Finite Groups – Decomposing Representations of Finite Groups

finite-groupsgr.group-theorymodular-representation-theoryrt.representation-theory

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.

Must there be $G$-invariant, proper subspaces $U,W \leq V$ such that $U + W = V$?

I do not require the sum to be direct. The question should be equivalent to asking:

Must $V$ have a nontrivial decomposable image?

I am able to prove this if $p$ does not divide $|G|$ by applying Maschke's theorem to decompose $V$ into a direct sum of finite-dimensional irreducible subrepresentations. Even if in general the answer is negative, I would like to know about additional cases in which the conclusion holds, to say:

Under what conditions on $G$ can we find such subspaces?

Interesting cases can be abelian, solvable or any other "nice" classes of groups.

Best Answer

I am not sure that I follow Rickard's argument, but here is a direct proof. Given an infinite dimensional module $V$ (over an arbitrary field.) for a finite group $G$, I want to argue first that there exists a proper $G$-submodule $U$ with finite codimension. Let $X < V$ be an arbitrary proper subspace with finite codimension. Then $U = \bigcap_{g \in G} X^g$ is $G$-invariant and is an intersection of finitely many proper subspaces with finite codimension, so is proper and has finite codimension. Next, let $Y \subseteq V$ be a finite dimensional subspace such that $U + Y = V$. Let $W = \sum_{g \in G} Y^g$. Then $W$ is $G$-invariant and finite dimensional, so $W < V$. Also, $U + W = V$, as wanted.

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[Math] How does one compute induced representations for modular representations

Frobenius reciprocity for Brauer characters is a little more complicated (a lot more complicated if you don't have complete tables). You need the projective characters to compute the multiplicities. I don't use anything special about the character being induced, though sometimes you can leverage that information (especially if you don't have complete tables).

In GAP, this is easily done:

g:=CharacterTable("ON");
# Let g be the O'Nan simple group
chi:=Sum([1..10],i->Random(Irr(g mod 2)));;
# chi is a random reducible Brauer character
ipr:=Irr(g)*DecompositionMatrix(g mod 2);;
# ipr is the list of projective Brauer characters
vec:=MatScalarProducts(ipr,[InducedClassFunction(chi,g)])[1];
# vec is the multiplicity of each Brauer character in chi
chi = vec * Irr( g mod 2 );
# should be true: we have decomposed it correctly.

To get the decomposition matrix, you must not only have the Brauer table but also the ordinary table. You don't need much from the subgroup H, just a character to induce and the element fusion from H into G.

This is theorem 2.13 on page 25 of Navarro's textbook on Characters and Blocks of Finite Groups.

By projective, I mean in the module theory sense, an indecomposable direct summand of the regular representation, usually these are denoted Φ_i corresponding to a Brauer character φ_i. I don't mean representation into projective groups, like PGL.

Finite Groups with No Irreducible Representations Other Than Characters

A group satisfies this property if and only if it is an extension of a $p$ group by an abelian group. The reason is the following: Assume that indeed all the irreducible representations of $G$ are one dimensional. This means that all elements of the form $x-1$ where $x$ belongs to $[G,G]$ are in the Jacobson radical, because they act trivially on all irreducible representations. But since this is a finite dimensional algebra over a field, the Jacobson radical is nilpotent. This means that the element $x-1$ is nilpotent for every $x\in [G,G]$, and this can only happen if the order of $x$ is a power of $p$. Therefore, $[G,G]$ is a $p$-group, and the quotient $G/[G,G]$ is abelian. In particular, this implies that the group is solvable.

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[Math] How does one compute induced representations for modular representations

Finite Groups with No Irreducible Representations Other Than Characters

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