From Parking Functions to Gelfand Pairs

TL;DR

This paper explores the connection between parking functions and Gelfand pairs, providing explicit decompositions of induced representations and introducing a new q-analogue of Catalan numbers.

Contribution

It establishes a novel link between parking functions and Gelfand pairs, including explicit representation decompositions and a new q-analogue of Catalan numbers.

Findings

Explicit decomposition of induced trivial representations into irreducibles.

Introduction of a new q-analogue of Catalan numbers.

Enhanced understanding of the algebraic structure related to parking functions.

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 $i=1,\...,n$. 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 $\frac{1}{n+1}{2n \choose n}$, $n\geq 1$.