2015/08/27 by Yosuke Ito, Ito, Yosuke, Soichi Okada +1
Computer Science · Mathematics · #05E10 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1508.06846
openalex publication_date 2015/08/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let W be an irreducible finite complex reflection group acting on a complex vector space V. For a positive integer k, we consider a class function φk given by φk(w) = kdim Vw for w ∈ W, where Vw is the fixed-point subspace of w. If W is the symmetric group of n letters and k=n+1, then φn+1 is the permutation character on (classical) parking functions. In this paper, we give a complete answer to the question when φk (resp. its q-analogue) is the character of a representation (resp. the graded character of a graded representation) of W. As a key to the proof in the symmetric group case, we find the greatest common divisors of specialized Schur functions. And we propose a unimodality conjecture of the coefficients of certain quotients of principally specialized Schur functions.