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

Abstract

Within the framework of McKay correspondence we determine, for every finite subgroup Γ of SL4C, how the finitedimensional irreducible representations of SL4C decompose under the action of Γ. Let h be a Cartan subalgebra of sl4C and let 1, 2, 3 be the corresponding fundamental weights. For (p, q, r) ∈ N 3 , the restriction πp,q,r|Γ of the irreducible representation πp,q,r of highest weight p 1 + q 2 + r 3 of SL4C decomposes as πp,q,r|Γ = l i=0 mi(p, q, r)γi, where {γ0,. .. , γ l } is the set of equivalence classes of irreducible finite-dimensional complex representations of Γ. We determine the multiplicities mi(p, q, r) and prove that the series PΓ(t, u, w)i = ∞ p=0 ∞ q=0 ∞ r=0 mi(p, q, r)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. У рамках вiдповiдностi Маккея для кожної скiнченної пiдгрупи Γ групи SL4C визначено, яким чином скiнченновимiрне незвiдне зображення SL4C розкладається пiд дiєю Γ. Нехай h-картанова пiдалгебра sl4C, а 1, 2, 3-вiдповiднi фундаментальнi ваги. Для (p, q, r) ∈ N 3 звуження πp,q,r|Γ незвiдного зображення πp,q,r найбiльшої ваги p 1 + q 2 + r 3 в SL4C розкладається у виглядi πp,q,r|Γ = l i=0 mi(p, q, r)γi, де {γ0,. .. , γ l }-множина класiв еквiвалентностi незвiдних скiнченновимiрних комплексних зображень Γ. Визначено кратностi mi(p, q, r) та доведено, що ряди PΓ(t, u, w)i = ∞ p=0 ∞ q=0 ∞ r=0 mi(p, q, r)t p u q w r є рацiональними функцiями. Це є узагальненням результатiв Костанта для SL2C, а також результатiв наших попереднiх робiт для SL3C.

Connectivity Properties of McKay Quivers

“…Others have generalised the correspondence, from both the representation-theoretic and algebro-geometric viewpoints. On the representation-theoretic side, Auslander and Reiten [1] provided a generalisation to arbitrary two-dimensional representations, Happel et al [8] to arbitrary fields whose characteristics do not divide |G|, Butin and Perets [4] to finite subgroups of SL 3 (C) and Butin [3] to finite subgroups of SL 4 (C).…”

Connectivity Properties of McKay Quivers

“…Others have generalised the correspondence, both from the representation-theoretic and algebro-geometric viewpoints. On the representation-theoretic side, Auslander and Reiten [1] provide a generalisation to arbitrary two-dimensional representations, Happel, Preiser and Ringel [9] to arbitrary fields whose characteristics do not divide |G|, Butin and Perets [4] to finite subgroups of SL 3 (C), and Butin [3] to finite subgroups of SL 4 (C).…”

Quivers supporting twisted Calabi-Yau algebras

“…Because H is triangular, H-MOD is a symmetric monoidal category [23]. Following [6], let {γ 1 , • • • , γ ℓ } be the set of equivalence classes of irreducible Hrepresentations and γ the natural representation. Let χ i be the character of γ i and χ the character of γ.…”