From Grassmann Necklaces to Restricted Permutations and Back Again

Abstract

We study the commutative algebras ZJK 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 Mm,n, GLn or SLn, then the algebras ZJK can always be constructed in terms of centres of localizations. The main purpose of the ZJK 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 × n matrices in terms of its restricted permutation.