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Kostant's generating functions and McKay-Slodowy correspondence

Naihuan Jing, Zhijun Li, Danxia Wang

TL;DR

This work extends Kostant's generating-function framework to the McKay-Slodowy setting, relating the decomposition of $S^k(V)$ under a normal subgroup pair $N\lhd G\leq\mathrm{SL}_2(\mathbb C)$ to affine Dynkin data via the affine Coxeter structure. It provides a uniform, rational description of Kostant-type generating functions and Kostant vectors for all multiply- and simply-laced cases, expressing them in terms of the affine Coxeter element and the adjacency matrix of the corresponding representation graphs. The key contributions are the explicit orbit-based expressions for Kostant vectors, a unified formula for generating functions with denominator $(1-t^a)(1-t^b)$ and the relation $ab=2|N|$, and a consolidated Poincaré-series framework $X(t)_i = z(t)_i/((1-t^a)(1-t^b))$ that captures multiplicities of every irreducible module in the symmetric algebra $S(V)$. These results illuminate the deep connections between finite subgroups of $\mathrm{SL}_2(\mathbb C)$, affine root systems, and invariant theory, enabling uniform computation of invariants and multiplicities across the McKay-Slodowy correspondence.

Abstract

Let $N\unlhd G$ be a pair of finite subgroups of $\mathrm{SL}_2(\mathbb{C})$ and $V$ a finite-dimensional fundamental $G$-module. We study Kostant's generating functions for the decomposition of the $\mathrm{SL}_2(\mathbb C)$-module $S^k(V)$ restricted to $N\lhd G$ in connection with the McKay-Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.

Kostant's generating functions and McKay-Slodowy correspondence

TL;DR

This work extends Kostant's generating-function framework to the McKay-Slodowy setting, relating the decomposition of under a normal subgroup pair to affine Dynkin data via the affine Coxeter structure. It provides a uniform, rational description of Kostant-type generating functions and Kostant vectors for all multiply- and simply-laced cases, expressing them in terms of the affine Coxeter element and the adjacency matrix of the corresponding representation graphs. The key contributions are the explicit orbit-based expressions for Kostant vectors, a unified formula for generating functions with denominator and the relation , and a consolidated Poincaré-series framework that captures multiplicities of every irreducible module in the symmetric algebra . These results illuminate the deep connections between finite subgroups of , affine root systems, and invariant theory, enabling uniform computation of invariants and multiplicities across the McKay-Slodowy correspondence.

Abstract

Let be a pair of finite subgroups of and a finite-dimensional fundamental -module. We study Kostant's generating functions for the decomposition of the -module restricted to in connection with the McKay-Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.
Paper Structure (6 sections, 14 theorems, 58 equations)

This paper contains 6 sections, 14 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. The generalizations by McKay-Slodowy correspondence
  3. Kostant's generating functions
  4. Kostant's vectors and the affine Coxeter elements
  5. Formulas for Kostant-type generating functions
  6. The unified Poincaré series for the Coxeter element on the root system associated to $N \unlhd G\leq \mathrm{SL}_2(\mathbb{C})$

Key Result

Lemma 2.1

Let $N\lhd G$ be a pair of finite subgroups in $\mathrm{SL}_2(\mathbb{C})$, and $\{\check\rho_j|j\in {\rm\check{I}}\}$ and $\{\hat{\phi}_j|j\in\hat{\mathrm{I}}\}$ the sets of the $N$-restriction modules and the induced $G$-modules respectively. Then where $d_j$ are the Kac symbols of the untwisted and twisted multiply laced affine Dynkin diagrams if $N\lhd G$ is $D_{n-1} \lhd D_{2(n-1)}$, $C_{2n

Theorems & Definitions (18)

  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Remark 3.6
  • ...and 8 more