The Combinatorics of Modules Supported in the Closure of a Nilpotent Conjugacy Class

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9800941 Shimozono The proposed research project is a combinatorial and algebraic study of the category of finitely generated graded modules over the coordinate ring of the affine space of n by n complex matrices, which are supported in the closure of a nilpotent conjugacy class and afford a compatible action of the general linear group. The Grothendieck group of this category is generated by a distinguished family of modules, whose isotypic components have Poincare polynomials with marvelous combinatorial properties. This subject provides a beautiful example of the interplay among classical subjects such as the geometry of conjugacy classes, characteristic-free invariant theory, and the combinatorics of Young tableaux, and the modern representation theory of quantum groups and two-dimensional solvable lattice models in statistical mechanics. This research is in the area of Algebraic Combinatorics, which is a discipline within mathematics which uses discrete structures and algorithms to analyze algebraic systems. It has recently enjoyed rapid growth and unprecedented success, since symbolic computation can often be employed to help solve its problems. This particular proposal has connections with physics, more specifically, with statistical mechanics.