McKay Centralizer Algebras

Georgia Benkart (1) , Tom Halverson (2)

1 Department of Mathematics [Madison]
2 Department of Mathematics, Statistics, and Computer Science [Saint-Paul]

For a finite subgroup G of the special unitary group SU2, we study the centralizer algebra Zk(G) = EndG(V⊗k) of G acting on the k-fold tensor product of its defining representation V = C2. The McKay corre- spondence relates the representation theory of these groups to an associated affine Dynkin diagram, and we use this connection to study the structure and representation theory of Zk(G) via the combinatorics of the Dynkin diagram. When G equals the binary tetrahedral, octahedral, or icosahedral group, we exhibit remarkable connections between Zk (G) and the Martin-Jones set partition algebras.

Domains: Combinatorics [math.CO]

Georgia Benkart, Tom Halverson. McKay Centralizer Algebras. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6360⟩. ⟨hal-02173744⟩