From Parking Functions to Gelfand Pairs

2010/02/08 by Kürşat Aker, Aker, Kürşat, Mahir Bilen Can +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT

paper · pdf · doi:10.48550/arxiv.1002.1519

14 pages, 1 figure

arxiv created 2010/09/25 · arxiv updated 2010/09/28

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 ≤ 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 (1)/(n+1)2n \choose n, n≥ 1.

Related

• Rational parking functions and Catalan numbers
• Characterization of the Gelfand pair: Quasi Gelfand pair
• Gelfand pairs admit an Iwasawa decomposition
• Evidence for parking conjectures
• Finite Gelfand pairs and cracking points of the symmetric groups