McKay trees

Avraham Aizenbud, Inna Entova-Aizenbud

Research output: Working paper/PreprintPreprint

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Abstract

Given a finite group $G$ and its representation $\rho$, the corresponding McKay graph is a graph $\Gamma(G,\rho)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $\pi,\tau$ of $\Gamma(G,\rho)$ is $\mathrm{dim} \mathrm{Hom}_G(\pi\otimes \rho, \tau) $. The collection of all McKay graphs for a given group $G$ encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of $SU(2)$ and the affine Dynkin diagrams of types $A, D, E$, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (undirected) forest graphs which are McKay graphs of finite groups; this classification turns out to be very concise. Moreover, we describe all pairs $(G,\rho)$ whose McKay graph is a tree.
Original languageEnglish GB
StatePublished - 1 Sep 2021

Keywords

  • Mathematics - Representation Theory
  • Mathematics - Combinatorics

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