NOTIONS OF PURITY AND THE COHOMOLOGY OF QUIVER MODULI

We explore several variations of the notion of purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the l-adic cohomology groups of the quiver moduli space is strongly pure.

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HOMOTOPY GROUPS OF MODULI SPACES OF STABLE QUIVER REPRESENTATIONS

International Journal of Mathematics ◽

10.1142/s0129167x1000646x ◽

2010 ◽

Vol 21 (09) ◽

pp. 1219-1238

Author(s):

GRAEME WILKIN

Keyword(s):

Euclidean Space ◽

Moduli Space ◽

Moduli Spaces ◽

Three Dimensional ◽

Dimensional Euclidean Space ◽

Homotopy Groups ◽

Quiver Representations ◽

Dimension Vector ◽

Fixed Dimension ◽

Low Dimensional

The purpose of this paper is to describe a method for computing homotopy groups of the space of α-stable representations of a quiver with fixed dimension vector and stability parameter α. 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 vector, and the choice of parameter. As a corollary we also compute low dimensional homotopy groups of the moduli space of α-stable representations of the quiver with fixed dimension vector, and apply the theory to the space of non-degenerate polygons in three-dimensional Euclidean space.

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Counting Representations of Valued Quivers over Finite Fields

Algebra Colloquium ◽

10.1142/s1005386706000605 ◽

2006 ◽

Vol 13 (04) ◽

pp. 655-666

Author(s):

Yanxin Wang

Keyword(s):

Finite Field ◽

Finite Fields ◽

Bilinear Form ◽

Cartan Matrix ◽

Dimension Vector ◽

Isomorphism Classes ◽

Indecomposable Representations ◽

Fixed Dimension ◽

Valued Graph

For a symmetrizable Borcherds–Cartan matrix A with integer entries and even diagonal entries, we show that there exists a k-species 𝓢 over the finite field k such that 𝓢 and the Borcherds–Cartan matrix provide the same bilinear form. We also show that the number of isomorphism classes of indecomposable representations of any valued graph with fixed dimension vector is a polynomial, and is independent of the orientation of the valued graph. This extends to the situation of valued graphs with loops.

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Stability conditions and braid group actions on affine An quivers

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501742 ◽

2021 ◽

pp. 2250174

Author(s):

Chien-Hsun Wang

Keyword(s):

Moduli Space ◽

Braid Group ◽

Group Actions ◽

Stability Conditions ◽

Quadratic Differentials ◽

Spherical Subgroup ◽

Type Formula ◽

Affine Type

We study stability conditions on the Calabi–Yau-[Formula: see text] categories associated to an affine type [Formula: see text] quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order [Formula: see text]. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type [Formula: see text] based on the Khovanov–Seidel–Thomas method.

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Kac Polynomials for Canonical Algebras

International Mathematics Research Notices ◽

10.1093/imrn/rnx244 ◽

2017 ◽

Vol 2019 (13) ◽

pp. 3981-4003

Author(s):

Pierre-Guy Plamondon ◽

Olivier Schiffmann

Keyword(s):

Finite Field ◽

Dimension Vector ◽

Indecomposable Representations ◽

Moduli Stacks ◽

Fixed Dimension ◽

Canonical Algebra

Abstract We prove that the number of geometrically indecomposable representations of fixed dimension vector $\mathbf{d}$ of a canonical algebra $C$ defined over a finite field $\mathbb{F}_q$ is given by a polynomial in $q$ (depending on $C$ and $\mathbf{d}$). We prove a similar result for squid algebras. Finally, we express the volume of the moduli stacks of representations of these algebras of a fixed dimension vector in terms of the corresponding Kac polynomials.

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