Vertices

We present some algorithmic approaches towards the determination of vertices of indecomposable and, in particular, simple modules for finite groups over fields of prime characteristic.

Theoretical Background

First of all, we provide the theoretical background needed for our computations. We suppose F to be a field of some prime characteristic p. Moreover, whenever G is some finite group, all FG-modules are understood to be finitely generated right modules for the group algebra FG.

Vertices and sources

In 1958/59 J.A. Green introduced the notion of vertices and sources. Given a finite group G, a subgroup H of G, an FG-module M is said to be relatively H-projective if M|IndHG (ResHG(M)). If the FG-module M is indecomposable and if P is a subgroup of G which is is minimal w.r.t the condition that M is relatively P-projective, then P is called a vertex of M.

The vertices of an indecomposable FG-module M are known to form a G-conjugacy class of p-subgroups of G. Given a vertex P of M, there is an indecomposable FP-module L such that M|IndHG(M). This module L has also vertex P, and is called a source of M. It is determined up to isomorphism and conjugation with elements in NG(P). For further details, see [G].

Simple modules

Now consider a simple FG-module D. The vertices of D have some properties wich are specific to simple modules and are not shared by arbitrary indecomposable FG-modules. Two of the most powerful results in this context are the following:

Theorem (Knörr 1979). Let P be a vertex of D. Then there is a defect group R of the block of FG containing D such that
CR(P)≤P≤R.

Theorem (Erdmann 1977). If D has a cyclic vertex P then P is a defect group of the block of FG containing D.

In consequence of these theorems, a simple FG-module belonging to a block with abelian defect groups has exactly these defect groups as its vertices, and a simple FG-module belonging to a block with non-cyclic defect groups cannot have cyclic vertices. These facts play an important role for our computations.

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Algorithms

Testing for relative projectivity

Suppose we are given a finite group G, a subgroup H of G and an FG-module M. In order to test whether M is relatively H-projective, we applied Higman's criterion:

Theorem. The FG-module M is relatively H-projective if and only if there is some f ∈ EndFH(M) such that
TrHG(f):= fg1+fg2+...+fgn=idM.

Here g1,...,gn are representatives for the right cosets of H in G.

The advantage of Higman's criterion for computational purposes is the transitivity of the relative trace map. Namely, for subgroups K ≤ H≤ G, we have TrKG = TrHG o TrKH.

Hence we may choose an appropriate chain of subgroups H=H0< H1< ... < Hk=G and successively compute TrHk-1Hk o ... o TrH0H1. This requires a time propotional to |Hk:Hk-1|+ ... + |H1:H0|, whereas the naive algorithm for computing TrHG requires a time proportional to |G:H|.

Suppose now that M is indecomposable of dimension d. Then its endomorphism algebra E:=EndFG is local. Since T:=TrHG (EndFH(M)) is ideal in E, M is relatively H-projective if and only if T⊄ J(E). Equivalently, if rk(TrHG(fi)=d, for some 1≤ i ≤ m, where f1 ,..., fm form an F-basis for EndFH(M).

The algorithms for carrying out the actual vertex computation had been developed by René Zimmermann in his PhD thesis [Z]. The MAGMA source code of his implementation is given here.

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Cyclic vertices

Consider a finite group G and an indecomposable FG-module M. Suppose that P is a p-subgroup of G containing some vertex Q of M. Set N:=ResPG(M). As far as the vertex computation is concerned, we may ignore all projective direct summands of N. In order to avoid the expensive computation of an explicit indecomposable direct sum decomposition of N, we applied the algorithm `ProjSummands' which splits indecomposable projective direct summands off N without computing a direct sum decomposition first. This algorithm is a Monte Carlo algorithm in that might not detect all indecomposable projective direct summands of N. However, its probability of success is greater than 99.9%.

Next suppose that M is simple. In consequence of Knörr's Theorem mentioned above, we may suppose further that M belongs to a block B of FG with non-abelian defect groups. Then, in particular, B does not have cyclic defect groups, and thus M does not have cyclic vertices, by Erdmann's Theorem. Therefore, we may also ingore all those indecomposable direct summands of N:=ResPG(M) which have cyclic vertices. The algorithm `CyclicVertexSummands' enables us to split such indecomposable direct summands off N. As the previous algorithm `ProjSummands', we need not compute an explicit indecomposable direct sum decomposition of N. Also `CyclicVertexSummands' is a Monte Carlo algorithm, since it may miss indecomposable direct summands of N with cyclic vertices. However, setting the bound b to 10, we achieve a probability of success greater than 99.9%.

For precise descriptions of both algorithms see [DKZ], Sec. 3. The MAGMA source codes of our implementations are given here:

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References

  • [DKZ] S. Danz, B. Külshammer, R. Zimmermann, On vertices of simple modules for symmetric groups of small degrees, J. Algebra 320 (2008), 680-707

  • [G] J.A. Green, On the indecomposable representations of a finite group, Math. Z. 70 (1959), 430-445

  • [Z] R. Zimmermann, Vertizes einfacher Moduln Symmetrischer Gruppen, PhD thesis (German), University of Jena, Jena, 2004