The Zrank Conjecture and Restricted Cauchy Matrices

Abstract

The rank of a skew partition $\lambda /\mu$, denoted $rank(\lambda /\mu )$, is the smallest number $r$ such that $\lambda /\mu$ is a disjoint union of $r$ border strips. Let $s_{\lambda /\mu }(1^t)$ denote the skew Schur function $s_{\lambda /\mu }$ evaluated at $x_1=...=x_t=1,x_i=0$ for $i>t$. The zrank of $\lambda /\mu$, denoted $zrank(\lambda /\mu )$, is the exponent of the largest power of $t$ dividing $s_{\lambda /\mu }(1^t)$. Stanley conjectured that $rank(\lambda /\mu )=zrank(\lambda /\mu )$. We show the equivalence between the validity of the zrank conjecture and the nonsingularity of restricted Cauchy matrices. In support of Stanley's conjecture we give affirmative answers for some special cases.

17pages, 6 figures