Branching Law for the Finite Subgroups of SL4 and the Related Generalized Poincaré Polynomials

Authors

  • F. Butin

Abstract

Within the framework of McKay correspondence, we determine, for every finite subgroup Γ of SL4, how the finite-dimensional irreducible representations of SL4 decompose under the action of Γ. Let h be a Cartan subalgebra of sl4 and let ϖ1,ϖ2, and ϖ3 be the corresponding fundamental weights. For (p,q,r)3, the restriction πp,q,r|Γ of the irreducible representation πp,q,r of the highest weight pϖ1+qϖ2+rϖ3 of SL4 decomposes as πp,q,r|Γ=li=0mi(p,q,r)γi, where {γ0,,γl} is the set of equivalence classes of irreducible finite-dimensional complex representations of Γ. We determine the multiplicities mi(p,q,r) and prove that the series PΓ(t,u,w)i=p=0q=0r=0mi(p,q,r)tpuqwr are rational functions. This generalizes the results of Kostant for SL2 and the results of our preceding works for SL3.

Published

25.10.2015

Issue

Section

Research articles

How to Cite

Butin, F. “Branching Law for the Finite Subgroups of SL4 and the Related Generalized Poincaré Polynomials”. Ukrains’kyi Matematychnyi Zhurnal, vol. 67, no. 10, Oct. 2015, pp. 1321-32, https://umj.imath.kiev.ua/index.php/umj/article/view/2069.