Simple modules in the Auslander-Reiten quiver of principal blocks with abelian defect groups

TL;DR

This paper studies the placement of simple modules in the Auslander-Reiten quiver of principal blocks with abelian defect groups, providing reductions to finite simple groups and specific results for characteristic 3.

Contribution

It introduces a reduction technique to finite simple groups and proves that simple modules lie at the end of their components in characteristic 3.

Findings

Reduction to finite simple groups

Simple modules in characteristic 3 lie at the end of their components

Provides structural insights into principal blocks with abelian defect groups

Abstract

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander-Reiten quiver of the principal block of a finite group with non-cyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$, we prove that simple modules in the principal block all lie at the end of their components

Equations

$$L_i(U):=U{J}^{i-1}/U{J}^i\text{ and }{S}_i(U):={\mathrm{soc}}^i(U)/{\mathrm{soc}}^{i-1}(U)$$

$$L_i(X){\downarrow }_N=L_i(X{\downarrow }_N)\text{ and }{S}_i(X){\downarrow }_N=S_i(X{\downarrow }_N)$$

$$L_i(Y){\uparrow }^G=L_i(Y{\uparrow }^G)\text{ and }{S}_i(Y){\uparrow }^G=S_i(Y{\uparrow }^G).$$

$$\mathcal{A}(M):0⟶{\Omega }^2(M)⟶X_M⟶M⟶0$$

$$\mathcal{A}({\Omega }^{-1}(S)):0⟶\Omega (S)⟶\mathcal{H}(P(S))\oplus P(S)⟶{\Omega }^{-1}(S)⟶0$$

$$P(S_2)=\boxed{\begin{array}{c}{S}_2\\ S_3\\ ⋮\\ S_n\\ S\\ S_2\end{array}},P(S_3)=\boxed{\begin{array}{c}{S}_3\\ ⋮\\ S_n\\ S\\ S_2\\ S_3\end{array}},\cdots ,P(S_n)=\boxed{\begin{array}{c}{S}_n\\ S\\ S_2\\ ⋮\\ S_{n-1}\\ S_n\end{array}}.$$

$$\left(\begin{array}{cccccccc}2& 1& \cdots & \cdots & 1& 0& \cdots & 0\\ 1& 2& \ddots & & ⋮& ⋮& & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮& ⋮& & ⋮\\ ⋮& & \ddots & 2& 1& 0& \cdots & 0\\ 1& \cdots & \cdots & 1& \ast & \cdots & \cdots & \ast \\ 0& \cdots & \cdots & 0& ⋮& & & ⋮\\ ⋮& & & ⋮& ⋮& & & ⋮\\ 0& \cdots & \cdots & 0& \ast & \cdots & \cdots & \ast \end{array}\right),$$

$$0⟶\Omega (S_2)⟶S\oplus P(S_2)⟶{\Omega }^{-1}(S_2)⟶0,$$

$$\displaystyle\mathcal{H}(P(S))\$$

$$W:=\boxed{\begin{array}{c}{S}_2\\ S_3\\ ⋮\\ S_n\\ S\end{array}}.$$

$$\mathcal{H}\supseteq U=\boxed{\begin{array}{c}{S}_2\\ S_3\\ ⋮\\ S_n\end{array}}.$$

$$P(S)\twoheadrightarrow \boxed{\begin{array}{c}S\\ S_2\\ S_3\\ ⋮\\ S_n\end{array}}\text{ and hence a kG-epimorphism }\psi :\mathcal{H}=P(S)J/S\twoheadrightarrow \boxed{\begin{array}{c}{S}_2\\ S_3\\ ⋮\\ S_n\end{array}}$$

$$1_{\tilde{b}}=1_B+1_{B_1}+\cdots +1_{B_n}$$

$$1_B\in 1_{\tilde{b}}Z(kG)\subseteq 1_{\tilde{b}}{C}_{kG}(H)=:C.$$

$$\begin{array}{rl}{1}_B\in C[\tilde{b}]& =Z(\tilde{b})\ast G[\tilde{b}]\text{ (where ∗ denotes the crossed product) }\\ & \subseteq Z(kH)\ast N\subseteq kN.\end{array}$$

$$1_b=1_B+1_{B_1^{\prime }}+\cdots +1_{B_t^{\prime }}$$

$$\mathcal{S}{\otimes }_k{Z}_j\cong \mathcal{S}$$

$$\mathrm{IBr}(B)$$

$$\mathrm{IBr}(b)$$

$${\mathcal{S}}_{ij}{\downarrow }_N={\mathcal{T}}_i\text{ and }{\mathcal{T}}_i{\uparrow }^G={\mathcal{S}}_{i1}\oplus \cdots \oplus {\mathcal{S}}_{iq},$$

$${\mathcal{S}}_i{\downarrow }_N={\mathcal{T}}_{i1}\oplus {\mathcal{T}}_{i2}\oplus \cdots \oplus {\mathcal{T}}_{iq}\text{ and }{\mathcal{T}}_{ij}{\uparrow }^G={\mathcal{S}}_i$$

$${\mathcal{S}}_{ij}={\mathcal{S}}_{i1}{\otimes }_k{Z}_j.$$

$$(\mathcal{S}{\otimes }_k{Z}_j){\downarrow }_N$$

$$=\mathcal{T}\oplus {\mathcal{T}}^g\oplus \cdots \oplus {\mathcal{T}}^{g^{q-1}}$$

$$P(\mathcal{S}){\downarrow }_N\cong P({\mathcal{T}}_1)\oplus \cdots \oplus P({\mathcal{T}}_q)\text{ and }\mathcal{H}(P(\mathcal{S})){\downarrow }_N\cong \underset{j=1}{\overset{q}{⨁}}\mathcal{H}(P({\mathcal{T}}_j)).$$

$$\mathcal{T}$$

$$=P(\mathcal{S}){\downarrow }_N/(P(\mathcal{S})kG\tilde{J})\text{ by Lemma refradical }$$

$$=P(\mathcal{S}){\downarrow }_N/P(\mathcal{S}){\downarrow }_N\tilde{J}.$$

$$\mathcal{H}(P(\mathcal{S})){\downarrow }_N=(P(\mathcal{S})J/\mathcal{S}){\downarrow }_N=\mathcal{H}(P(\mathcal{T})).$$

$$\mathcal{H}(P(S)){\downarrow }_N\cong \mathcal{H}(P(T)).$$

$$S{\downarrow }_N=T_{11}\oplus \cdots \oplus T_{1q}\text{ and}{T}_{1j}{\uparrow }^G=S\text{ for each }1\le j\le q,$$

$$S_i{\downarrow }_N=T_{i1}\oplus T_{i2}\oplus \cdots \oplus T_{iq}.$$

Full text

The authors gratefully acknowledge financial support by the funding scheme TU Nachwuchsring Individual Funding 2016 granted to the second author by the TU Kaiserslautern. The first author was also supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)15K04776, 2015–2018.

1. Introduction

The position of simple modules in the stable Auslander-Reiten quiver of the group algebra $kG$ over a field $k$ of characteristic $p$ of a finite group $G$ of order divisible by $p$ is a question that was partially investigated in the 1980’s and the 1990’s in a series of articles by different authors. We refer the reader in particular to [Kaw97, KMU00, KMU01, BU01] and the references therein. The aim of this note is to come back to the following question:

Question A**.**

Let $B$ be a wild $p$-block of $kG$. Under which conditions do all simple $B$-modules lie at the end of their connected components in the stable Auslander-Reiten quiver of $kG$?

A main reason of interest in this question lies in the fact that a simple $kG$-module lies at the end of its component if and only if the heart of its projective cover is indecomposable.

In this article, we focus attention on the case in which the principal block $B_{0}(kG)$ is of wild representation type with abelian defect groups and the prime $p$ is odd, which amounts to requiring that the $p$-rank of $G$ is at least $2$. The case $p=2$ was treated by Kawata-Michler-Uno in [KMU00, Theorem 5]. We aim at extending their results and part of their methods to arbitrary primes. Further, we note that the cases when $B_{0}(kG)$ is of finite or tame representation type are well-understood. In the former case, the distance of a simple module to the rim of its connected component (a tube of shape $({\mathbb{Z}}/\!e{\mathbb{Z}})A_{m}$) is a function of its position in the Brauer tree of the block, while in the later case the position of the simple modules in their connected components is given by Erdmann’s work on tame blocks [Erd90].

Assuming the field $k$ is algebraically closed we prove the following results:

Theorem B**.**

Let $G$ be a finite group and $N\trianglelefteq G$ a normal subgroup such that $G/N$ is solvable of $p^{\prime}$-order. Let $B$ and $b$ be wild blocks of $kG$ and $kN$ respectively such that $1_{B}=1_{b}$. If every simple $b$-module lies at the end of its connected component in the stable Auslander-Reiten quiver of $kN$, then every simple $B$-module lies at the end of its connected component in the stable Auslander-Reiten quiver of $kG$.

Theorem C**.**

Let $p$ be an odd prime. Let $G$ be a finite group with non-cyclic abelian Sylow $p$-subgroups and $O_{p^{\prime}}(G)=1$. Write $O^{p^{\prime}}(G)=Q\times H_{1}\times\cdots\times H_{m}$ ($m\geq 0$), where $Q$ is an abelian $p$-group and $H_{i}$ is a non-abelian finite simple group with non-trivial Sylow $p$-subgroups for each $1\leq i\leq m$. Assume that one of the following conditions is satisfied:

1. (i)

$Q\neq 1$; or 2. (ii)

$Q=1$* and $m\geq 2$; or* 3. (iii)

$Q=1$, $m=1$ and every simple $B_{0}(kH_{1})$-module lies at the end of its connected component in the stable Auslander-Reiten quiver of $kH_{1}$.

Then every simple $B_{0}(kG)$-module lies at the end of its connected component in the stable Auslander-Reiten quiver of $kG$.

Corollary D**.**

Let $p$ be an odd prime. Assume that every simple $B_{0}(kH)$-module lies at the end of its connected component in the stable Auslander-Reiten quiver of $kH$ for every non-abelian finite simple group $H$ with non-cyclic abelian Sylow $p$-subgroups. Then every simple $B_{0}(kG)$-module lies at the end of its connected component in the stable Auslander-Reiten quiver of $kG$ for any finite group $G$ with non-cyclic abelian Sylow $p$-subgroups.

We note that if $p=2$, then the analogues of Theorem C and Corollary D were essentially proven by Kawata-Michler-Uno [KMU00], although not stated in these terms. As a corollary, we also obtain the equivalent of [KMU00, Theorem 5(a)] for the prime $3$.

Theorem E**.**

Assume $p=3$. Let $G$ be a finite group with abelian Sylow $3$-subgroups. If $B_{0}(kG)$ is a wild $3$- block, then every simple $B_{0}(kG)$-module lies at the end of its connected component in the stable Auslander-Reiten quiver of $kG$.

The paper is organised as follows: in Section 2, we recall the state of knowledge on the subject and extend a result of Kawata’s [Kaw97, Theorem 1.5] to describe more precisely the indecomposable summands of the heart of the projective cover of a simple module not lying on the rim of its component. In Section 3, we consider groups having a solvable quotient of $p^{\prime}$-order and prove Theorem B. In Sections 4 and 5, we proceed to a reduction of Question A for principal blocks to the case of finite non-abelian simple groups and prove Theorem C and Corollary D. Finally in Section 6 we deal with the case $p=3$ and prove Theorem E.

2. Notation and preliminary results

Throughout, we assume that $k$ is an algebraically closed field of characteristic $p>0$, and we let $G$ denote a finite group of order divisible $p$. All modules are assumed to be finitely generated right modules. For a $p$-block $B$, we write $1_{B}$ for the corresponding block idempotent and ${\mathrm{IBr}}(B)$ for the set of isomorphism classes of simple $kB$- modules. Furthermore, unless otherwise specified, we assume $B_{0}:=B_{0}(kG)$, the principal block of $kG$, is wild. (When the defect groups of $B$ are abelian, we may therefore assume that a Sylow $p$-subgroup of $G$ is non-cyclic, or equivalently that the $p$-rank of $G$ is at least $2$). We denote by $k_{G}$ the trivial $kG$-module.

We let $J:=J(kG)$ denote the Jacobson radical of $kG$. For a $kG$-module $U$, we define $J(kG)^{0}:=kG$ and for any integer $i\geq 0$, $\mathrm{soc}^{i}(U):=\{u\in U\,|\,u\,J^{i}=\{0\}\}$, then inductively for any $i\geq 1$, we write

[TABLE]

for the $i$-th Loewy layer and the $i$-th socle layer of $U$, respectively. We use throughout without further mention the following well-known properties:

Lemma 2.1**.**

Assume $N\trianglelefteq G$ of index prime to $p$.

1. (a)

We have $J=\tilde{J}\,kG=kG\,\tilde{J}$ where $\tilde{J}:=J(kN)$. 2. (b)

Let $X$ be a $kG$-module and $Y$ a $kN$-module, then for any $i\geq 1$ we have

[TABLE]

and

[TABLE]

Proof.

Part (a) is a well-known result of Villamayor [Vil59] and part (b) follows from (a). ∎

Given a $kG$-module $M$, we denote by $\Omega^{n}(M)$ $(n\in{\mathbb{Z}})$ its $n$-th Heller translate. Given a simple $kG$-module $S$, we denote by $P(S)$ its projective cover and by $\mathcal{H}(P(S))$ the heart of $P(S)$, that is $\mathcal{H}(P(S))=P(S)J/\operatorname{soc}(P(S))$.

We let $\Gamma_{s}(kG)$, resp. $\Gamma_{s}(B)$, denote the stable Auslander-Reiten quiver of $kG$, resp. of the $p$-block $B$, and denote by $\Gamma_{s}(M)$ the connected component of $\Gamma_{s}(kG)$ containing a given indecomposable $kG$-module $M$. Moreover, by convention, we use the terminology AR-component to refer to a connected component of $\Gamma_{s}(kG)$.

Erdmann [Erd95] proved that all components of the stable Auslander-Reiten quiver belonging to a wild block have tree class $A_{\infty}$, that is are of the form ${\mathbb{Z}}A_{\infty}$ or infinite tubes ${\mathbb{Z}}A_{\infty}/\langle\tau^{a}\rangle$ of rank $a$, where $\tau=\Omega^{2}$ is the Auslander-Reiten shift.

In a component with tree class $A_{\infty}$ an indecomposable non-projective $kG$-module $M$ is said to lie at the end (or on the rim) of its AR-component if the projective-free part of the middle term $X_{M}$ of the Auslander-Reiten sequence

[TABLE]

terminating at $M$ is indecomposable. For a non-projective simple $kG$-module $S$, the Auslander-Reiten sequence terminating at $\Omega^{-1}(S)$ is of the form

[TABLE]

and is called the standard sequence associated to $S$. In this set up, clearly a simple module $S$ lies at the end of its component if and only if $\mathcal{H}(P(S))$ is indecomposable, and $S$ lies in a tube if and only if $S$ is periodic (i.e. $\Omega$-periodic).

We also recall that for a selfinjective algebra the shape of the components of the stable Auslander-Reiten quiver is an invariant of its Morita equivalence class. By the above, the property of lying on the rim of its AR-component for a non-projective simple module is also invariant under Morita equivalence.

Simple $kG$-modules are known to lie on the rim of their $AR$-components in the following cases:

Theorem 2.2**.**

Let $B$ be a wild $p$-block of $kG$. Then every simple $B$-module lies at the end of its AR-component in all of the following cases:

1. (a)

$G$* has a non-trivial normal $p$-subgroup ([Kaw97, Theorem 2.1]);* 2. (b)

$G$* is $p$-solvable ([Kaw97, Corollary 2.2]);* 3. (c)

$G$* is a perfect finite group of Lie type in the defining characteristic and $B$ has full defect ([KMU01, Theorem]);* 4. (d)

$G$* has an abelian Sylow $2$-subgroup and $B$ is the principal $2$-block. [KMU00, Theorem 5];* 5. (e)

$G$* is a symmetric group, an alternating group or a Schur cover of the latter groups, and the defect of $B$ is divisible by $p^{3}$ ([BU01, §5]).*

Moreover, we will use the following computational criterion throughout:

Theorem 2.3** (Kawata’s Criterion on Cartan matrices [Kaw97, Theorem 1.5]).**

Let $B$ be a wild $p$-block of $kG$. Suppose that there exists a simple $B$-module $S$ lying on the $n$-th row from the end of $\Gamma_{s}(S)$, where $n\geq 2$ is minimal with this property. Then there exist pairwise non-isomorphic simple $B$-modules $S_{2},\ldots,S_{n}$ with the following properties:

1. (a)

For each 2\leq i\leq n$$,S_{i}\cong\Omega^{2(i-2)}(S_{2}) and lies at the end of $\Gamma_{s}(\Omega(S))$; 2. (b)

The projective covers of $P(S_{i})$ of the simple modules $S_{i}\ (2\leq i\leq n)$ are uniserial of length $n+1$ with the following Loewy structure:

[TABLE]

The Cartan matrix of $B$ is given by

[TABLE]

where the columns are labelled by $S_{n},\ldots,S_{2},S,\ldots$ in this order.

Remark 2.4*.*

1. (a)

If the Cartan matrix of a block has the shape of Theorem 2.3(b) above with $n=2$, then the simple module $S$ corresponding to the second column lies on the 2nd row of its AR-component. Indeed in this case $P(S_{2})=\boxed{\begin{smallmatrix}S_{2}\\ S\\ S_{2}\end{smallmatrix}}$ and the standard sequence associated to $S_{2}$ is

[TABLE]

so that $S_{2}$ lies at the end of its AR-component and $S$ on the 2nd row of its AR-component.

A converse to Kawata’s Criterion need not be true in general for an $n\geq 3$. 2. (b)

The above was used to produce two counter-examples of simple modules not lying at the end of their AR-components. Namely, the group $F_{4}(2)$ for $p=5$ has a simple module in the principal block of dimension 875823 lying on the the 2nd row of its AR-component, and the group $2.Ru$ for $p=3$ has a faithful simple module also lying on the 2nd row. See [KMU01, §4]. Both counter-examples are obtained thanks to the decomposition matrices of these groups computed by G. Hiß.

We can now improve Kawata’s result by describing more accurately the structure of the heart of the projective cover of the simple module $S$ lying on the $n$-th row of its AR-component.

Corollary 2.5**.**

With the assumptions and the notation of Theorem 2.3, we have that the heart of the projective cover of the simple module $S$ is decomposable and has a uniserial indecomposable summand of length $n-1$. More precisely

[TABLE]

where $V\,{\not=}\,\{0\}$ is an indecomposable $kG$-module.

Proof.

Let $S_{2},\ldots,S_{n}$ be the simple $kG$-modules given by Theorem 2.3 and set $\mathcal{H}:=\mathcal{H}(P(S))$. By the structure of $P(S_{2})$ (see Theorem 2.3(b)), there exists a uniserial $kG$-module

[TABLE]

We may assume that $W\subseteq P(S)$ since ${\mathrm{soc}}(W)=S$, and hence $W\subseteq P(S)J$ since $S_{2}\,{\not\cong}\,S$. This implies that $W/S\subseteq P(S)J/S=\mathcal{H}$. Namely, there exists a $kG$-submodule $U$ of $\mathcal{H}$ with

[TABLE]

On the other hand, by the structure of $P(S_{n})$ (see Theorem 2.3(b)) there is a $kG$-epimorphism

[TABLE]

since $S_{n}\,{\not\cong}\,S$. Set $V:=\ker(\psi)$. Since the entry $(1,n)$ of the Cartan matrix is equal to one, then by definition of $\psi$, and by Theorem 2.3(b) we know that ${\mathrm{soc}}(U)\cap{\mathrm{soc}}(V)=\{0\}$, hence $U\cap V=\{0\}$. Thus we have a direct sum $U\oplus V\subseteq\mathcal{H}$. Then, by the definitions of $U$ and $V$ and by counting the number of composition factors of $\mathcal{H}$, we obtain $U\oplus V=\mathcal{H}$. Clearly $V\,{\not=}\,\{0\}$ by assumption. ∎

Finally given a normal subgroup $H\trianglelefteq G$ and $\tilde{b}$ a $p$-block of $H$, we will use the group $G[\tilde{b}]$ defined by Dade [Dad73].

Lemma 2.6** (Dade).**

Let $H\trianglelefteq G$ such that $p\,{\not|}\,|G/H|$, and let $P$ be a Sylow $p$-subgroup of $H$. Let $\tilde{b}:=B_{0}(kH)$ and $B:=B_{0}(kG)$ be the principal blocks of $kH$ and $kG$, respectively. Set $N:=H\,C_{G}(P)$. Then the following hold:

1. (a)

*The block $\tilde{b}$ is $G$-invariant. the Frattini argument, that is ”Sylow’s theorem”). * 2. (b)

$N=G[\tilde{b}]$* and $N\trianglelefteq G$.* 3. (c)

If $b$ denotes the principal block of $kN$, then $1_{B}=1_{b}$.

Proof.

(a) Obvious since $\tilde{b}$ is the principal block.

(b) This follows from [Dad73, Corollary 12.6] since $\tilde{b}$ is the principal block (see [Dad77, Proof of Lemma 3]). As $\tilde{b}$ is $G$-invariant, the fact that $G[\tilde{b}]\trianglelefteq G$ follows from [Dad73, Proposition 2.17].

(c) (see also [Kue95, p.303 line 10]) As $\tilde{b}$ is $G$-invariant, $1_{\tilde{b}}$ is an idempotent of $Z(kG)$ and we can write

[TABLE]

for an integer $n\geq 0$ and for distinct non-principal blocks $B_{1},\ldots,B_{n}$ of $kG$. Thus, $1_{\tilde{b}}\,1_{B}=1_{B}$. Namely,

[TABLE]

This implies $1_{B}\in Z(C)$ since $1_{B}\in Z(kG)$. Hence it follows from [Kue90, Corollary 4] and part (b) that

[TABLE]

Thus $1_{B}\in Z(kN)$. On the other hand, since $b$ is the principal block of $kN$, we have $1_{b}$ is $G$-invariant, so that $1_{b}\in Z(kG)$. Hence as above we can write

[TABLE]

for an integer $t\geq 0$ and for some distinct non-principal blocks $B^{\prime}_{1},\cdots,B^{\prime}_{t}$ of $kG$. Set $\tilde{e}:=1_{b}-1_{B}\in Z(kN)$ since $1_{B}\in Z(kN)$. Therefore $1_{b}=1_{B}+\tilde{e}$ is a decomposition of $1_{b}$ into orthogonal idempotents of $Z(kN)$. This implies that $\tilde{e}=0$, and hence $1_{b}=1_{B}$. ∎

3. Groups having a solvable quotient of $p^{\prime}$-order

Hypothesis 3.1**.**

Assume that:

1. (a)

$G$ is a finite group of order divisible by $p$ and $N\trianglelefteq G$ is a normal subgroup such that $|G/N|=:q$ is a prime number with $q\neq p$, and we set $G/N=:\langle gN\rangle$ for an element $g\in G\backslash N$. 2. (b)

$B$ and $b$ are wild blocks of $kG$ and $kN$ respectively such that $1_{B}=1_{b}$.

Lemma 3.2**.**

Assume Hypothesis 3.1 holds. Let $\zeta\in k^{\times}$ be a primitive $q$-th root of unity in $k$, and for each $1\leq j\leq q$ let $Z_{j}$ be the one-dimensional $k(G/N)$-module defined by $Z_{j}:=\langle\alpha_{j}\rangle_{k}$ and $\alpha_{j}{\cdot}gN:=\zeta^{j-1}\alpha_{j}$, so that in particular $Z_{1}=k_{G/N}$. The following holds:

1. (a)

If $\mathcal{S}\in{\mathrm{IBr}}(B)$ is such that $\mathcal{S}{\downarrow}_{N}$ is not simple, then for each $1\leq j\leq q$,

[TABLE]

as $kG$-modules, where we see $Z_{j}$ as a $kG$-module via inflation. 2. (b)

There are integers $m\geq 1$ and $\ell\geq 0$ such that

[TABLE]

where for each $1\leq i\leq m$ and each $1\leq j\leq q$,

[TABLE]

and for each $m+1\leq i\leq m+\ell$ and each $1\leq j\leq q$,

[TABLE]

where we may assume that $\mathcal{T}_{ij}:={\mathcal{T}_{i1}}^{g^{j-1}}$.

Moreover, we can assume that for each $1\leq j\leq q$,

[TABLE]

Proof.

(a) Let $1\leq j\leq q$. By assumption and Clifford’s theory we have that

[TABLE]

for some $\mathcal{T}\in{\mathrm{IBr}}(b)$. Hence $\mathcal{T}{\uparrow}^{G}\cong\mathcal{S}$, and $\mathcal{T}{\uparrow}^{G}\cong\mathcal{S}\otimes_{k}Z_{j}$ for each $1\leq j\leq q$.

(b) As by Hypothesis 3.1 the quotient $G/N$ is cyclic, the claim follows from the result of Schur-Clifford [NT88, Chap. 3 Corollary 5.9 and Problem 11(i)]. ∎

Lemma 3.3**.**

Assume Hypothesis 3.1 holds. Let $\mathcal{S}\in{\mathrm{IBr}}(B)$.

1. (a)

If $\mathcal{S}{\downarrow}_{N}=:\mathcal{T}$ is simple, then $P(\mathcal{S}){\downarrow}_{N}\cong P(\mathcal{T})$ and $\mathcal{H}(P(\mathcal{S})){\downarrow}_{N}\cong\mathcal{H}(P(\mathcal{T}))$. 2. (b)

If $\mathcal{S}{\downarrow}_{N}$ is not simple, then we can write $\mathcal{S}{\downarrow}_{N}=\mathcal{T}_{1}\oplus\mathcal{T}_{2}\oplus\cdots\oplus\mathcal{T}_{q}$ with $\mathcal{T}_{j}:={\mathcal{T}_{1}}^{g^{j-1}}$ for each $1\leq j\leq q$ and we have that

[TABLE]

Proof.

(a) Obviously

[TABLE]

Hence the top of $P(\mathcal{S}){\downarrow}_{N}$ is $\mathcal{T}$, which implies that $P(\mathcal{S}){\downarrow}_{N}\cong P(\mathcal{T})$. Therefore,

[TABLE]

(b) Similar to (a). ∎

Proposition 3.4**.**

Assume Hypothesis 3.1 holds. If every simple module $T\in{\mathrm{IBr}}(b)$ lies at the end of its AR-component, then every simple module $S\in{\mathrm{IBr}}(B)$ lies at the end of its AR-component.

Proof.

Let $S\in{\mathrm{IBr}}(B)$ be a simple module. First assume that $S{\downarrow}_{N}=:T\in{\mathrm{IBr}}(b)$ is simple. Then by Lemma 3.3(a)

[TABLE]

But by assumption $\mathcal{H}(P(T))$ is indecomposable, therefore so is $\mathcal{H}(P(S))$.

We assume now for the rest of the proof that $S{\downarrow}_{N}$ is not simple. If $S$ lies at the end of its AR-component, then there is nothing to do. Therefore we now also assume that $S$ lies on the $n$-th row from the bottom of $\Gamma_{s}(S)$ for an integer $n\geq 2$, minimal (as in Kawata’s Criterion on Cartan matrices). By Lemma 3.2(b),

[TABLE]

where $T_{1j}:={T_{11}}^{g^{j-1}}$ for $1\leq j\leq q$ are non-isomorphic simple modules in ${\mathrm{IBr}}(b)$. We also set $T_{1}:=T_{11}$.

Let $S_{2},\ldots,S_{n}$ be the simple modules given by Theorem 2.3.

Claim 1. If the modules $S_{2}{\downarrow}_{N},\ldots,S_{n}{\downarrow}_{N}$ are all non-simple, then we have a contradiction.

Proof of Claim 1. By assumption and Lemma 3.2, we can write

[TABLE]

For each $2\leq i\leq n$ we define $T_{i}\in{\mathrm{IBr}}(b)$ by $T_{ij}={T_{i}}^{g^{j-1}}$, where $1\leq j\leq q$. We claim that

[TABLE]

Indeed, we know by Theorem 2.3(b) and Lemma 3.3(b) that

[TABLE]

where the boxes mean the Loewy and socle series of the $kN$-modules. Since the left-hand-side is a direct sum of exactly $q$ indecomposable $kN$-modules that are $\langle g\rangle$-conjugate to each other, by interchanging the indices of $T_{3},\ldots,T_{n},T_{1}$, we may assume that the PIM $P(T_{2})$ has the desired structure. Then automatically the structures of $P(T_{3}),\ldots,P(T_{n})$ are as claimed.

Now, using a similar argument as above, we also obtain

[TABLE]

where the last equality holds by Corollary 2.5. Hence we have

[TABLE]

since $P(T_{2}),\ldots,P(T_{n})$ are uniserial by the above.

But we are assuming that $T_{2},\ldots,T_{n}$ lie at the end of their AR-components, so that $\mathcal{H}(P(T_{2})),\ldots,\mathcal{H}(P(T_{n}))$ are indecomposable. Therefore the right-hand side term in the later equation has exactly $q$ indecomposable direct summands. This implies that $V=\{0\}$, hence a contradiction.

Claim 2. If the modules $S_{2}{\downarrow}_{N},\ldots,S_{n}{\downarrow}_{N}$ are all simple, then we have a contradiction.

Proof of Claim 2. Set $T_{i}:=S_{i}{\downarrow}_{N}$ for $2\leq i\leq n$. We have

[TABLE]

By the assumption and Lemma 3.2, for each $2\leq i\leq n$ we can write ${T_{i}}\!\uparrow^{G}=S_{i1}\oplus\cdots\oplus S_{iq}$ with $S_{ij}:=S_{i1}\otimes_{k}Z_{j}$ for $1\leq j\leq q$. In particular $S_{i1}=S_{i}$ for each $2\leq i\leq n$. By Theorem 2.3(b)

[TABLE]

so Lemma 3.2(a) implies that

[TABLE]

This yields that there exists a $kG$-module

[TABLE]

with simple socle isomorphic to $S$. Therefore $W/S$ has a proper uniserial submodule

[TABLE]

Now by Corollary 2.5, $U|\mathcal{H}(P(S))$, so that by Lemma 3.2(a)

[TABLE]

for each $1\leq j\leq q$. Therefore $q=2$ since $\mathcal{H}(P(S))$ has exactly two non-projective indecomposable direct summands by the assumption that $S$ does not lie at the end of its AR-component. Notice that this already provides a contradiction in case the characteristic of $k$ is $2$, since we assume $q\neq p$. So we now assume that $p\geq 3$. Then, the Loewy and socle structures of PIMs $P(S)$, $P(S_{i})$ and $P(S_{i2})$ for $2\leq i\leq n$ are:

[TABLE]

Now considering the restrictions $S{\downarrow}_{N}$ and $S_{i}{\downarrow}_{N}$ for $2\leq i\leq n$, we obtain by Lemma 3.3 that the Loewy and socle structures of the PIMs $P(T_{1})$, $P(T_{1}^{g})$ and $P(T_{i})$ for each $2~{}\leq~{}i~{}\leq~{}n$ are

[TABLE]

since $T_{1}\,{\not\cong}\,{T_{1}}^{g}$. Now, as the dimension of any PIM for $kN$ is divisible by $|N|_{p}=:p^{a}$ for an integer $a\geq 1$, and since $\dim\,T_{1}=\dim\,{T_{1}}^{g}$, we have for each $2\leq i\leq n$

[TABLE]

so that

[TABLE]

This implies that

[TABLE]

since $p\,{\not=}\,2$ (since $q=2$). Thus, $\dim\,T_{i}\equiv 0$ (mod $p^{a}$) for any $1\leq i\leq n$. Now, looking at the composition factors of PIMs $P(T_{1}),P({T_{1}}^{g}),P(T_{2}),\ldots,P(T_{n})$, we know that ${\mathrm{IBr}}(b)=\{T_{1},{T_{1}}^{g},T_{2},\ldots,T_{n}\}$, which implies that $p^{a}\,|\,\dim\,\mathcal{T}$ for any $\mathcal{T}\in{\mathrm{IBr}}(b)$. Now it follows from Brauer’s result [NT88, Chap.3, Theorem 6.25] that there is a simple $\mathcal{T}\in{\mathrm{IBr}}(b)$ such that $\nu_{p}(\dim\,\mathcal{T})=a-d(b)$ (where $d(b)$ is the defect of $b$). Hence we have a contradiction since $b$ is a wild block, i.e. of positive defect.

Claim 3.

1. (a)

If there is an integer $2\leq m\leq n-1$ such that $S_{2}{\downarrow}_{N},\ldots,S_{m}{\downarrow}_{N}$ are not simple and $S_{m+1}{\downarrow}_{N}$ is simple, then we have a contradiction. 2. (b)

If there is an integer $2\leq m\leq n-1$ such that $S_{2}{\downarrow}_{N},\ldots,S_{m}{\downarrow}_{N}$ are simple and $S_{m+1}{\downarrow}_{N}$ is not simple, then we have a contradiction.

Proof of Claim 3. (a) Set $T_{m+1}:=S_{m+1}{\downarrow}_{N}$. By Lemma 3.2 there exists a simple module $T_{m}\in{\mathrm{IBr}}(b)$ with $S_{m}{\downarrow}_{N}=T_{m}\oplus{T_{m}}^{g}\oplus\cdots\oplus{T_{m}}^{g^{q-1}}$. Then, by Lemma 3.2,

[TABLE]

where $S_{m+1,j}:=S_{m+1}\otimes_{k}Z_{j}$ for each $1\leq j\leq q$ and $T_{m}{\uparrow}^{G}=S_{m}$. By the structure of $P(S)$, we have that $\operatorname{Ext}^{1}_{kG}(S_{m},S_{m+1})\neq 0$. Therefore by Eckmann- Shapiro’s Lemma we have that $\operatorname{Ext}^{1}_{kN}(T_{m},T_{m+1})\neq 0$. Thus there exists a $kN$-module with Loewy structure

[TABLE]

So it follows from Lemma (a) that

[TABLE]

where the right-hand side box is the Loewy and socle series. But $P(S_{m})$ is uniserial by Theorem 2.3(b), so applying again Lemma (a), we must have $q=1$, which contradicts the assumption that $q$ is prime.

(b) follows in a similar fashion using a dual argument.

Altogether, Claims 1-3 prove that the simple modules $S_{2},\ldots,S_{n}$ cannot exist, therefore $S$ must lie at the end of its AR-component. ∎

As a consequence of the above discussion we obtain Theorem B of the Introduction.

Proof of Theorem B.

Because $G/N$ is solvable of order prime to $p$, it follows by induction on $|G/N|$, that we may assume that $|G/N|$ is a prime distinct from $p$. Then Proposition 3.4 yields the result. ∎

4. The principal block of $O^{p^{\prime}}(G)$

From now on, we assume that $p\geq 3$ and $G$ is a finite group with non-cyclic abelian Sylow $p$-subgroups. Because we consider the principal block only, we assume that ${O_{p^{\prime}}(G)=1}$. The structure of $O^{p^{\prime}}(G)$ is given by the following well-known result of Fong-Harris [FH93].

Lemma 4.1** ([FH93, 5A–5C]).**

Let $p$ be an odd prime. Let $G$ be a finite group with a non-trivial abelian Sylow $p$-subgroup. Then

[TABLE]

where $m$ is a non-negative integer (i.e. possibly $O^{p^{\prime}}(G/O_{p^{\prime}}(G))\cong Q$), $Q$ is an abelian $p$-group, and for each $1\leq i\leq m$, $H_{i}$ is a non-abelian simple group with non-trivial Sylow $p$-subgroups.

Therefore, we fix the notation $O^{p^{\prime}}(G)=Q\times H_{1}\times\cdots\times H_{m}$, where $Q$ is an abelian $p$-group, and $H_{1},\ldots,H_{m}$ are non-abelian simple groups with non-trivial Sylow $p$-subgroups as given by the lemma.

4.1. Simple modules in infinite tubes ${\mathbb{Z}}A_{\infty}/\langle\tau^{a}\rangle$

Lemma 4.2** ([KMU00, Lemma 5.2] generalised version).**

Let $H=\tilde{H}_{1}\times\cdots\times\tilde{H}_{m}$ $(m\geq 1)$ be a finite group such that $p\mid|\tilde{H}_{i}|$ for each $1\leq i\leq m$. If $B_{0}(kH)$ is a wild block and contains a periodic simple module, then $m=1$.

Proof.

Let $S$ be a simple periodic $B_{0}(kH)$-module. Then we may write $S=S_{1}\otimes_{k}\cdots\otimes_{k}\,S_{m}$ where $S_{i}$ is a simple $B_{0}(k\tilde{H}_{i})$-module for each $1\leq i\leq m$. Then, by iterating [KMU00, Lemma 2.2], there exists an index $1\leq i_{o}\leq m$ such that $S_{i_{0}}$ is periodic and $S_{j}$ is a projective $k\tilde{H}_{j}$-module for each $1\leq j\neq i_{o}\leq m$. But $B_{0}(k\tilde{H}_{j})$ cannot contain a simple projective module, since we assume that $p\mid|\tilde{H}_{i}|$ for each $1\leq i\leq m$. Hence this forces $H=\tilde{H}_{i_{0}}$, i.e. $m=1$. ∎

As a consequence, the existence of simple periodic modules in the principal block lying in tubes drastically restricts the possible structure of $O^{p^{\prime}}(G)$.

Corollary 4.3**.**

If $B_{0}(kG)$ contains a periodic simple module, then $O^{p^{\prime}}(G)=H_{1}$ is a non-abelian finite simple group with non-cyclic abelian Sylow $p$-subgroups.

Proof.

By Lemma 4.2, either $O^{p^{\prime}}(G)=Q$ or $O^{p^{\prime}}(G)=H_{1}$. But the former cannot happen. Indeed, the indecomposable direct summands of the restriction to $O^{p^{\prime}}(G)$ of a simple periodic $kG$-module are all simple periodic modules, however the unique simple $kQ$-module is the trivial module, which is not periodic since we assume that $B_{0}(kG)$ is wild, and hence $Q$ is non-cyclic. The leaves only the possibility $O^{p^{\prime}}(G)=H_{1}$, and the $p$-rank of $H_{1}$ must be at least $2$ again because we assume that $B_{0}(kG)$ is wild. ∎

This immediately leads to the following reduction to non-abelian simple groups:

Corollary 4.4**.**

Assume that every periodic simple $B_{0}(kH)$-module lies at the end of its AR-component for every non-abelian finite simple group $H$ with non-cyclic abelian Sylow $p$-subgroups. Then every simple periodic $B_{0}(kO^{p^{\prime}}(G))$-module lies at the end of its AR-component for any finite group $G$ with $O_{p^{\prime}}(G)=1$ and non-cyclic abelian Sylow $p$-subgroups.

4.2. Simple modules in ${\mathbb{Z}}A_{\infty}$-components

Lemma 4.5**.**

Let $H=\tilde{H}_{1}\times\cdots\times\tilde{H}_{m}$ $(m\geq 1)$ be a finite group with abelian Sylow $p$-subgroups such that $p\mid|\tilde{H}_{i}|$ for each $1\leq i\leq m$. If $B_{0}(kH)$ is a wild block containing a non-periodic simple module $S$ not lying at the end of its AR-component, then $m=1$.

This lemma and its proof below generalises parts of the proof of [KMU00, Theorem 5(i)].

Proof.

Assume that $m\geq 2$. Then by Theorem 2.3(b), there exists a simple $B_{0}(kH)$-module $T$ lying at the end of $\Gamma_{s}(\Omega(S))$. By Knörr’s Theorem [Knö79, 3.7 Corollary], we know that the vertices of the simple modules in $B_{0}(kH)$ are the Sylow $p$-subgroups of $H$, because they are abelian. Now by assumption $\Gamma_{s}(S)\cong{\mathbb{Z}}A_{\infty}$, which implies that all the modules in $\Gamma_{s}(S)$ and $\Gamma_{s}(\Omega(S))$ have the Sylow $p$-subgroups as their vertices by [OU94, Theorem]. So all the modules in $\Gamma_{s}(S)$ and $\Gamma_{s}(\Omega(S))$ are not projective relatively to the subgroup $N:=\tilde{H}_{1}\times\cdots\times\tilde{H}_{m-1}$ as it does not contain a Sylow $p$-subgroup of $H$. Thus, as $p\neq 2$, all the simple direct summands of $S{\downarrow}_{N}$ belong to blocks of defect zero by [KMU00, Lemma 1.4]. But

[TABLE]

and there exist a simple $B_{0}(kN)$-module $S_{0}$ and a simple $B_{0}(k\tilde{H}_{m})$-module $S_{m}$ such that

[TABLE]

By the above, $S_{0}$ is a projective $kN$-module (indeed $S{\downarrow}_{N}=(\dim_{k}S_{m})S_{0}$), hence

[TABLE]

is projective relatively to $\tilde{H}_{m}$ and therefore so is $S$ seen as the above tensor product. This contradicts the fact that the vertices of $S$ are the Sylow $p$-subgroups of $H$. Hence we conclude that $S$ must lie at the end of $\Gamma_{s}(S)$. ∎

Proposition 4.6**.**

Let $G$ be a finite group with $O_{p^{\prime}}(G)=1$ and non-cyclic abelian Sylow $p$-subgroups. Assume moreover that one of Conditions (i), (ii), or (iii) of Theorem C is satisfied. Then every non-periodic simple $B_{0}(kO^{p^{\prime}}(G))$-module lies at the end of its AR-component.

Proof.

We have $O^{p^{\prime}}(G)=Q$ or $O^{p^{\prime}}(G)=Q\times H_{1}\times\cdots\times H_{m}$, where $Q$ is an abelian $p$-group and $H_{i}$ is a non-abelian finite simple group with non-trivial Sylow $p$-subgroups for each $1\leq i\leq m$.

If (i) holds, that is $Q\neq 1$, then by Theorem 2.2(a), all simple $B_{0}(kO^{p^{\prime}}(G))$-modules lie at the end of their AR-components. Therefore, we assume for the rest of the proof that $Q=1$.

Next if (ii) holds, that is $m\geq 2$, the claim follows from Lemma 4.5.

Finally if (iii) holds, that is $O^{p^{\prime}}(G)=H_{1}$, then $H_{1}$ must have a non-cyclic Sylow $p$-subgroup, therefore all simple $B_{0}(kO^{p^{\prime}}(G))$-modules lie at the end of their $AR$-components by assumption. ∎

5. Reduction to $O^{p^{\prime}}(G)$

We continue assuming that $G$ is a finite group with non-cyclic abelian Sylow $p$-subgroups such that ${O_{p^{\prime}}(G)=1}$, unless otherwise stated. We now prove that an answer to Question A is detected by restriction to the normal subgroup $O^{p^{\prime}}(G)$ of $G$.

We set $H:=O^{p^{\prime}}(G)$, let $P\in\operatorname{Syl}_{p}(H)$ be a Sylow $p$-subgroup, and set $N:=HC_{G}(P)$. Moreover we set $B:=B_{0}(kG)$, $b:=B_{0}(kN)$ and $\tilde{b}:=B_{0}(kH)$. Then $N$ is Dade’s Group $G[\tilde{b}]$ and $N\trianglelefteq G$, see Lemma 2.6.

First of all Question A has an affirmative answer for the group $N$ if and only if it has an affirmative answer for the group $H$.

Lemma 5.1**.**

With the above notation, every simple $b$-module lies at the end of its AR-component if and only if every simple $\tilde{b}$-module lies at the end of its AR-component.

Proof.

By the Alperin-Dade Theorem [Dad77, Theorem], the blocks $b$ and $\tilde{b}$ are isomorphic as $k$-algebras, hence Morita equivalent. But for a simple module, lying at the end of its AR-component is a property preserved by Morita equivalence. ∎

Proposition 5.2**.**

If every simple $\tilde{b}$-module lies at the end of its AR-component, then every simple $B$-module lies at the end of its AR-component.

Proof.

Let $S$ be a simple $B$-module and let $T$ be a simple direct summand of $S{\downarrow}_{H}$. Then $T$ is periodic if and only if $S$ is. Therefore $\Gamma_{s}(S)~{}\cong~{}{\mathbb{Z}}A_{\infty}$ if and only if $\Gamma_{s}(T)~{}\cong~{}{\mathbb{Z}}A_{\infty}$, and $\Gamma_{s}(S)$ is an infinite tube with tree class $A_{\infty}$ if and only if $\Gamma_{s}(T)$ is an infinite tube with tree class $A_{\infty}$.

In case $\Gamma_{s}(S)~{}\cong~{}{\mathbb{Z}}A_{\infty}$, then $S$ lies at the end of $\Gamma_{s}(S)$ if and only if $T$ lies at the end of $\Gamma_{s}(T)$ by [KMU00, Lemma 1.5].

In case $\Gamma_{s}(S)$ is an infinite tube with tree class $A_{\infty}$, then by Corollary 4.3, $H$ is a non-abelian finite simple group with non-cyclic abelian Sylow $p$-subgroups. Now, by Schreier’s conjecture (now proven by the Classification of Finite Simple Groups, see [GLS2, Definition 2.1] [GLS3, Theorem 7.1.1]), we know that $G/H$ is a solvable $p^{\prime}$-subgroup of $\operatorname{Out}(H)$. Now by Lemma 5.1, we may assume $H=N$ and by Lemma 2.6(c) we have $1_{B}=1_{b}$. Therefore Theorem B implies that $S$ lies at the end of $\Gamma_{s}(S)$ because every simple $b$-module lies at the end of its AR-component. ∎

As a corollary, we obtain Theorem C of the Introduction.

Proof of Theorem C.

Let $G$ be a finite group with non-cyclic abelian Sylow $p$-subgroups. As $B_{0}(kG)$ and $B_{0}(kG/O_{p^{\prime}}(G))$ are Morita equivalent, we may assume that $O_{p^{\prime}}(G)~{}=~{}1$. Therefore, by Proposition 5.2, every simple $B_{0}(kG)$-module lies at the end of its AR-component if every simple $kB_{0}(O^{p^{\prime}})(G)$-module lies at the end of its AR-component. Now if $B_{0}(G)$ contains a periodic simple module, then by Corollary 4.3 we must have that $O^{p^{\prime}}(G)=H_{1}$ is a non-abelian finite simple group with non-cyclic abelian Sylow $p$-subgroups, then the claim holds by Corollary 4.4. Therefore we may assume that $B_{0}(kG)$, and hence $B_{0}(kO^{p^{\prime}}(G))$, contains no periodic simple module. In this case, if one of Conditions (i),(ii), or (iii) holds, then the claim follows from Proposition 4.6. ∎

Now Corollary D is a direct consequence of Theorem C.

6. Principal $3$-blocks

We now fix $p:=3$, and continue assuming that $G$ is a finite group with non-cyclic Sylow $3$-subgroups, so that $B_{0}(kG)$ is wild. We may also assume that $O_{3^{\prime}}(G)=1$.

We start by investigating principal $3$-blocks of non-abelian finite simple groups with abelian defect group. To this aim, we recall that the list of non-abelian finite simple groups with abelian Sylow $3$-subgroups is known by the classification of finite simple groups and was determined by Paul Fong (in an unpublished manuscript).

Proposition 6.1** ([KY10, Proposition 4.3]).**

If $G$ is a non-abelian finite simple group with non-cyclic abelian Sylow $3$-subgroup, then $G$ is one of:

1. (i)

${\mathfrak{A}}_{7}$, ${\mathfrak{A}}_{8}$, $M_{11}$, $M_{22}$, $M_{23}$, $HS$, $O^{\prime}N$; 2. (ii)

$\operatorname{PSL}_{3}(q)$* for a prime power $q$ such that $3||(q-1)$;* 3. (iii)

$\operatorname{PSU}_{3}(q^{2})$* for a prime power $q$ such that $3||(q+1)$;* 4. (iv)

$\operatorname{PSp}_{4}(q)$* for a prime power $q$ such that $3|(q-1)$;* 5. (v)

$\operatorname{PSp}_{4}(q)$* for a prime power $q$ such that $q>2$ and $3|(q+1)$;* 6. (vi)

$\operatorname{PSL}_{4}(q)$* for a prime power $q$ such that $q>2$ and $3|(q+1)$;* 7. (vii)

$\operatorname{PSU}_{4}(q^{2})$* for a prime power $q$ such that $3|(q-1)$;* 8. (viii)

$\operatorname{PSL}_{5}(q)$* for a prime power $q$ such that $3|(q+1)$;* 9. (ix)

$\operatorname{PSU}_{5}(q^{2})$* for a prime power $q$ such that $3|(q-1)$; or* 10. (x)

$\operatorname{PSL}_{2}(3^{n})$* for an integer $n\geq 2$.*

As a consequence we obtain:

Proposition 6.2**.**

If $G$ is a non-abelian finite simple group with non-cyclic abelian Sylow $3$-subgroups, then every simple $B_{0}(kG)$-modules lies at the end of its component in $\Gamma_{s}(B_{0}(kG))$.

Proof.

Let $P\in\operatorname{Syl}_{3}(G)$, and set $N:=N_{G}(P)$ and $B_{0}:=B_{0}(kG)$. We go through the list of groups in Proposition 6.1.

In case (i), in all cases all simple $B_{0}$-modules lie at the end of their component in $\Gamma_{s}(B_{0})$ by Theorem 2.3(b): indeed if $G$ is one of ${\mathfrak{A}}_{8}$, $M_{22}$ or $O^{\prime}N$, then one checks from GAP [GAP13] that the Cartan matrix of $B_{0}$ has no diagonal entry equal to $2$. If $G$ is one of ${\mathfrak{A}}_{7}$, $M_{11}$, $M_{23}$, or $HS$, then one checks from GAP [GAP13] that the Cartan matrix of $B_{0}$ does not have the shape of Theorem 2.3(b) either.

In case (ii), then the Cartan matrix of $B_{0}$ is computed in [Kun00, Table 2] and does not satisfy Theorem 2.3(b).

Next if $G$ is one of the groups listed in Proposition 6.1(iii),(iv),(vii), or (ix), then it is proven in [KY10, Lemma 3.7] that $B_{0}$ is Puig equivalent to $B_{0}(kN)$. But $N$ has a non-trivial normal Sylow $3$-subgroup, therefore all simple $B_{0}(kN)$-modules lie at the end of their components in $\Gamma_{s}(B_{0}(kN))$ by Theorem 2.2(a), and therefore so do the simple $B_{0}$-modules via the latter Puig (Morita) equivalence.

In case (v), the decomposition numbers of $B_{0}$ were computed by White and Okuyama-Waki. If $q$ is even then we read from [Whi95, Table II] that each column of the decomposition matrix of $B_{0}$ has at least $3$ positive entries. If $q$ is odd, then the decomposition matrix of $B_{0}$ is given in [Whi90, Theorem 4.2] up to two parameters $\alpha$ and $\beta$. But [OW98, Theorem 2.3] proves that $\alpha\in\{1,2\}$. This is enough to see that each column of the decomposition matrix of $B_{0}$ has at least $3$ positive entries. Therefore in both cases all the diagonal entries of the Cartan matrix of $B_{0}$ are at least $3$.

In case (vi) and (viii), we proceed as follows. For $n\in\{4,5\}$ fixed, we may regard $B_{0}(k\operatorname{PSL}_{n}(q))$ as the principal block of $\operatorname{SL}_{n}(q)$ as $3\nmid|Z(\operatorname{SL}_{n}(q))|$. Then we check that the Cartan matrix of $B_{0}(k\operatorname{GL}_{n}(q))$ does not satisfy Theorem 2.3(b). To this end we use the information on the decomposition numbers of $B_{0}(k\operatorname{GL}_{n}(q))$ provided in [Jam90, Appendix I]. In both cases, it is enough to consider only the square submatrix $\Delta_{n,0}$ of the decomposition matrix of $B_{0}(k\operatorname{GL}_{n}(q))$ whose rows are indexed by the unipotent characters. Both in case $n=4$ and $n=5$, there are five modular characters in the principal block (using [FS82]) and

[TABLE]

(See e.g. [KM00, Proposition 3.1 and Proposition 4.1].) It follows that the Cartan integers of $B_{0}(\operatorname{GL}_{n}(q))$ have lower bounds given by the entries of the following matrices:

[TABLE]

Therefore the Cartan matrix of $B_{0}(k\operatorname{GL}_{n}(q))$ cannot satisfy Theorem 2.3(b), and we conclude that all simple $B_{0}(k\operatorname{GL}_{n}(q))$-modules lie at the end of their AR-components. Now, from the known values of the unipotent characters of $\operatorname{GL}_{n}(q)$, we easily check that the dimension of the simple modules in $B_{0}(k\operatorname{GL}_{n}(q))$ are prime to $3$, hence they cannot be periodic by [Car79], as $3^{(a-1)}$ must divide the dimension of any simple periodic module, where $a:=$ the $p$-rank of the group, but in our case $a\geq 2$. Therefore every simple $B_{0}(k\operatorname{SL}_{n}(q))$-module lies at the end of its AR-component by [KMU00, Lemma 1.5].

Finally, if $G=\operatorname{PSL}_{2}(3^{n})$ for some integer $n\geq 2$, then the claim follows from Theorem 2.2(c) as $G$ is a finite simple group of Lie type in defining characteristic. ∎

As a corollary we obtain Theorem E of the Introduction.

Proof of Theorem E.

The claim now follows from Corollary D together with Proposition 6.1. ∎

Acknowledgements. Both authors gratefully acknowledge financial support and the hospitality provided by the Centre Interfacultaire Bernoulli (CIB) of the École Polytechnique Fédérale de Lausanne during the writing period of this article. The first author is grateful to the hospitality of the Department of Mathematics in TU Kaiserslautern.