# Tensor Powers of the Defining Representation of \$\$S_n\$\$ S n

Tensor Powers of the Defining Representation of \$\$S_n\$\$ S n We give a decomposition formula for tensor powers of the defining representation of \$\$S_n\$\$ S n and apply it to bound the mixing time of a Markov chain on \$\$S_n\$\$ S n . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

# Tensor Powers of the Defining Representation of \$\$S_n\$\$ S n

, Volume 30 (3) – Feb 18, 2016

## Tensor Powers of the Defining Representation of \$\$S_n\$\$ S n

J Theor Probab (2017) 30:1191–1199 DOI 10.1007/s10959-016-0673-9 Tensor Powers of the Deﬁning Representation of S Shanshan Ding Received: 21 August 2015 / Revised: 31 January 2016 / Published online: 18 February 2016 © Springer Science+Business Media New York 2016 Abstract We give a decomposition formula for tensor powers of the deﬁning repre- sentation of S and apply it to bound the mixing time of a Markov chain on S . n n Keywords Markov chain · Mixing time · Kronecker coefﬁcients Mathematics Subject Classiﬁcation (2010) 20C30 · 60J10 · 05E10 1 Introduction The deﬁning, or permutation, representation of S is the n-dimensional representation where 1 σ( j ) = i ((σ )) = (1.1) i, j 0 otherwise. Since the ﬁxed points of σ can be read off of the matrix diagonal, the character of  at σ , χ (σ ), is precisely the number of ﬁxed points of σ . The irreducible representations, or irreps for short, of S are parameterized by the partitions of n, and  decomposes as (n−1,1) (n) S ⊕ S . Note that χ (σ ) is one less than the number of ﬁxed points of σ . (n−1,1) (n−1,1) In the terminology of [7], we call the (n − 1)-dimensional irrep S the standard representation of S . A classic question in the representation theory of symmetric groups is how tensor products of representations decompose as direct sums of irreps. In Sect. 2 we will Shanshan Ding dish@sas.upenn.edu Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA 123 1192 J Theor Probab (2017) 30:1191–1199 present a neat formula for the decomposition of tensor powers of  and, as corollary, (n−1,1) that of tensor powers of S . Our study of tensor powers of  arose from an investigation in the mixing time of the Markov chain on S formed by applying a single uniformly chosen n-cycle to a deck of n cards and following up with repeated random...
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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-016-0673-9
Publisher site
See Article on Publisher Site

### Abstract

We give a decomposition formula for tensor powers of the defining representation of \$\$S_n\$\$ S n and apply it to bound the mixing time of a Markov chain on \$\$S_n\$\$ S n .

### Journal

Journal of Theoretical ProbabilitySpringer Journals

Published: Feb 18, 2016

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