Lenagan, T. H.; Rigal, L. Quantum analogues of Schubert varieties in the Grassmannian. (English) Zbl 1134.16018 Glasg. Math. J. 50, No. 1, 55-70 (2008). 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. Reviewer: Vyacheslav A. Artamonov (Moskva) Cited in 6 Documents 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) Keywords:rings arising in quantum group theory; generic coordinate algebras; quantum Grassmannians; quantum minors; quantum Schubert varieties; normal domains Citations:Zbl 1108.16026 PDFBibTeX XMLCite \textit{T. H. Lenagan} and \textit{L. Rigal}, Glasg. Math. J. 50, No. 1, 55--70 (2008; Zbl 1134.16018) Full Text: DOI arXiv References: [1] DOI: 10.1017/S0013091502000809 · Zbl 1050.16014 · doi:10.1017/S0013091502000809 [2] Lakshmibai, Special functions pp 149– (1990) [3] DOI: 10.1007/BF02101594 · Zbl 0829.15024 · doi:10.1007/BF02101594 [4] DOI: 10.1142/S0219498804000630 · Zbl 1081.16048 · doi:10.1142/S0219498804000630 [5] DOI: 10.1006/aima.1993.1065 · Zbl 0793.05143 · doi:10.1006/aima.1993.1065 [6] DOI: 10.1016/j.jalgebra.2005.10.021 · Zbl 1108.16026 · doi:10.1016/j.jalgebra.2005.10.021 [7] DOI: 10.1006/jabr.1996.6885 · Zbl 0937.16049 · doi:10.1006/jabr.1996.6885 [8] DOI: 10.1006/aima.1999.1897 · Zbl 0959.16008 · doi:10.1006/aima.1999.1897 [9] Rigal, Proc. Edinburgh Math. Soc 42 pp 621– (1999) [10] Parshall, Mem. Amer. Math. Soc. 89 pp 439– (1991) [11] Gonciulea, Actualit?s Math?matiques (2001) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.