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Branching Law for the Finite Subgroups of SL4ℂ and the Related Generalized Poincaré Polynomials

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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 1 +  2 +  3 of SL4ℂ decomposes as π p,q,r | Γ  = ⊕  l i = 0 m i (pqr 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

$$ {P}_{\varGamma }{\left(t,u,w\right)}_i={\displaystyle \sum_{p=0}^{\infty }{\displaystyle \sum_{q=0}^{\infty }{\displaystyle \sum_{r=0}^{\infty }{m}_i\left(p,q,r\right){t}^p{u}^q{w}^r}}} $$

are rational functions. This generalizes the results of Kostant for SL2ℂ and the results of our preceding works for SL3ℂ.

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References

  1. T. Bridgeland, A. King, and M. Reid, “The McKay correspondence as an equivalence of derived categories,” J. Amer. Math. Soc., 14, 535–554 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Butin and G. S. Perets, “McKay correspondence and the branching law for finite subgroups of SL3ℂ,” J. Group Theory, 17, Issue 2, 191–251 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Gomi, I. Nakamura, and K. Shinoda, “Coinvariant algebras of finite subgroups of SL3ℂ,” Can. J. Math., 56, 495–528 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Gonzalez-Sprinberg and J.-L. Verdier, “Construction géométrique de la correspondance de McKay,” An. Sci. de l’E. N. S., 4ème Sér., 16, No. 3, 409–449 (1983).

  5. A. Hanany and Y.-H. He, “A monograph on the classification of the discrete subgroups of SU(4),” JHEP, 27 (2001).

  6. B. Kostant, “The McKay correspondence, the coxeter element and representation theory,” SMF. Astérisque, Hors Sér., 209–255 (1985).

  7. B. Kostant, “The Coxeter element and the branching law for the finite subgroups of SU(2),” Coxeter Legacy, Amer. Math. Soc., Providence, RI (2006), pp. 63–70.

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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

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