Graded characters of modules supported in the closure of a nilpotent conjugacy class

We study the Poincare polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka-Foulkes and are q-analogues of Littlewood-Richardson coefficients corresponding to arbitrary tensor products of irreducibles. Many properties and formulas for these polynomials are derived, such as a generalized Morris recurrence, q-Kostant formula, and a conjectural formula in terms of catabolizable tableaux and charge.Comment: 31 pages, AMS-LaTe

Local cohomology with support in ideals of symmetric minors and Pfaffians

We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group GL on X. We describe the D-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the case of odd minors, this was obtained by Barile in the 90s. Another consequence of our work is that we obtain a description of the decomposition into irreducible GL-representations of the local cohomology modules (the analogous problem in the case when X is the vector space of m x n matrices was treated in earlier work of the authors)

Geometry of orbit closures for the representations associated to gradings of Lie algebras of types E7E_7

This paper is a continuation of arXiv:1201.1102. We investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on the simple Lie algebra of type $E_7$. The methods for classifying the orbits for these actions were developed by Vinberg . We give the orbit descriptions, the degeneration partial orders, and indicate normality of the orbit closures. We also investigate the rational singularities, Cohen-Macaulay and Gorenstein properties for the orbit closures. We give the information on the defining ideals of orbit closures.Comment: arXiv admin note: substantial text overlap with arXiv:1201.1102 some corrections in degeneration order

Extending Upper Cluster Algebras

Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose that there is some cluster variable $x_e$ such that ${R}_{{x}_e} = S[{x}_e^{\pm 1}]$. We try to understand under which conditions ${R}$ is an upper cluster algebra, and how the quiver of $R$ relates to that of $S$. Moreover, if the restriction of $(\Delta,W)$ to some subquiver is a cluster model, we give a sufficient condition for $(\Delta,W)$ itself being a cluster model. As an application, we show that the semi-invariant ring of any complete $m$-tuple flags is an upper cluster algebra whose quiver is explicitly given. Moreover, the quiver with its rigid potential is a polyhedral cluster model.Comment: 27 pages,7 figures. Comments are welcom

The Combinatorics of Quiver Representations

We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically review and develop the necessary methods (exceptional and Schur sequences, orthogonal categories, semi-stable decompositions, GIT quotients for quivers). In the Appendix we include a version of Belkale's geometric proof of Fulton's conjecture that works for arbitrary quivers.Comment: 63 page

Gale-Robinson quivers: from representations to combinatorial formulas

We investigate a family of representations of Gale-Robinson quivers that are geared towards providing concrete information about the corresponding cluster algebras. In this way, we provide a representation theoretic explanation for known combinatorial formulas for the Gale-Robinson sequence and also obtain similar formulas for several other cluster variables.Comment: 27 pages, 6 figure

A Fitting Lemma for Z/2-graded modules

We study the annihilator of the cokernel of a map of free Z/2-graded modules over a Z/2-graded skew-commutative algebra in characteristic 0 and define analogues of its Fitting ideals. We show that in the ``generic'' case the annihilator is given by a Fitting ideal, and explain relations between the Fitting ideal and the annihilator that hold in general. Our results generalize the classical Fitting Lemma, and extend the key result of Green [1999]. They depend on the Berele-Regev theory of representations of general linear Lie super-algebras.Comment: 14 pages Plain TeX; uses diagrams.te

Isotropic Schur roots

In this paper, we study the isotropic Schur roots of an acyclic quiver $Q$ with $n$ vertices. We study the perpendicular category $\mathcal{A}(d)$ of a dimension vector $d$ and give a complete description of it when $d$ is an isotropic Schur $\delta$. This is done by using exceptional sequences and by defining a subcategory $\mathcal{R}(Q,\delta)$ attached to the pair $(Q,\delta)$. The latter category is always equivalent to the category of representations of a connected acyclic quiver $Q_{\mathcal{R}}$ of tame type, having a unique isotropic Schur root, say $\delta_{\mathcal{R}}$. The understanding of the simple objects in $\mathcal{A}(\delta)$ allows us to get a finite set of generators for the ring of semi-invariants SI$(Q,\delta)$ of $Q$ of dimension vector $\delta$. The relations among these generators come from the representation theory of the category $\mathcal{R}(Q,\delta)$ and from a beautiful description of the cone of dimension vectors of $\mathcal{A}(\delta)$. Indeed, we show that SI$(Q,\delta)$ is isomorphic to the ring of semi-invariants SI$(Q_{\mathcal{R}},\delta_{\mathcal{R}})$ to which we adjoin variables. In particular, using a result of Skowro\'nski and Weyman, the ring SI$(Q,\delta)$ is a polynomial ring or a hypersurface. Finally, we provide an algorithm for finding all isotropic Schur roots of $Q$. This is done by an action of the braid group $B_{n-1}$ on some exceptional sequences. This action admits finitely many orbits, each such orbit corresponding to an isotropic Schur root of a tame full subquiver of $Q$.Comment: 31 page

Generic decompositions and semi-invariants for string algebras

We investigate the rings of semi-invariants for tame string algebras A(n) of non-polynomial growth. We are interested in dimension vectors of band modules. We use geometric technique related to the description of coordinate rings on varieties of complexes. The fascinating combinatorics emerges, showing that our rings of invariants are the rings of some toric varieties. We show that for $n\le 6$ the rings of semi-invariants are complete intersections but we show an example for $n=7$ that this is not the case in general

Noncommutative desingularization of orbit closures for some representations of GLnGL_n

We describe noncommutative desingularizations of determinantal varieties, determinantal varieties defined by minors of generic symmetric matrices, and pfaffian varieties defined by pfaffians of generic anti-symmetric matrices. For maximal minors of square matrices and symmetric matrices, this gives a non-commutative crepant resolution. Along the way, we describe a method to calculate the quiver with relations for any non-commutative desingularizations coming from exceptional collections over partial flag varieties.Comment: Significant revisions, 39 page