Doubly stochastic matrices and Schur-Weyl duality for partition algebras

TL;DR

This paper establishes a new basis for the centralizer algebra of the partition algebra using permutations with specific subsequence properties, linking combinatorics, matrix theory, and Schur-Weyl duality.

Contribution

It introduces a novel basis for the centralizer algebra of the partition algebra via permutations with increasing or decreasing subsequences, connecting to doubly stochastic matrices.

Findings

New basis for the centralizer algebra of the partition algebra

Connection between permutation properties and Kronecker powers

Results on the set of doubly stochastic matrices within the algebra

Abstract

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the linear span of that set. Thanks to a known Schur--Weyl duality, this gives a new basis for the centralizer algebra of the partition algebra acting on the $r$th tensor power of a vector space. We give some related results on the set of doubly stochastic matrices in that algebra.