Some remarks on varieties of pairs of commuting upper triangular matrices and an interpretation of commuting varieties

上三角矩阵的交换概形与表示同调

Commuting Schemes of Upper Triangular Matrices and Representation Homology

本研究建立了与上三角幂幺矩阵群概形$U_n$相关的高亏格广义交换概形$C_g(U_n)$和黎曼曲面的$U_n$系数表示同调的联系，并给出了判定$C(U_n)$是否为完全交的数值准则。

Some schemes related to the commuting variety

The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX. We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal}, which we prove to be a reduced complete intersection, one component of which is the commuting variety. (We conjecture there to be only one other component.) The diagonal commutator scheme has a flat degeneration to the scheme {(X,Y) : XY lower triangular, YX upper triangular}, which is again a reduced complete intersection, this time with n! components (one for each permutation). The degrees of these components give interesting invariants of pe…

Commuting matrices and volumes of linear stacks

A conjecture by Higman asserts that the number of conjugacy classes in the unipotent group of upper triangular matrices over a finite field depends polynomially on the number of elements of the field. We will study several alternative counting problems arising from quiver representations and prove explicit formulas relating the corresponding invariants to the invariants of Higman's conjecture. To do this, we develop a general framework of linear stacks over small etale sites and study volumes of these stacks and of their substacks of absolutely indecomposable objects.

Commuting difference operators and the combinatorial Gale transform

We study the spectral theory of $n$-periodic strictly triangular difference operators $L=T^{-k-1}+\sum_{j=1}^k a_i^j T^{-j}$ and the spectral theory of the "superperiodic" operators for which all solutions of the equation $(L+1)ψ=0$ are (anti)periodic. We show that for a superperiodic operator $L$ there exists a unique superperiodic operator ${\cal L}$ of order $(n-k-1)$ which commutes with $L$ and show that the duality $L\leftrightarrow {\cal L}$ coincides up to a certain involution with the combinatorial Gale transform recently introduced in [21].

Irregular locus of the commuting variety of reductive symmetric Lie algebras and rigid pairs

The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain som…

Isospectral commuting variety, the Harish-Chandra D-module, and principal nilpotent pairs

Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We relate gr(M), an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g x g x h x h which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of X_norm, the normalization of the isospectral commuting variety. It follows, thanks to the theory of Hodge modules, that th…