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2014 Posets, tensor products and Schur positivity
Vyjayanthi Chari, Ghislain Fourier, Daisuke Sagaki
Algebra Number Theory 8(4): 933-961 (2014). 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 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.

Citation

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Vyjayanthi Chari. Ghislain Fourier. Daisuke Sagaki. "Posets, tensor products and Schur positivity." Algebra Number Theory 8 (4) 933 - 961, 2014. https://doi.org/10.2140/ant.2014.8.933

Information

Received: 12 April 2013; Accepted: 15 August 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1320.17004
MathSciNet: MR3248990
Digital Object Identifier: 10.2140/ant.2014.8.933

Subjects:
Primary: 17B67

Rights: Copyright © 2014 Mathematical Sciences Publishers

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