2
$\begingroup$

An isotypic (maybe reducible) representation V of GL(V) may be represented by its highest weight subspace HW(V). We have dim HW(V) equal to the multiplicity of the irreducible representation inside V and we can identify GL(V) morphisms between two such components with (linear) morphisms between highest weight spaces.

I would like to ask if there is a similar theory for other (e.g. finite) groups. More precisely, is there a nice way to choose a subspace of a representation with the two properties above.

(Of course one can do it naively by choosing explicitely a decomposition. Then choose one vector in the irreducible component and the corresponding vectors (unique up to scalar by Schur lemma) in other components - the vector space spanned by all these vectors will have the desired property. However, I would like to know if there is a better way - an analogue of highest weight vectors. Of course some choices have to be made, but maybe they are more natural then choosing a decomposition and a vector inside, as this is in fact choosing a basis, not just a vector subspace)

$\endgroup$
3
  • 1
    $\begingroup$ An obstruction to a satisfying theory of weights for finite groups is the following. In the case of, say, compact connected Lie groups $G$, we need to pick a maximal torus $T$ to get things started. One reason looking at the action of $T$ works out is that the conjugates of $T$ cover $G$, so in particular the restriction to $T$ completely determines the character of a representation. But the conjugates of a proper subgroup of a finite group can never cover it (exercise). $\endgroup$ Sep 6, 2014 at 4:25
  • 1
    $\begingroup$ The standard answer to your question in the finite case is Deligne-Lusztig theory: en.wikipedia.org/wiki/Deligne-Lusztig_theory Naturally, this does not cover all finite groups, but it does work for a large and interesting class, including 16 of the 18 infinite families of finite simple groups. $\endgroup$
    – S. Carnahan
    Sep 6, 2014 at 4:37
  • $\begingroup$ The question implicitly limits attention to fields of characteristic 0, which is only part of the story for finite groups. There probably is no satisfactory general answer to the question (even for finite groups), but keep in mind Alperin's "weight" conjecture over a field of characteristic dividing the finite group order. This is inspired by highest weight theory for Lie groups (in characteristic 0), which has a precise analogue for finite groups of Lie type in the defining characteristic $p>0$. $\endgroup$ Sep 6, 2014 at 15:05

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.