Quantum Analogues of Schubert Varieties in the Grassmannian

Abstract

Abstract. We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.2000 Mathematics Subject Classification. 16W35; 16P40; 16S38; 17B37; 20G42.Introduction. Since the appearance of quantum groups in the eighties, there have been several attempts to define quantum analogues of coordinate rings of grassmannian varieties and, more generally, of flag varieties. Here, we are interested in such deformations for grassmannian varieties and we follow the approach of Lakshmibai and Reshetikhin (see [8]). Hence, we start with the (usual) quantum deformation, denoted by O q (M m,n (k)), of the coordinate ring of m × n matrices. Then, denoting by G m,n (k) the grassmanian of m-dimensional subspaces in k n , the quantum deformation of the coordinate ring of G m,n (k) that we consider is the k-subalgebra of O q (M m,n (k)) generated by the maximal quantum minors. We denote this algebra by O q (G m,n (k)) and call it the quantum grassmannian for simplicity. Precise definitions are recalled in Section 1.Our main interest, in this paper, is the study of a family of quotients of O q (G m,n (k)) that appear in [8] and are natural quantum analogues of coordinate rings of Schubert varieties in G m,n (k). These quantum Schubert varieties have already been studied, to some extent, in [10]. There, they were used as a central tool to show that O q (G m,n (k)) is a quantum graded algebra with a straightening law. Details on the notion of quantum graded algebra with a straightening law (quantum graded A.S.L. for short) can be

Quantum analogues of Richardson varieties in the grassmannian and their toric degeneration

“…It is the subalgebra of O q (M v,u (k)) generated by the u × u quantum minors of O q (M v,u (k)). Notice that we adopt here a convention exchanging rows and columns with respect to the convention of [LR1] and [LR2]. However, embedding all the relevant algebras in O q (M v (k)) and using the transpose automorphism introduced above shows that the two different conventions lead to isomorphic algebras.…”

“…Namely, the fact that quantum analogues of Richardson varieties degenerate to quantum analogues of toric varieties. (A much more elementary proof of the fact that quantum analogues of Schubert varities are integral domains is given in [LR2]. )…”

Prime ideals in the quantum grassmannian

“…Most proofs will be omitted since these results already appear in [12,15,16]. Appropriate references will be given in the text.…”

“…The quantum Schubert variety S(γ) associated to γ is S(γ) := O q (G m,n (k))/ Π γ m,n . (Note that S(γ) was denoted by O q (G m,n (k)) γ in [16].) This definition is inspired by the classical description of the coordinate rings of Schubert varieties in the grassmannian.…”

“…Subsequently, some of their properties have been established in [16]. In this paper, we define quantum Schubert cells as noncommutative dehomogenisations of quantum Schubert varieties.…”