Commutation relations among quantum minors in \({\mathcal O}_q(M_n(k))\).
Let \({\mathcal O}_q(M_n(k))\) be the coordinate algebra of \((n\times n)\)-matrices with the standard set of generators \(a_{ij}\), \(1\leqslant i,j\leqslant n\). Choose two subsets of indices \(I=\{i_1,\dots,i_r\}\), \(J=\{j_1,\dots,j_r\}\), where \(1\leqslant i_1<\cdots<i_r\leqslant n\), \(1\leqslant j_1<\cdots<j_r\leqslant n\). Then the quantum row minor \([I,J]\) is defined as \[ [I,J]=\sum_{\sigma\in S_r} (-q)^{l(\sigma)}a_{i_1\sigma(j_1)}\cdots a_{i_r\sigma(j_r)}. \] The aim of the paper is to find presentations of the product \([I',J'][I,J]\) of minors as a linear combination of products of two minors which are less than \([I',J'],[I,J]\) with respect to some order.
scientific article; zbMATH DE number 2136741