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 \( \mathfrak{h} \) be a Cartan subalgebra of \( \mathfrak{s}\mathfrak{l} \) 4ℂ and let ϖ 1, ϖ 2, and ϖ 3 be the corresponding fundamental weights. For (p, q, r) ∈ ℕ3, the restriction \( \pi \) p,q,r | Γ of the irreducible representation \( \pi \) p,q,r of the highest weight pϖ 1 + qϖ 2 + rϖ 3 of SL4ℂ decomposes as π p,q,r | Γ = ⊕ l i = 0 m i (p, q, r)γ i , where {\( \gamma \) 0,…, \( \gamma \) 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
are rational functions. This generalizes the results of Kostant for SL2ℂ and the results of our preceding works for SL3ℂ.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 10, pp. 1321–1332, October, 2015.
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Butin, F. Branching Law for the Finite Subgroups of SL4ℂ and the Related Generalized Poincaré Polynomials. Ukr Math J 67, 1484–1497 (2016). https://doi.org/10.1007/s11253-016-1167-8
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DOI: https://doi.org/10.1007/s11253-016-1167-8