2023/10/17 by Burcroff, Amanda
#05E10 #52B05 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2310.11437
Kostka coefficients appear in the representation theory of the general linear group and enumerate semistandard Young tableaux of fixed shape and content. The r-Kostka cone is the real polyhedral cone generated by pairs of partitions with at most r parts, written as non-increasing r-tuples, such that the corresponding Kostka coefficient is nonzero. We provide several results showing that its faces have interesting structural and enumerative properties. We show that the d-faces of the r-Kostka cone can be determined from those of the (3d+3)-Kostka cone, allowing us to characterize its 2-faces and enumerate its d-faces for d ≤ 4. We provide tight asymptotics for the number of d-faces for arbitrary d and determine the maximum number of extremal rays contained in a d-face for d < r. We then make progress towards a generalization of the Gao-Kiers-Orelowitz-Yong Width Bound on initial entries of partitions (λ,μ) appearing in the Hilbert basis of the λ1-Kostka cone. We show that at least 93.7% of integer pairs λ1 ≥ μ1 > 0 appear as the initial entries of partitions (λ,μ) comprising a Hilbert basis element of the r-Kostka cone for every r > λ1. We conclude with a conjecture about a curious h-vector phenomenon.