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Degenerate Affine Flag Varieties and Quiver Grassmannians

Alexander Pütz

Abstract

We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

Degenerate Affine Flag Varieties and Quiver Grassmannians

Abstract

We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

Paper Structure

This paper contains 28 sections, 24 theorems, 84 equations.

Key Result

Theorem 1.3

Let $Q$ be a finite connected quiver and $\mathrm{I}$ an admissible ideal of the path algebra $\Bbbk Q$. The indecomposable injective representation of the bound quiver $(Q,\mathrm{I})$ ending at vertex $j \in Q_0$ is denoted by $I_j$. Then where and $\mathrm{M}^s_{\mathbf{e},\mathbf{d}}(Q,\mathrm{I})$ is the geometric quotient of $\mathrm{R}^s_{\mathbf{e},\mathbf{d}}(Q,\mathrm{I})$ by the group

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: FFR2017KaPe1986
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 48 more