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Quantum analogues of Schubert varieties in the Grassmannian. (English) Zbl 1134.16018

Let \(\mathcal O_q(M_{m,n}(k))\) be a generic coordinate \(k\)-algebra on rectangular matrices of size \(m\times n\). It is assumed that \(m\leqslant n\). If \(I=\{i_1<\cdots<i_u\}\), \(K=\{k_1<\cdots<k_v\}\subset M=\{1,\dots,m\}\) and \(J=\{j_1<\cdots<j_u\}\), \(L=\{l_1<\cdots<l_v\}\subset\{1,\dots,n\}\), then \((I,J)\leqslant_{st}(K,L)\) if and only if \(u\leqslant v\) and \(i_s\leqslant k_s\), \(j_s\leqslant l_s\) for all possible indices \(s\). Denote by \(\Pi_{m,n}\) the set of all pairs of indices \((I,J)\) in which \(u=m\).
The quantum Grassmannian \(\mathcal O_q(G_{m,n}(k))\) is the subalgebra in \(\mathcal O_q(M_{m,n}(k))\) generated by all \(m\times m\) quantum minors. In the authors’ previous paper [J. Algebra 301, No. 2, 670-702 (2006; Zbl 1108.16026)], it is shown that there is a vector basis in \(\mathcal O_q(G_{m,n}(k))\) consisting of monomials \([I_1\mid M]\cdots [I_t\mid M]\) such that \((I_1,M)\leqslant_{st}\cdots\leqslant_{st}(I_t,M)\).
Take \(\gamma\in\Pi_{m,n}\) and denote by \(\Pi_{m,n}^\gamma\) the set of all \(\alpha\in\Pi_{m,n}\) such that \(\alpha\ngeqslant_{st}\gamma\). The quantum Schubert variety \(\mathcal O_q(G_{m,n}(k))_\gamma\) associated with \(\gamma\) is the algebra \(\mathcal O_q(G_{m,n}(k))\) factorized by the ideal generated by \(\Pi_{m,n}^\gamma\). It is shown that \[ \mathcal O_q(G_{m,n}(k))[Y^{\pm 1};\varphi]\simeq\mathcal O_q(G_{m,n}(k))_\gamma[\gamma^{-1}]. \] It is given a criterion under which a quantum Schubert variety has left and right finite injective dimensions. Let \(I_t\) be the ideal in \(\mathcal O_q(M_{m,n}(k))\) generated by all minors of a fixed size \(t\). It is shown that \(\mathcal O_q(M_{m,n}(k))/I_t\) is a normal domain.

MSC:

16W35 Ring-theoretic aspects of quantum groups (MSC2000)
14M15 Grassmannians, Schubert varieties, flag manifolds
16U30 Divisibility, noncommutative UFDs
16S38 Rings arising from noncommutative algebraic geometry
16P40 Noetherian rings and modules (associative rings and algebras)

Citations:

Zbl 1108.16026
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References:

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