Branching Law for the Finite Subgroups of SL4ℂ and the Related Generalized Poincaré Polynomials
Abstract
Within the framework of McKay correspondence, we determine, for every finite subgroup Γ of SL4ℂ, how the finite-dimensional irreducible representations of SL4ℂ decompose under the action of Γ. Let h be a Cartan subalgebra of sl4ℂ and let ϖ1,ϖ2, and ϖ3 be the corresponding fundamental weights. For (p,q,r)∈ℕ3, the restriction πp,q,r|Γ of the irreducible representation πp,q,r of the highest weight pϖ1 + qϖ2 + rϖ3 of SL4ℂ 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=0∞∑q=0∞∑r=0mi(p,q,r)tpuqwr are rational functions. This generalizes the results of Kostant for SL2ℂ and the results of our preceding works for SL3ℂ.Published
25.10.2015
Issue
Section
Research articles
How to Cite
Butin, F. “Branching Law for the Finite Subgroups of SL4ℂ 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.