Bases for coordinate rings of conjugacy classes of nilpotent matrices
Let \(\mathfrak g\) be the Lie algebra of \(\text{GL}_n(\mathbb{C})\), and \(U\) the subgroup of upper triangular matrices. In this work the authors first construct a basis \(\mathcal B\) of the space \(\mathbb{C}[{\mathfrak g}]^U\) of highest weight vectors in the coordinate ring \(\mathbb{C}[{\mathfrak g}]\). This involves bitableaux. Now let \(\mu\) be a partition of \(n\) whose transpose \(\mu^t\) lists the sizes of the Jordan blocks of a nilpotent \(n\) by \(n\) matrix. Let \(X_\mu\) be the closure of the conjugacy class of that matrix. They try to find a subset \(\mathcal B_\mu\) of \(\mathcal B\) which yields a basis for the space of highest weight vectors \(\mathbb{C}[X_\mu]^U\). For the largest nilpotent conjugacy class we have \(\mu=(1^n)\) and they find such a subset using Kostant's description in terms of invariants of the ring \(\mathbb{C}[X_{(1^n)}]\). Character theory and combinatorics then lead them to suggest a choice of \(\mathcal B_\mu\) for any \(\mu\). All this also relies on work of \textit{W. Hesselink} [Comment. Math. Helv. 54, 105-110 (1979; Zbl 0395.14014)], \textit{R. K. Gupta} [Bull. Am. Math. Soc., New Ser. 16, 287-291 (1987; Zbl 0648.22011)], \textit{A. Lascoux} [in Proceedings of the Hyderabad conference on algebraic groups held at the School of Mathematics and Computer/Information Sciences of the University of Hyderabad, India, December 1989. Madras: Manoj Prakashan. 323-347 (1991; Zbl 0823.20012)], \textit{A. Lascoux, M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. A 286, 323-324 (1978; Zbl 0374.20010)]. To describe their guess the authors introduce the catabolism type for certain column strict tableaux. When \(\mu\) is of hook shape or when \(\mu\) has two parts, they prove their choice is correct and they enlarge the basis of \(\mathbb{C}[X_\mu]^U\) to an explicit basis of \(\mathbb{C}[X_\mu]\). Further they show this basis works in any characteristic. That exploits the fact that \(\mathbb{Z}[X_\mu]\) is a \(\text{GL}_n\) module with good filtration, by \textit{S. Donkin} [Invent. Math. 101, No. 3, 717-736 (1990; Zbl 0822.20045)]. That helps both in the base change to other characteristic and in the enlargement of the basis. The authors have made an effort to explain all the ingredients.
scientific article; zbMATH DE number 1356291
scientific article