Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

Abstract

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over Q.

A graph theoretic method for determining generating sets of prime ideals in quantum matrices

“…His work implies a correspondence between Cauchon diagrams and the collection of totally nonnegative matrices over R (that is, matrices all of whose minors are nonnegative). The connection between Postnikov's work and the ideal structure of A has recently been developed by Goodearl, Launois and Lenagan [7,8]. In view of this and the results of this paper, it is perhaps not surprising that Talaska [20] has independently been able to give an explicit description of the correspondence between Postnikov's L-diagram and totally-nonnegative-matrices using the classic version of Lindström's Lemma.…”

“…Launois [15] originally proved the following result for K = C, but by results of Goodearl, Launois and Lenagan [8] it suffices to set K to be any field of characteristic zero. Launois [14] has given an algorithm which takes as input the Cauchon diagram corresponding to an H-invariant prime ideal I, and outputs a matrix whose vanishing quantum minors correspond to quantum minors of X A which generate I.…”

“…Finally, we note that by a result of Launois [16] (see also [7,8]), the poset structure of H-spec(A) is order-isomorphic to a subset of (n + m)-permutations ordered with the Bruhat order. Yakimov [22] has recently developed an alternate approach to determine generators for H-primes based on such permutations.…”

From Grassmann Necklaces to Restricted Permutations and Back Again

“…We do this by constructing explicit finitely generated denominator sets for each pair of H-primes J K: this is the content of Theorem 1.1 below. We achieve this in the first instance by exploiting the connection between H-primes in quantum matrices and cells of totally nonnegative real matrices from [10,11]: for background and definitions, see §2. 3.…”

“…The structure of this paper is as follows. In §2 we introduce the required notation and background information, including details of the remarkable connection between the study of H-primes and total nonnegativity obtained in [10,11]. In §3 we construct our denominator sets E JK using Grassmann necklaces and the language of total nonnegativity, and hence prove Theorem 1.1.…”