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Chari, Vyjayanthi, Fourier, Ghislain and Sagaki, Daisuke (2014). Posets, tensor products and Schur positivity. Algebr. Number Theory, 8 (4). S. 933 - 962. BERKELEY: MATHEMATICAL SCIENCE PUBL. ISSN 1944-7833

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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 similar to be the equivalence relation defined by the preorder and P+ (lambda, k)/similar to be the corresponding poset of equivalence classes. 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)/similar to 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. (2005) 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 <=. 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 along with the partial order <= on P+ (lambda, k)/similar to. In particular, if g is of type A(n), this means that the difference of the characters is Schur positive.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Chari, Vyjayanthi UNSPECIFIED UNSPECIFIED UNSPECIFIED Fourier, Ghislain UNSPECIFIED UNSPECIFIED UNSPECIFIED Sagaki, Daisuke UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-449341
DOI:	10.2140/ant.2014.8.933
Journal or Publication Title:	Algebr. Number Theory
Volume:	8
Number:	4
Page Range:	S. 933 - 962
Date:	2014
Publisher:	MATHEMATICAL SCIENCE PUBL
Place of Publication:	BERKELEY
ISSN:	1944-7833
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language LITTLEWOOD-RICHARDSON COEFFICIENTS; CLASSICAL LIE-ALGEBRAS; LOG-CONCAVITY; REPRESENTATIONS; CHARACTERS; RULE Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/44934