# On the Existence of Generalized Parking Spaces for Complex Reflection Groups

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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 $\varphi_k$ given by $\varphi_k(w) = k^{\dim V^w}$ for $w \in W$, where $V^w$ is the fixed-point subspace of $w$. If $W$ is the symmetric group of $n$ letters and $k=n+1$, then $\varphi_{n+1}$ is the permutation character on (classical) parking functions. In this paper, we give a complete answer to the question when $\varphi_k$ (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.

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@article{Ito2015, title = {On the Existence of Generalized Parking Spaces for Complex Reflection Groups}, author = {Ito, Yosuke and Okada, Soichi}, url = {http://arxiv.org/abs/1508.06846 http://arxiv.org/pdf/1508.06846v1}, urldate = {2015-09-06}, abstract = {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 {\$}\varphi{\_}k{\$} given by {\$}\varphi{\_}k(w) = k^{\{}\dim V^w{\}}{\$} for {\$}w \in W{\$}, where {\$}V^w{\$} is the fixed-point subspace of {\$}w{\$}. If {\$}W{\$} is the symmetric group of {\$}n{\$} letters and {\$}k=n+1{\$}, then {\$}\varphi{\_}{\{}n+1{\}}{\$} is the permutation character on (classical) parking functions. In this paper, we give a complete answer to the question when {\$}\varphi{\_}k{\$} (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.}, comment = {}, month = {aug}, year = 2015, archivePrefix = {arXiv}, eprint = {1508.06846}, primaryClass = {math.CO}, collections = {Attention-grabbing titles,The groups group} }