Branching Law for the Finite Subgroups of SL_4ℂ and the Related Generalized Poincaré Polynomials
Within the framework of McKay correspondence, we determine, for every finite subgroup Γ of SL_4ℂ, how the finite-dimensional irreducible representations of SL_4ℂ decompose under the action of Γ. Let h be a Cartan subalgebra of sl_4ℂ and let ϖ_1, ϖ_2, and ϖ_3 be the corresponding fundamental weights. For (p, q, r) ∈ ℕ^3, the restriction π_p,q,r | Γ of the irreducible representation π_p,q,r of the highest weight pϖ_1 + qϖ_2 + rϖ_3 of SL_4ℂ decomposes as π_p, q, r | Γ = ⊕_i = 0^l m_i (p, q, r)γ_i, where {γ_0,…, γ_l} is the set of equivalence classes of irreducible finite-dimensional complex representations of Γ. We determine the multiplicities m_i (p, q, r) and prove that the series P_(t,u,w)_i=_p=0^∞ _q=0^∞ _r=0^∞ m_i(p,q,r)t^pu^qw^r are rational functions. This generalizes the results of Kostant for SL_2ℂ and the results of our preceding works for SL_3ℂ., number=10, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Butin, F., year=2015, month=Oct., pages=1321–1332