Branching Law for the Finite Subgroups of 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 ${\pi }_{p,q,r}{|}_{\Gamma }={⨁}_{i=0}^l{m}_i(p,q,r){\gamma }_i.$ We determine the multiplicities $m_i(p,q,r)$ and prove that the series $P_{\Gamma }(t,u,w{)}_i={\sum }_{p=0}^{\infty }{\sum }_{q=0}^{\infty }{\sum }_{r=0}^{\infty }{m}_i(p,q,r)t^p{u}^q{w}^r$ are rational functions. This generalizes results from Kostant for ${\mathbf{SL}}_2\mathbb{C}$ and our preceding works about ${\mathbf{SL}}_3\mathbb{C}$.

arXiv comment9 pages

References in corpus (1)

• A Monograph on the Classification of the Discrete Subgroups of SU(4)