On a theorem of L. Mirsky on even doubly-stochastic matrices (Q1061816)

Let \(A_ n\) denote the alternating group of degree n. An \(n\times n\) doubly stochastic matrix is said to be even if it can be written as a convex combination of even permutation matrices. \textit{L. Mirsky} [Math. Ann. 144, 418-421 (1961; Zbl 0101.255)] showed that the condition that \(\sum^{n}_{i=1}a_{i\sigma (i)}-3a_{j\sigma (j)}\leq n-3\) holds for all \(\sigma \in A_ n\) and all \(j=1,...,n\), is necessary for an \(n\times n\) doubly stochastic matrix A to be even. In this paper it is shown to be sufficient only if \(n\leq 3\).