}$|⁠. As such,
$$ \begin{align*} \dim_{\mathbb{K}}{\mathbb{K}}{\mathcal{F}}&=4n+1=\dim_{{\mathbb{F}}}{{\mathbb{F}}}{\mathcal{F}},\\ \dim_{\mathbb{K}}\textsf{Rad}({\mathbb{K}}{\mathcal{F}})&=3n-1=\dim_{{\mathbb{F}}}\textsf{Rad}({{\mathbb{F}}}{\mathcal{F}}),\\ \dim_{\mathbb{K}}\textsf{Rad}^{2}({\mathbb{K}}{\mathcal{F}})&=n-1=\dim_{{\mathbb{F}}}\textsf{Rad}^{2}({{\mathbb{F}}}{\mathcal{F}}),\\ \dim_{\mathbb{K}}\textsf{Rad}^{r}({\mathbb{K}}{\mathcal{F}})&=0=\dim_{{\mathbb{F}}}\textsf{Rad}^{r}({{\mathbb{F}}}{\mathcal{F}}) \end{align*} $$
for all |$r\geqslant 3$|⁠. Thus, |${\mathcal{O}}{\mathcal{F}}$| satisfies Hypothesis E using Lemma 6.2 ((v) |$\Rightarrow $| (i)). In particular, both Theorem 2.3(ii) and (iii) apply for |${\mathcal{O}}{\mathcal{F}}$|⁠.

 

Question 13.4.

Let |$B$| be a connected CW left regular band. Does |${{\mathcal{O}} B}$| satisfy Hypothesis E?

Funding

The second author would like to thank the Edinburgh Napier University for their support at the beginning for this project.

Communicated by Prof. Weiqiang Wang

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