Posets, tensor products and Schur positivity

DOI10.2140/ANT.2014.8.933zbMATH Open1320.17004arXiv1210.6184OpenAlexW2004327230MaRDI QIDQ399375

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Publication date: 19 August 2014

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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.

Full work available at URL: https://arxiv.org/abs/1210.6184

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