More tight monomials in quantized enveloping algebras
Let \(U=U_q(g)\) be the quantum group associated to a semisimple Lie algebra \(g\) of rank \(n\) by Drinfeld and Jimbo. The negative part \(U^-\) of \(U\) has a canonical basis \(B\) with some nice properties. To calculate the elements of \(B\) explicitly is a hard problem; it was solved completely only in types \(A_1\) and \(A_2\). In [Isr. Math. Conf. Proc. 7, 117-132 (1993)], \textit{G. Lusztig} describes a method to calculate certain elements of \(B\). Fixing an orientation of the Dynkin diagram of \(g\), Lusztig associates a quadratic form and shows that a monomial is tight (resp. semi-tight) provided this quadratic form satisfies a certain positivity condition (resp. nonnegativity condition) and that the positivity condition for tightness always holds in type \(A_3\), and asks when one has (semi-)tightness in type \(A_n\). In this paper, the author shows that for a certain orientation of the Dynkin diagram, the positivity condition is always satisfied in type \(A_4\). It is also demonstrated that Lusztig's comment that things get more complicated in cases with higher rank in type \(A\), is true. In particular, the author exhibits a reduced expression for \(w_0\) in type \(A_r\) \((r\geq 6)\) with a quadratic form that does not even satisfy the conditions for semi-tightness, for any orientation of the Dynkin diagram. In Section 2, the author describes the situation he is working in, and in Section 3, he summarizes the results about tight monomials from Lusztig's paper which are needed. In Section 4, the author shows how they can be applied to the type \(A_4\) case, and in Section 5, he explains how some techniques of linear programming providing spanning sets in a certain sense for cones can be applied to understanding the quadratic form. These techniques are applied to a tractable reduced expression in type \(A_5\). In Section 6, the counterexample mentioned above is given. In Section 7, the author gives a general description in type \(A_n\) of the spanning vectors used in Section 5, showing how they can be parametrized in a natural way by cycles in symmetric groups. These vectors seem to be interesting combinatorially and also should be of use in further understanding the quadratic form.
scientific article; zbMATH DE number 1218681