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Weyl-invariant subspaces are (usually) not generic

Riku Kurama, Ruoxi Li, Henry Talbott, Rachel Webb

Abstract

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $θ$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_θ(G) \subseteq V$ of semistable points. We give necessary, sufficient, and in some cases equivalent conditions for the existence of $θ$ such that a maximal torus $T$ of $G$ acts on $V^{ss}_θ(T)$ with finite stabilizers. In such cases, the stack quotient $[V^{ss}_θ(G)/G]$ is is known to be Deligne-Mumford. Our proof uses the combinatorial structure of the weights of irreducible representations of semisimple groups. As an application we generalize the Grassmannian flop example of Donovan-Segal.

Weyl-invariant subspaces are (usually) not generic

Abstract

Let be a linear representation of a connected complex reductive group . Given a choice of character of , Geometric Invariant Theory defines a locus of semistable points. We give necessary, sufficient, and in some cases equivalent conditions for the existence of such that a maximal torus of acts on with finite stabilizers. In such cases, the stack quotient is is known to be Deligne-Mumford. Our proof uses the combinatorial structure of the weights of irreducible representations of semisimple groups. As an application we generalize the Grassmannian flop example of Donovan-Segal.

Paper Structure

This paper contains 18 sections, 33 theorems, 97 equations, 1 table.

Table of Contents

  1. Introduction
  2. Weyl-genericity
  3. Representations of semisimple groups
  4. Applications
  5. Acknowledgements
  6. Background
  7. Reductive groups and their representations
  8. Convex geometry
  9. Twisted affine GIT
  10. Affine GIT
  11. Affine VGIT
  12. VGIT and abelianization
  13. Degeneracies of representations
  14. Weyl-genericity: necessary and sufficient conditions
  15. Necessary conditions
  16. ...and 3 more sections

Key Result

Proposition 1.3.1

If $(V, G)$ is a Weyl-generic representation, then for $r > \dim G$ there exists $\theta \in \chi(G)$ for which the stack quotient $[(V^{\oplus r})^{ss}_\theta(G)/G]$ is a nonempty Deligne-Mumford stack.

Theorems & Definitions (88)

  • Definition 1.1.1
  • Definition 1.2.1: Definition \ref{['def:degeni']} and Lemma \ref{['lem:degen']}
  • Proposition 1.3.1: Corollary \ref{['cor:oplus']}
  • Theorem 1.3.2: Theorem \ref{['thm:sufficient']}
  • Proposition 1.3.3: Example \ref{['ex:gr-flop']}
  • Theorem 2.1.1
  • proof
  • Example 2.1.2
  • Definition 2.1.3
  • Lemma 2.1.4: Stembridge
  • ...and 78 more