GIT-equivalence and semi-stable subcategories of quiver representations

In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case in [14]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.     

Indecomposable representations for real roots of a wild quiver

For a given quiver and dimension vector, Kac has shown that there is exactly one indecomposable representation up to isomorphism if and only if this dimension vector is a positive real root. However, it is not clear how to compute these indecomposable representations in an explicit and minimal way, and the properties of these representations are mostly unknown. In this note we study representations of a particular wild quiver. We define operations which act on representations of this quiver, and using these operations we construct indecomposable representations for positive real roots, compute their endomorphism rings and show that these representations are tree representations. The operations correspond to the fundamental reflections in the Weyl group of the quiver. Our results are independent of the characteristic of the field.     

The maximal sizes of faces and vertices in an arrangement

Let A be an arragement of n lines in the plane. Suppose that F₁,…,F_r are faces of A and that V,…,V_s are vertices of A. Suppose also that each F_i is a (V_j) of the lines of A intersect at V_j. Then we show that

$\text{∑}\text{i=1}\text{r}\text{t(F}{}_{\mathrm{i}}\text{+}\text{∑}\text{j=1}\text{s}\text{t(V}{}_{\mathrm{j}}\text{)?n+4}\text{r}\text{2}\text{+}\text{s}\text{2}\text{+ 2rs}$

.     

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we show that there is always an open adjointorbit in the nilpotent radical of a seaweed Lie algebra in gl_n(k),thus answering positively in this gl_n(k) case to a questionraised independently by Michel Duflo and Dmitri Panyushev. Theproof gives an explicit construction, using -filtered modulesof quasi-hereditary algebras arising from quotients of the doubleof quivers of type A. An example of a seaweed Lie algebra ina simple Lie algebra of type E₈ not admitting an open orbitin its nilpotent radical is given.     

Berezin kernels and maximal degenerate representations associated with Riemannian symmetric spaces of Hermitian type

We introduce the Berezin kernels for Riemannian symmetric spaces of Hermitian type by restricting the maximal degenerate representations of the corresponding noncompactly causal Lie groups. Bibliography: 20 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 11–21.This revised version was published online in April 2005 with a corrected cover date and article title.     

Quiver varieties and finite dimensional representations of quantum affine algebras

We study finite dimensional representations of the quantum affine algebra using geometry of quiver varieties introduced by the author.

As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.

     

Matrices with zero line sums and maximal rank

An n×n sign pattern admits a matrix with zero-line-sums and rank n?1 if and only if it is fully indecomposable and every arc of an associated directed bipartite graph lies on a circuit. This proves a conjecture of Fiedler and Grone made in the study of sign patterns of inverse-positive matrices.     

On the maximal volume of convex bodies with few vertices

The problem of determining the largest volume of a (d + 2)-point set in E^d of unit diameter is settled. The extremal polytopes are described completely.     

On maximal stable orders on an inverse semigroup of finite rank with zero

We consider maximal stable orders on semigroups that belong to a certain class of inverse semigroups of finite rank. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1035–1041, August, 2008.     

Algebraic varieties with automorphism groups of maximal rank

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 (Zhang in Adv Math 225:2332–2340, 2010) though it is not used in the present paper.     

Hochschild Cohomology of an Algebra Associated with a Circular Quiver

In this article we show that an algebra A = K Γ/(f(X ^s )) has a periodic projective bimodule resolution of period 2, where KΓ is the path algebra of the circular quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K and X is the sum of all arrows in KΓ. Moreover, by means of this projective bimodule resolution, we compute the Hochschild cohomology group of A, and we give a presentation of the Hochschild cohomology ring HH?(A) by the generators and the relations in the case K is a field.     

A polyhedron of genus 4 with minimal number of vertices and maximal symmetry

We construct a symmetric polyhedron of genus 4 in R ³ with 11 vertices. This shows that for given genus g the minimal numbers of vertices of combinatorial manifolds and of polyhedra coincide in the first previously unknown case g=4 also. We show that our polyhedron has the maximal symmetry for the given genus and minimal number of vertices.     

On rank three graphs with a large eigenvalue

Suppose a rank three graph has parameters n, k, λ, μ, and eigenvalues k, s, ?r. Assume that s is larger than a certain function of μ and r and that the graph has a rank three permutation group acting on it; then the graph is a partial geometry. This supplements a theorem of R.C. Bose.