Posets, tensor products and Schur positivity

Chari, V., Fourier, G. and Sagaki, D. (2014) Posets, tensor products and Schur positivity. Algebra and Number Theory, 8(4), pp. 933-961. (doi: 10.2140/ant.2014.8.933)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Fourier, Dr Ghislain
• Authors: Chari, V., Fourier, G., and Sagaki, D.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Algebra and Number Theory
• Publisher: Mathematical Sciences Publishers
• ISSN: 1937-0652
• ISSN (Online): 1944-7833