Doubly stochastic matrices and Schur-Weyl duality for partition algebras

Stephen Doty

Published 2021-08-31Version 1

In 1916 D\'{e}nes K\"{o}nig proved that the permanent of a doubly stochastic matrix is positive; equivalently, the matrix has a positive diagonal. K\"{o}nig's result is equivalent to Birkhoff's 1946 result, that the set of $n \times n$ doubly stochastic matrices is the convex hull of the $n \times n$ permutation matrices. We conjecture that K\"{o}nig's result extends to doubly stochastic matrices in the $\mathbb{R}$-span of $r$th tensor powers of $n \times n$ permutation matrices. The conjecture implies a simple new algorithmic proof that Schur--Weyl duality for partition algebras (originally proved over $\mathbb{C}$ by V.F.R. Jones) holds over any commutative ring.