Cells in certain sets of matrices
It is well-known that for an associative algebra \({\mathcal A}\) over a field \(K\) and a basis \({\mathcal B}\) for \({\mathcal A}\) as a \(K\)-vector space, \({\mathcal B}\) is divided into cells [\textit{G. Lusztig}, Introduction to quantum groups, Birkhäuser, Boston (1993; Zbl 0788.17010)]. In certain nice circumstances, cells are important in the study of representation theory and can be classified combinatorially. In this paper, the author considers two more examples in the case of type \(A\), namely, when \({\mathcal A}\) is a \(q\)-Schur algebra \({\mathcal S}_q (n,r)\) or a modified quantized enveloping algebra \(\dot U\) of type \(A\) and \({\mathcal B}\) is the \(K\)-\(L\) basis of \({\mathcal S}_q (n,r)\) or the canonical basis of \(\dot U\). These bases are indexed by certain sets of matrices. So the cell decomposition of \({\mathcal B}\) induces a cell decomposition of these matrix sets. The author gives a combinatorial description for those two-sided cells. In Section 1, a result of \textit{Curtis Greene} [J. Comb. Theory, Ser. A 20, 69-79 (1979; Zbl 0323.06002)] is first generalized, which associates partitions to finite posets, to a result on a certain matrix semigroup \(M(n)\), namely one associates partitions to the matrices in \(M(n)\). Thus, \(M(n)\) is decomposed into subsets in terms of partitions. The main result is Theorem 2.1, which shows that these subsets are two-sided cells of the \(K\)-\(L\) bases of \(q\)-Schur algebras. This theorem is proved in Section 3. In Section 4 the author applies the result to modified quantum groups \(\dot U\) of type \(A\).
scientific article; zbMATH DE number 1044120
scientific article