From parking functions to Gelfand pairs

Aker, Kürşat; Can, Mahi̇r

doi:10.1090/S0002-9939-2011-11010-4

From parking functions to Gelfand pairs
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by Kürşat Aker and Mahi̇r Bi̇len Can PDF

Proc. Amer. Math. Soc. 140 (2012), 1113-1124 Request permission

Abstract:

A pair $(G,K)$ of a group and its subgroup is called a Gelfand pair if the induced trivial representation of $K$ on $G$ is multiplicity free. Let $(a_j)$ be a sequence of positive integers of length $n$, and let $(b_i)$ be its non-decreasing rearrangement. The sequence $(a_i)$ is called a parking function of length $n$ if $b_i \leq i$ for all $i=1,\dots ,n$. In this paper we study certain Gelfand pairs in relation with parking functions. In particular, we find explicit descriptions of the decomposition of the associated induced trivial representations into irreducibles. We obtain and study a new $q$-analogue of the Catalan numbers $\frac {1}{n+1}{ 2n \choose n }$, $n\geq 1$.

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