Supersolvability in Representation Theory

Supersolvable lattices, introduced by R. Stanley and advanced by P. McNamara, are precisely those posets
that admit a good 0-Hecke algebra action on their maximal chains. In this talk, we present our
results on supersolvability of certain lattices that arise naturally in the context of representation theory.
More precisely, suppose we have an irreducible representation V of a semisimple algebraic group G.
We are interested in the following affine algebraic variety M=Zariski closure of G in End(V).
It is shown by Mohan Putcha that the GxG orbits in M are parametrized by a sublattice L
of the face lattice of a polyhedral cone arising from the embedding of a maximal torus of G in M.
We determine irreducible V for which the lattice L is supersolvable.