From Grassmann necklaces to restricted permutations and back again

Abstract

We study the commutative algebras $Z_{JK}$ appearing in Brown and Goodearl's extension of the $\mathcal{H}$-stratification framework, and show that if $A$ is the single parameter quantized coordinate ring of $M_{m,n}$, $G{L}_n$ or $S{L}_n$, then the algebras $Z_{JK}$ can always be constructed in terms of centres of localizations. The main purpose of the $Z_{JK}$ is to study the structure of the topological space $spec(A)$, which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real $m\times n$ matrices in terms of its restricted permutation.

Same results, different order: many of the proofs are unchanged, but the exposition has been overhauled and the Grassmann necklaces now take centre stage. Updated to include references to arXiv:1602.05052