Tensor Powers of the Defining Representation of
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 , 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 firstname.lastname@example.org 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...