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

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...
Loading next page...
 
/lp/springer_journal/tensor-powers-of-the-defining-representation-of-s-n-s-n-vtntrvdDd0
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off