Posets, Tensor Products and Schur positivity

TL;DR

This paper introduces a new preorder on tuples of dominant weights related to Lie algebras, demonstrating how tensor product dimensions and characters behave along this order, with special cases showing Schur positivity.

Contribution

It defines a novel preorder on weight tuples, characterizes its structure in special cases, and links it to tensor product dimensions and Schur positivity for Lie algebra modules.

Findings

The poset of weight tuples coincides with S_k-orbits in specific cases.

Tensor product dimensions increase along the partial order.

Differences of characters are Schur positive in certain cases.

Abstract

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then $P(\lambda, k)/\sim$ coincides with the set of S_k-orbits in $P(\lambda,k)$, where S_k acts on $P(\lambda, k)$ as the permutations of components. If g is of type A_n and k=2, we show that the S_2-orbit of the row shuffle defined by Fomin et al is the unique maximal element in the poset. Given an element of $P(\lambda, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, \lambda, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when \lambda is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (\lambda is general), there exists an inclusion of tensor products of g-modules along with the partial order. In particular, if g is of type A_n, this means that the difference of the characters is Schur positive.