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Ukrainian Mathematical Journal

, Volume 67, Issue 10, pp 1484–1497 | Cite as

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

  • F. Butin
Article
  • 38 Downloads
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  = ⊕  i = 0 l 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.

Keywords

Irreducible Representation Formal Power Series Cartan Subalgebra Minimal Resolution Trivial Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • F. Butin
    • 1
  1. 1.University of LyonLyonFrance

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