Vol. 8, No. 4, 2014

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Posets, tensor products and Schur positivity

Vyjayanthi Chari, Ghislain Fourier and Daisuke Sagaki

Vol. 8 (2014), No. 4, 933–961
Abstract

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight λ, we define a preorder on the set P+(λ,k) of k-tuples of dominant weights which add up to λ. Let be the equivalence relation defined by the preorder and P+(λ,k) be the corresponding poset of equivalence classes. We show that if λ is a multiple of a fundamental weight (and k is general) or if k = 2 (and λ is general), then P+(λ,k) coincides with the set of Sk-orbits in P+(λ,k), where Sk acts on P+(λ,k) as the permutations of components. If g is of type An and k = 2, we show that the S2-orbit of the row shuffle defined by Fomin et al. (2005) is the unique maximal element in the poset.

Given an element of P+(λ,k), consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, λ, and k) the dimension of this tensor product increases along . We also show that in the case when λ is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A2 and k = 2 (λ is general), there exists an inclusion of tensor products along with the partial order on P+(λ,k) . In particular, if g is of type An, this means that the difference of the characters is Schur positive.

Keywords
Schur positivity, tensor products, posets, Lie algebras
Mathematical Subject Classification 2010
Primary: 17B67
Milestones
Received: 12 April 2013
Accepted: 15 August 2013
Published: 10 August 2014
Authors
Vyjayanthi Chari
Department of Mathematics
University of California
Riverside, CA 92521
United States
Ghislain Fourier
Mathematisches Institut
Universität zu Köln
Germany
Daisuke Sagaki
Department of Mathematics
University of Tsukuba
Japan