PBW bases of \(q\)-Schur algebras (Q2822019)

Let \(\mathbf{U}(n, r)\) denote the \(q\)-Schur algebra over \(\mathbb Q(v)\), where \(v\) is an indeterminate, introduced by \textit{R. Dipper} and \textit{G. James} [Proc. Lond. Math. Soc. (3) 59, No. 1, 23--50 (1989; Zbl 0711.20007); Trans. Am. Math. Soc. 327, No. 1, 251--282 (1991; Zbl 0798.20009)] and \textit{M. Jimbo} [Lett. Math. Phys. 11, 247--252 (1986; Zbl 0602.17005)]. Let also \(\mathbf{U}_{\mathcal Z}(n, r)\) denote the integral \(q\)-Schur algebra, where \(\mathcal Z = \mathbb Z[v, v^{-1}]\).NEWLINENEWLINEIn their paper [Int. Math. Res. Not. 2002, No. 36, 1907--1944 (2002; Zbl 1059.16015)] \textit{S. Doty} and \textit{A. Giaquinto} constructed a certain set \(\mathcal P_{i_0}\), \(1\leq i_0 \leq n\), as a truncated form of Lusztig's analogue for \(\mathbf{U}_{\mathcal Z}\) (\(\mathcal Z\)-form of the quantized enveloping algebra \(\mathbf{U} = \mathbf{U}(\mathfrak{gl}_n)\) over \(\mathbb Q(v)\)) of Kostant's basis for \(U_{\mathbb Z}\) (\(\mathbb Z\)-form of the universal enveloping algebra \(U = U(\mathfrak{gl}_n)\) ). They conjectured [loc. cit.] that the set \(\mathcal P_{i_0}\) is a \(\mathcal Z\)-basis of the integral \(q\)-Schur algebra.NEWLINENEWLINEIn the paper under review the authors construct new monomial bases for \(\mathbf{U}(n, r)\) and show that \(\mathcal P_{i_0}\) is indeed a \(\mathbb Q(v)\)-basis of \(\mathbf{U}(n, r)\). They also show by an example (with \(n = 3\), \(r = 2\) and \(i_0 = 3\)) that \(\mathcal P_{i_0}\) is not a \(\mathcal Z\)-basis of \(\mathbf{U}_{\mathcal Z}(n, r)\) in general.