On Stanley's Inequalities for Character Multiplicities

TL;DR

This paper explores inequalities related to the number of orbits of automorphism groups acting on ranked posets, connecting classical inequalities to prime divisors of the group order and providing stronger bounds.

Contribution

It generalizes Stanley's inequalities for character multiplicities, introducing new bounds based on prime divisors of the automorphism group's order.

Findings

New inequalities depend on prime divisors of the group order.

These inequalities often provide stronger bounds than classical ones.

Connection established between orbit counts and representation theory.

Abstract

Let G be a group of automorphisms of a ranked poset Q and let N_{k} denote the number of orbits on the elements of rank k in Q. What can be said about the N_{k} for standard posets, such as finite projective spaces or the Boolean lattice? We discuss the connection of this question to the representation theory of the group, and in particular to the inequalities of Livingstone-Wagner and Stanley. We show that these are special cases of more general inequalities which depend on the prime divisors of the group order. The new inequalities often yield stronger bounds depending on the order of the group.