Suppose I have a subgroup $H$ of $GL(V)$ such that $H$ acts irreducibly on all the exterior powers of $V$. Is there any sort of characterization of such things? (I am intentionally not specifying the coefficients, since the results are presumably depend on these).
$\begingroup$
$\endgroup$
5
asked Nov 19, 2013 at 16:37
-
$\begingroup$ Do you mean such that H acts irreducibly on all exterior powers of V? $\endgroup$– Noah SnyderCommented Nov 19, 2013 at 16:40
-
$\begingroup$ @NoahSnyder Yes, that's correct. $\endgroup$– Igor RivinCommented Nov 19, 2013 at 16:42
-
$\begingroup$ Related mathoverflow.net/questions/32401 $\endgroup$– David E SpeyerCommented Nov 19, 2013 at 18:11
-
3$\begingroup$ An example is $S^n$ acting on $n$-tuples of numbers that sum to $0$, over any characteristic zero field. $\endgroup$– David E SpeyerCommented Nov 19, 2013 at 18:17
-
$\begingroup$ @Igor: For an arbitrary subgroup $H$ I'm skeptical about finding a decent characterization, but maybe it's reasonable for more restrictive classes of groups. Certainly there are problems with some of the classical linear groups, but not others; so a characterization would have to be rather subtle even over a field of characteristic 0. $\endgroup$– Jim HumphreysCommented Nov 19, 2013 at 18:47
Add a comment
|