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Stable Graded Multiplicities for Harmonics on a Cyclic Quiver

Andrew Frohmader, Alexander Heaton

Abstract

We consider Vinberg $θ$-groups associated to a cyclic quiver on $k$ nodes. Let $K$ be the product of the general linear groups associated to each node. Then $K$ acts naturally on $\oplus \text{Hom}(V_i, V_{i+1})$ and by Vinberg's theory the polynomials are free over the invariants. We therefore consider the harmonics as a representation of $K$, and give a combinatorial formula for the stable graded multiplicity of each $K$-type. A key lemma provides a combinatorial separation of variables that allows us to cancel the invariants and obtain generalized exponents for the harmonics.

Stable Graded Multiplicities for Harmonics on a Cyclic Quiver

Abstract

We consider Vinberg -groups associated to a cyclic quiver on nodes. Let be the product of the general linear groups associated to each node. Then acts naturally on and by Vinberg's theory the polynomials are free over the invariants. We therefore consider the harmonics as a representation of , and give a combinatorial formula for the stable graded multiplicity of each -type. A key lemma provides a combinatorial separation of variables that allows us to cancel the invariants and obtain generalized exponents for the harmonics.
Paper Structure (6 sections, 16 theorems, 57 equations, 1 figure)

This paper contains 6 sections, 16 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Vinberg Pairs
  4. Partitions, Tableaux, and $\textbf{GL}_n$ representations
  5. The action of $K^2$
  6. Stable Multiplicities via Branching

Key Result

Proposition 3.1

GoodmanWallach2009 The irreducible representations of $\textbf{GL}_{n_1} \times \textbf{GL}_{n_2} \times \dots \times \textbf{GL}_{n_l}$ are the representations $V_1 \otimes V_2 \otimes \dots \otimes V_l$ where $V_i$ is an irreducible representation of $\textbf{GL}_{n_i}$.

Figures (1)

  • Figure 1: Cyclic quiver on $k$ nodes

Theorems & Definitions (32)

  • Proposition 3.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1: Stable Branching Rule
  • Theorem 4.2
  • proof
  • Corollary 4.1
  • Proposition 4.1
  • proof
  • ...and 22 more