Doubly stochastic matrices and Schur-Weyl duality for partition algebras (Q2112559)

Summary: We prove that the permutations of \(\{1,\ldots, n\}\) 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.