From Parking Functions to Gelfand Pairs

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 calle...

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Bibliographic Details
Published in:arXiv.org 2010-09
Main Authors: Aker, Kürşat, Mahir Bilen Can
Format: Article
Language:English
Subjects:
Integers
Parking
Representations
Subgroups
Online Access:Get full text
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Summary: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 \(\frac{1}{n+1}{2n \choose n}\), \(n\geq 1\).
ISSN:2331-8422