# Branching Law for the Finite Subgroups of SL_{4}ℂ and the Related Generalized Poincaré Polynomials

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Within the framework of McKay correspondence, we determine, for every finite subgroup Γ of SL
are rational functions. This generalizes the results of Kostant for SL

_{4}ℂ, how the finite-dimensional irreducible representations of SL_{4}ℂ decompose under the action of Γ.Let \( \mathfrak{h} \) be a Cartan subalgebra of \( \mathfrak{s}\mathfrak{l} \)_{4}ℂ and let*ϖ*_{1},*ϖ*_{2}, and*ϖ*_{3}be the corresponding fundamental weights. For (*p, q, r*) ∈ ℕ^{3}, 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 SL_{4}ℂ decomposes as π_{ p,q,r }|_{ Γ }= ⊕_{ i = 0}^{ l }*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$$ {P}_{\varGamma }{\left(t,u,w\right)}_i={\displaystyle \sum_{p=0}^{\infty }{\displaystyle \sum_{q=0}^{\infty }{\displaystyle \sum_{r=0}^{\infty }{m}_i\left(p,q,r\right){t}^p{u}^q{w}^r}}} $$

_{2}ℂ and the results of our preceding works for SL_{3}ℂ.## Keywords

Irreducible Representation Formal Power Series Cartan Subalgebra Minimal Resolution Trivial Representation
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## References

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