Vertex and Source of an Indecomposable Module#

For a finite group $G$ and a finite field $K$, a vertex of an indecomposable $K[G]$-modules $M$ is a subgroup $V$ such that there exists an indecomposable $K[V]$-module $S$ such that $M$ is a direct summand of the induced module $S^G$, and $V$ is minimal subject to this condition. The module $S$ in the above is called a source for $M$. A vertex is defined up to conjugacy in $G$, and, for fixed vertex $V$, a source is defined up to conjugacy in the normalizer of $V$. The projective indecomposable modules of $G$ are exactly those with vertex the trivial subgroup. If $K$ has characteristic $p$, then a vertex is a $p$-subgroup of $G$.

These concepts were introduced by J.A. Green in [Green, 1958/59]. The algorithms used by Magma follow those given in [Danz et al., 2008].

Vertex(M : parameters): ModGrp -> Grp#

H: Grp                    Default: 

Let $G$ be a finite group, $K$ be a finite field and $M$ an indecomposable $K[G]$-module. This intrinsic returns a representative $V$ of the conjugacy class of vertices for $M$, where $V$ a subgroup of $G$.

The parameter H may be used to give a subgroup of $G$ that contains the vertex of $M$, such as a defect group for the block of $M$. By default the value of $H$ is set to a Sylow $p$-subgroup of $G$.

Source(M : parameters): ModGrp -> ModGrp, ModGrp#

H: Grp                    Default: 

Let $G$ be a finite group, $K$ be a finite field and $M$ an indecomposable $K[G]$-module. This intrinsic returns a source module $S$ for $M$. The action group for $S$ is a vertex for $M$. The Green correspondent of $M$ is the second return value. The parameter $H$ is same as for the Vertex intrinsic above, and is passed to it.

Example: Vertices and Sources in $M_{22}.2$ (ex-5a661a)#

We compute the vertices and sources of the absolutely irreducible 2-modular $G$-modules for the group $M_{22}.2$. We can compare the results with those of [Danz and Külshammer, 2009].

> G := PermutationGroup("M22d2",1);
> irrs := AbsolutelyIrreducibleModules(G, GF(2));
> irrs;
[
  GModule of dimension 1 over GF(2),
  GModule of dimension 10 over GF(2),
  GModule of dimension 10 over GF(2),
  GModule of dimension 34 over GF(2),
  GModule of dimension 98 over GF(2),
  GModule of dimension 140 over GF(2)
]
> time verts := [Vertex(m):  m in irrs];
Time: 2.330
> [#v : v in verts];
[ 256, 256, 256, 256, 256, 128 ]

Run in calculator

The orders of the vertices tally with those given in [Danz and Külshammer, 2009]. A Sylow 2-subgroup of $G$ has order 256, and a vertex for the 140 dimensional module is a Sylow subgroup of the socle $M_{22}$. This time we work out the sources.

> time srcs := [Source(m) : m in irrs];
Time: 2.690
> srcs;
[
  GModule of dimension 1 over GF(2),
  GModule of dimension 10 over GF(2),
  GModule of dimension 10 over GF(2),
  GModule of dimension 34 over GF(2),
  GModule of dimension 98 over GF(2),
  GModule of dimension 140 over GF(2)
]

Run in calculator

The article cited gives these results except for the last, where the source is stated to have dimension 70. The article assumes an algebraically closed field however, which is not the case with $GF(2)$.

> m := irrs[#irrs];
> m4 := ChangeRing(m, GF(4));
> time Source(m4);
GModule of dimension 70 over GF(2^2)
Time: 6.590

Run in calculator

This resolves our differences: though the irreducible module over $GF(2)$ is absolutely irreducible, the restriction to a vertex is indecomposable, but not absolutely indecomposable, and a field extension of degree 2 yields an absolutely indecomposable source module.