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 permutations.
The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this paper we generalise the commuting variety by using the commuting distance of matrices. We show that over an algebraically closed field, each of our sets does indeed form a variety. We compute the dimension of the distance-$2$ commuting variety and characterize its irreducible components. We also work over other fields, showing that the distance-$2$ commuting…
The commuting variety of a reductive Lie algebra ${\goth g}$ is the underlying variety of a well defined subscheme of $\gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities.
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…
We observe that the degree of the commuting variety and other related varieties occur as coefficients in the leading eigenvector of an integrable loop model based on the Brauer algebra.
We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. We ask for similar criteria in the realm of tropical linear algebra, giving conditions for two tropical matrices that are polytropes to commute. From the algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimension 2 and 3.
We study certain "σ-commuting varieties" associated with a pair of commuting involutions of a semisimple Lie algebra $\g$. The usual commuting variety of $\g$ and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of σ-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the σ-commuting variety is isomorphic to the commuting variety of a simple Jordan algebra. As a by-product of our theory, we show that if $J$ is the Jordan algebra of…
Let $X$ be an affine algebraic variety endowed with an action of complexity one of an algebraic torus $\mathbb{T}$. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions $\mathbb{K}[X]$ can be described in terms of proper polyhedral divisors corresponding to $\mathbb{T}$-variety $X$. We prove that homogeneous locally nilpotent derivations commute if an only if some combinatorial criterion holds. These results are used to describe actions of unipotent groups of dimension two on affine $\mathbb{T}$-varieties.
We study the anti-commuting variety which consists of pairs of anti-commuting $n\times n$ matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT quotient of the anti-commuting variety with respect to the conjugation action of $GL_n$ is shown to be of pure dimension $n$. We also show the semi-nilpotent anti-commuting variety (in which one matrix is required to be nilpotent) is of pure dimension $n^2$ and describe its irreducible components.
The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the…
It is known that the variety of pairs of n x n commuting upper triangular matrices isn't a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n > m. We also show that m < 18 and that it could be found by determining the dimension of the variety of pairs of commuting strictly upper triangular matrices. Then we define a natural map from the variety of pairs of commuting n x n matrices onto a subvariety defined by linear equations of the grassmannian of subspaces of codimension 2 of a vector space of dimension n x n.
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 give an explicit description of gr(M), the associated graded module with respect to a canonical Hodge filtration on M. The description involves the isospectral commuting variety, a subvariety X of g x g x h x h which is a finite extension of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of the normalization of X. It follows, thanks to the theory of polarized …
Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g x g such that x and y are nilpotents and are in the same Borel sualgebra. The main result of this paper is that N is a closed and irreducible subvariety of g x g, its normalisation has rational singularities and its normalization morphism is bijective.
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…
It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of parabolic subalgebras of $M\_n(\K)$ and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the numerical conditions under which they are irreducible. In some reducible cases, we describe the irreducible components and their dimension.
We study a scheme M closely related to the set of pairs of n by n-matrices with rank 1 commutator. We show that M is a reduced complete intersection with n+1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Nil in M. We introduce a category, C, of D-modules whose characteristic variety is contained in Nil. Simple objects of that category are analogous to Lusztig's character sheaves. We construct a functor of Quantum Hamiltonian reduction from category C to the category O for type A rational Cherednik algebra.
Let $\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\ge 2$, let $C_r(\N_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that $C_2(\N_n)$ is irreducible for every $n$. We study in this note the reducibility of $C_r(\N_n)$ for various values of $n$ and $r$. In particular it will be shown that the reducibility of $C_r(\mathfrak{gl}_n)$, the variety of commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r(\N_n)$ under certain condition. Then we prove that $C_r(\N_n)$ is reduc…
We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational.
Here we prove that a commuting variety associated with a symmetric pair (g, g_0) is irreducible for (so_{n+m}, so_n + so_m) and reducible for (gl_{n+m}, {gl}_n + gl_m) with n>m, (so_{2n}, gl_n) with odd n, (E_6, {so}_{10} + k).
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $\mathcal C(\mathfrak p) = \{(X,Y) \in \mathfrak p \times \mathfrak p \mid [X,Y] = 0\}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $\mathcal C(\mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $\mathfrak p$. As a conseque…
A multiplication on the 2D cohomological Hall algebra (CoHA) of the variety of commuting matrices was described by Schiffman and Vasserot. This construction can be generalised to other varieties that exist as the zero-locus of a function on a smooth ambient variety. On the 2D CoHA of the character variety of the fundamental group of a genus $g$ Riemann surface, we compare the multiplication induced by the standard presentation and that of a brane tiling presentation.
Let $N(d,n)$ be the variety of all $d$-tuples of commuting nilpotent $n\times n$ matrices. It is well-known that $N(d,n)$ is irreducible if $d=2$, if $n\le 3$ or if $d=3$ and $n=4$. On the other hand $N(3,n)$ is known to be reducible for $n\ge 13$. We study in this paper the reducibility of $N(d,n)$ for various values of $d$ and $n$. In particular, we prove that $N(d,n)$ is reducible for all $d,n\ge 4$. In the case $d=3$, we show that it is irreducible for $n\le 6$.
We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of size 8 matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Białynicki-Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our…
We introduce the_Brauer loop scheme_ E := {M in M_N(C) : M\cp M = 0}, where \cp is a certain degeneration of the ordinary matrix product. Its components of top dimension, floor(N^2/2), correspond to involutions πin S_N having one or no fixed points. In the case N even, this scheme contains the upper-upper scheme from [Knutson '04] as a union of (N/2)! of its components. One of those is a degeneration of the_commuting variety_ of pairs of commuting matrices. The_Brauer loop model_ is a quantum integrable stochastic process introduced in [de Gier--Nienhuis '04], and some of the entries of its Pe…
We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper.
