McKay trees
Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 513-531.

Given a finite group G and its representation ρ, the corresponding McKay graph is a graph Γ(G,ρ) whose vertices are the irreducible representations of G; the number of edges between two vertices π,τ of Γ(G,ρ) is dimHom G (πρ,τ). 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) trees which are McKay graphs of finite groups and describe the corresponding pairs (G,ρ); this classification turns out to be very concise.

Moreover, we give a partial classification of McKay graphs which are forests, and construct some non-trivial examples of such forests.

Published online:
DOI: 10.5802/alco.270
Classification: 20C15
Keywords: Representation theory, groups, McKay graphs
Aizenbud, Avraham 1; Entova-Aizenbud, Inna 2

1 Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
2 Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Aizenbud, Avraham; Entova-Aizenbud, Inna. McKay trees. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 513-531. doi : 10.5802/alco.270.

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