Doubly stochastic matrices and Schur-Weyl duality for partition algebras

DOI10.37236/10831arXiv2109.00107MaRDI QIDQ6376518

S. R. Doty

Publication date: 31 August 2021

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 $nimesn$ 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.

Mathematics Subject Classification ID

Factorials, binomial coefficients, combinatorial functions (05A10) Combinatorial aspects of matrices (incidence, Hadamard, etc.) (05B20) Permutations, words, matrices (05A05) Hecke algebras and their representations (20C08) Representations of finite symmetric groups (20C30) Stochastic matrices (15B51) Symmetric groups (20B30) Combinatorial aspects of commutative algebra (05E40)

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