Vol. 8, No. 4, 2014

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Posets, tensor products and Schur positivity

Vyjayanthi Chari, Ghislain Fourier and Daisuke Sagaki

Vol. 8 (2014), No. 4, 933–961

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.

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