Degrees of stretched Kostka coefficients |
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Authors: | Tyrrell B McAllister |
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Institution: | (1) Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands |
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Abstract: | Given a partition λ and a composition β, the stretched Kostka coefficient
is the map n
↦
K
n
λ,n
β
sending each positive integer n to the Kostka coefficient indexed by n
λ and n
β. Kirillov and Reshetikhin (J. Soviet Math. 41(2), 925–955, 1988) have shown that stretched Kostka coefficients are polynomial functions of n. King, Tollu, and Toumazet have conjectured that these polynomials always have nonnegative coefficients (CRM Proc. Lecture
Notes 34, 99–112, 2004), and they have given a conjectural expression for their degrees (Séminaire Lotharingien de Combinatoire 54A, 2006).
We prove the values conjectured by King, Tollu, and Toumazet for the degrees of stretched Kostka coefficients. Our proof depends
upon the polyhedral geometry of Gelfand–Tsetlin polytopes and uses tilings of GT-patterns, a combinatorial structure introduced
in De Loera and McAllister, (Discret. Comput. Geom. 32(4), 459–470, 2004).
Research supported by NSF VIGRE Grant No. DMS-0135345 and by NWO Mathematics Cluster DIAMANT. |
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Keywords: | Kostka coefficient Representation theory Gelfand– Tsetlin polytope |
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