From Grassmann necklaces to restricted permutations and back again

TL;DR

This paper explores the structure of certain commutative algebras related to quantum coordinate rings, using combinatorial techniques like Grassmann necklaces and restricted permutations to understand their spectra.

Contribution

It introduces methods to construct the centers of localizations of these algebras using Grassmann necklaces and restricted permutations, linking combinatorics with algebraic structure.

Findings

Constructed denominator sets via two techniques, Grassmann necklaces and restricted permutations.

Established the equivalence of the two methods for constructing these sets.

Derived a formula relating Grassmann necklaces to restricted permutations for totally nonnegative matrices.

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 $M_{m,n}$, $GL_n$ or $SL_n$, 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.