A note on vertices of indecomposable tensor products

TL;DR

This paper provides a sufficient criterion for when the vertices of two indecomposable modules over a finite group algebra generate a Sylow p-subgroup, extending Navarro's earlier result for simple modules.

Contribution

It generalizes Navarro's result by establishing a criterion applicable to indecomposable modules beyond simple modules over finite p-solvable groups.

Findings

Provides a sufficient condition for vertices to generate a Sylow p-subgroup

Extends Navarro's result to a broader class of modules

Offers insights into the structure of indecomposable modules

Abstract

G. Navarro raised the question under what circumstancs two vertices of two indecomposable modules over a finite group algebra generate a Sylow $p$-subgroup. The present note provides a sufficient criterion for when this is the case. This generalises a result by Navarro for simple modules over finite $p$-solvable groups, which is the main motivation for this note.

Equations

$$|G:Q\cap R{|}_p={\mathrm{rk}}_{\mathcal{O}}(W{)}_p={\mathrm{rk}}_{\mathcal{O}}(U{)}_p\cdot {\mathrm{rk}}_{\mathcal{O}}(V{)}_p\ge |G:Q{|}_p\cdot |G:R{|}_p.$$

$$|Q:Q\cap R|\ge |G:R{|}_p=|G{|}_p/|R|.$$

$$|G:Q{|}_p={\mathrm{rk}}_{\mathcal{O}}(U{)}_p,$$

$$|G:R{|}_p={\mathrm{rk}}_{\mathcal{O}}(V{)}_p,$$

$$|G:S{|}_p={\mathrm{rk}}_{\mathcal{O}}(W{)}_p.$$

$${\dim }_k(M_U{)}_p=|N_G(Q,X)/Q{|}_p,$$

$${\dim }_k(M_V{)}_p=|N_G(R,Y)/R{|}_p,$$

$${\dim }_k(M_W{)}_p=|N_G(S,Z)/S{|}_p.$$

$${\dim }_k(M_W{)}_p=|N_G(S,Z)/S{|}_p$$

Full text

Let $p$ be a prime and ${\mathcal{O}}$ a complete local principal ideal domain with residue field $k$ of characteristic $p$. We allow the case ${\mathcal{O}}=k$, unless stated otherwise. We assume that $k$ is large enough for the modules and their sources that appear in this note to be absolutely indecomposable; the point of this hypothesis is that it ensures that indecomposable modules over finite group algebras have multiplicity modules (see [9, §5.7] for background material). If the field of fractions $K$ of ${\mathcal{O}}$ has characteristic zero, then we also assume that $K$ is large enough so that the irreducible characters over $K$ that arise below are absolutely irreducible. In some of the results in the literature cited below, $k$ is algebraically closed for convenience, but it is easy to see that ‘large enough’ in the sense above will do for the quoted results. Modules are finitely generated left modules.

The proof of Theorem 1 yields some further technical information regarding which intersections of vertices of $U$ and $V$ do yield vertices of $W$ (cf. Remark 9), and some information regarding the $p$-parts of the ranks of $U$, $V$, $W$ and the dimensions of their multiplicity modules (cf. Remark 10). The proof shows also that the hypothesis on the simplicity of the multiplicity module of $W$ can be replaced by a slightly weaker condition on the dimension of the multiplicity module of $W$ (cf. Remark 11). See Külshammer [8, Proposition 2.1] for a sufficient criterion for when the tensor product of two modules is indecomposable. We note some immediate consequences of Theorem 1. By a result of Knörr (in the proof of [7, Proposition 3.1], also described in [9, Corollary 5.7.9]), simple $kG$-modules and ${\mathcal{O}G}$-lattices with irreducible characters have simple multiplicity modules. Thus Theorem 1 implies the following two results.

By a result of Puig in unpublished notes from 1988 (see e. g. [15, Theorem (30.5)] or [9, Theorem 10.6.8]), if $G$ is a finite $p$-solvable group, then simple $kG$-modules have endopermutation sources; in particular, their sources have dimensions prime to $p$. Thus Corollary 2 implies the following result due to Navarro (and this is the main motivation for this note).

For the proof of Theorem 1, we collect a few elementary observations on vertices and sources which imply that the first two statements in Theorem 1 hold under weaker hypotheses. We refer to [9, §5.1] for definitions and basic properties of Green’s theory of vertices and sources of modules from [6]; these - as well as a number of arguments in this note - depend on the fact, used without further mention, that the Krull-Schmidt theorem holds in the context of the present note. The following is well-known.

Let $G$ be a finite group and let $U$, $V$ be indecomposable ${\mathcal{O}}$-free ${\mathcal{O}G}$-modules, and let $Q$, $R$ be vertices of $U$, $V$, respectively. Then for every indecomposable direct summand $W$ of $U\otimes_{\mathcal{O}}V$ there is $x\in$ $G$ such that $Q\cap{{}^{x}R}$ contains a vertex of $W$.

1. (i)

For any $x\in$ $G$ there exists an indecomposable direct summand $W$ of $U\otimes_{\mathcal{O}}V$ such that $Q\cap{{}^{x}R}$ is contained in a vertex of $W$. 2. (ii)

If $x\in G$ is chosen such that $Q\cap{{}^{x}R}$ has maximal order amongst the subgroups of the form $Q\cap{{}^{y}R}$, where $y\in G$, then $U\otimes_{\mathcal{O}}V$ has an indecomposable direct summand $W$ with vertex $Q\cap{{}^{x}R}$ and with sources of ${\mathcal{O}}$-rank prime to $p$. 3. (iii)

Suppose that $U\otimes_{\mathcal{O}}V$ is indecomposable. Then the sources of $U\otimes_{\mathcal{O}}V$ have ${\mathcal{O}}$-rank prime to $p$, and for any $x\in$ $G$, the subgroup $Q\cap{{}^{x}R}$ is contained in a vertex of $U\otimes_{\mathcal{O}}V$. If $Q\cap{{}^{x}R}$ has maximal order amongst all subgroups of this form, then $Q\cap{{}^{x}R}$ is a vertex of $U\otimes_{\mathcal{O}}V$.

For $m$ a positive integer, we denote by $m_{p}$ the highest power of $p$ which divides $m$, and we denote by $\mathrm{rk}_{\mathcal{O}}(U)$ the ${\mathcal{O}}$-rank of a free ${\mathcal{O}}$-module $U$.

See e. g. [9, Theorem 5.12.13] or [12, Ch. 7, Theorem 7.5] for proofs of this Lemma. The key ingredient for the proof of Theorem 1 is the following result due to Knörr, which is a criterion for when the inequality in Lemma 7 is an equality.

The statement in [7, Theorem 4.5] does not mention multiplicity modules explicitly, but the hypotheses in the theorem ensure the simplicity of multiplicity modules, and this is all that is used in the proof of [7, Theorem 4.5]. See [14, §9] for more general background material on multiplicity modules and characterisations of simple multiplicity modules. A description of some of this material closer to the terminology used above is given in [9, Theorems 5.7.7, 5.12.15].

Since $W$ has a simple multiplicity module, it follows from Theorem 8 and Lemma 7 that we have

[TABLE]

The left side is also equal to $|G:Q|_{p}\cdot|Q:Q\cap R|$, hence cancelling $|G:Q|_{p}$ yields

[TABLE]

Now $|Q:Q\cap R|=$ $|QR|/|R|$, so together this yields $|QR|\geq$ $|G|_{p}$.

Let $P$ be a Sylow $p$-subgroup of $G$ containing $Q$. Let $x\in$ $G$ such that ${{}^{x}{R}}\subseteq$ $P$. It follows from Lemma 6 (ii) that $Q\cap{{}^{x}{R}}$ is contained in a vertex of $W$. In particular, we have $|Q\cap{{}^{x}{R}}|\leq$ $|Q\cap R|$. Thus $|Q({{}^{x}R})|\geq|QR|\geq$ $|P|$. But since both $Q$, ${{}^{x}R}$ are contained in $P$, we also have $|Q({{}^{x}R})|\leq|P|$. Thus all inequalities in this proof are equalities. This forces $Q({{}^{x}R})=P$ and $|Q\cap{{}^{x}R}|=$ $|Q\cap R|$, so $Q$ and ${{}^{x}R}$ (instead of $R$) satisfy all conclusions. ∎

[TABLE]

[TABLE]

[TABLE]

Moreover, by [9, Theorem 5.12.15] the associated multiplicity modules $M_{U}$, $M_{V}$, $M_{W}$ of $U$, $V$, $W$, respectively, satisfy

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for the multiplicity module $M_{W}$ of $W$ associated with a vertexs-souce pair $(S,Z)$ of $W$ (this is the last equality in Remark 10). By a result of Knörr [7, Proposition 4.2], this equality holds if $M_{W}$ is simple (essentially because $M_{W}$ is then a projective simple module of a finite central $p^{\prime}$-extension of $N_{G}(S,Z)/S$), but the simplicity of $M_{W}$ is not necessary in general for this equality. In particular, even if $M_{W}$ is simple, we do not know whether this implies that $M_{U}$, $M_{V}$ are necessarily simple as well.