Branching Law for the Finite Subgroups of SL(4,C)

TL;DR

This paper determines how finite subgroups of SL(4,C) decompose irreducible representations and proves that the associated generating series are rational functions, extending previous results for lower dimensions.

Contribution

It provides explicit multiplicity formulas for the decomposition of SL(4,C) representations under finite subgroup actions and establishes the rationality of their generating series.

Findings

Decomposition formulas for irreducible representations of SL(4,C)

Rationality of the series P_(t,u,w)_i

Generalization of Kostant's results to SL(4,C)

Abstract

In the framework of McKay correspondence we determine, for every finite subgroup $\Gamma$ of $\mathbf{SL}_4\mathbb{C}$, how the finite dimensional irreducible representations of $\mathbf{SL}_4\mathbb{C}$ decompose under the action of $\Gamma$. Let $\go{h}$ be a Cartan subalgebra of $\go{sl}_4\mathbb{C}$ and let $\varpi_1,\,\varpi_2,\,\varpi_3$ be the corresponding fundamental weights. For $(p,q,r)\in \mathbb{N}^3$, the restriction $\pi_{p,q,r}|_\Gamma$ of the irreducible representation $\pi_{p,q,r}$ of highest weight $p\varpi_1+q\varpi_2+r\varpi_3$ of $\mathbf{SL}_4\mathbb{C}$ decomposes as ${\pi_{p,q,r}}|_{\Gamma}=\bigoplus_{i=0}^l m_i(p,q,r)\gamma_i.$ We determine the multiplicities $m_i(p,q,r)$ and prove that the series $P_\Gamma(t,u,w)_i=\sum_{p=0}^\infty\sum_{q=0}^\infty\sum_{r=0}^\infty m_i(p,q,r)t^pu^qw^r$ are rational functions. This generalizes results from Kostant for $\mathbf{SL}_2\mathbb{C}$ and our preceding works about $\mathbf{SL}_3\mathbb{C}$.