Abstract
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.
Highlights
In 1980, John McKay [13] discovered that there is a natural one-to-one correspondence between the finite subgroups of the special unitary group SU2 and the -laced affine Dynkin diagrams
The representation graph RV(G) has vertices indexed by the λ ∈ Λ(G) and aλ,μ edges from λ to μ if Gμ occurs in Gλ ⊗ V with multiplicity aλ,μ
Felix Klein had determined that a finite subgroup of SU2 must be one of the following: (a) a cyclic group Cn of order n, (b) a binary dihedral group Dn of order 4n, or (c) one of the 3 exceptional groups: the binary tetrahedral group T of order 24, the binary octahedral group O of order 48, or the binary icosahedral group I of order 120
Summary
To cite this version: Georgia Benkart, Tom Halverson. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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