We count the Fq-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras – one is one-point extended from a quiver Q, and the other is the Dynkin A2 tensored with Q. For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper but algebraic manipulations in Hall algebra will b...

We study quantum geometry of Nakajima quiver varieties two different types - framed A-type quivers and ADHM quivers. While these spaces look completely we find a surprising connection between equivariant K-theories thereof with nontrivial match their parameters. In particular, demonstrate that K-theory $A_n$ in certain $n\to\infty$ limit reproduces the Hilbert scheme points on $\mathbb{C}^2$. a...

We consider a large class of models where an SU(5) gauge symmetry and a FroggattNielsen (FN) Abelian flavor symmetry arise from a quiver gauge theory. Such quiver models are very restrictive and therefore have strong predictive power. In particular, under mild assumptions neutrino mass anarchy is predicted.

The quiver Hopf algebras are classified by means of ramification system with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Nichols type Hopf algebras. 2000 Mathematics Subject Classification: 16W30, 16G10 keywords: Quiver, Hopf algebra, Hopf bimodule, Nichols algebras

We construct bar-invariant Z[q 1 2 ]−bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of [14],[4] and [11] respectively. As a byproduct, we prove the positivity of the elements in these bases.

A factorization formula for certain automorphisms of a Poisson algebra associated with a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulas for Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman [4].

The purpose of this paper is to describe the topology of the space of strictly stable representations of a quiver with fixed dimension vector, where stability is defined with respect to a given parameter (which may be generic or non-generic). The main result is that the homotopy groups of this space are trivial up to a certain dimension, which depends on the quiver, the choice of dimension vect...

Buch and Fulton [7] conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. Knutson, Miller and Shimozono [21] proved this conjecture as an immediate consequence of their “component formula”. We present an alternative proof of the component formula by substituting combinatorics for Gröbner degeneration [21, 20]. We relate the component formula t...

We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GLn.C/ with fixed generic semisimple conjugacy classes at k punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper....

We introduce a new algorithm for computing zigzag persistence, designed in the same spirit as the standard persistence algorithm. Our algorithm reduces a single matrix, maintains an explicit set of chains encoding the persistent homology of the current zigzag, and updates it under simplex insertions and removals. The total worst-case running time matches the usual cubic bound. A noticeable diff...