Representation theory MOC

McKay quiver

The McKay quiver¹ $Γ 𝐺 (𝑉)$ of $𝐺$ at an $H$ -module $𝑉$ is a quiver such that #m/def/rep2

the vertices of $Γ 𝑉 (𝐺)$ are the simple modules ${𝑆 𝑖} 𝑡 𝑖 = 1$ ;
the edge set $𝑉 𝑖 \to 𝑉 𝑗$ is $ℕ 𝑛$ , where $𝑛$ is the multiplicity of $𝑉 𝑗$ in the Jordan-Hölder decomposition of $𝑆 𝑖 \otimes 𝕂 𝑉$ .

The McKay matrix $𝐌 𝑉 (𝐺)$ is the adjacency matrix of $Γ 𝐺 (𝑉)$ .

Special cases

McKay quiver of a Chevalley $𝕂$ -bimonoid at a semisimple module

If $H$ has the Chevalley property and $𝑉$ is semisimple, then we have

𝑆 𝑖 \otimes 𝑉 ≅ H 𝑡 ⨁ 𝑗 = 1 𝑆 Γ 𝑉 (𝐺) (𝑆 𝑖, 𝑆 𝑗) 𝑗

which characterizes $Γ 𝑉 (𝐺)$ up to quiver isomorphism.

McKay quiver of a semisimple $𝕂$ -bimonoid

if $H$ is semisimple, then the hypotheses of McKay quiver of a Chevalley $ mathbb{K}$-bimonoid at a semisimple module always hold.

McKay quiver of a group in nice characteristic

This is the original case proposed by John McKay.² If $H = 𝕂 𝐺$ is a group algebra in “nice characteristic” (i.e. Maschke's theorem holds), then in particular the hypothesis of McKay quiver of a semisimple $ mathbb{K}$-bimonoid holds.

Spectrum of a McKay quiver of a group

#state/develop | #lang/en | #SemBr

məˈkaɪ, for John McKay. ↩
1980. Graphs, singularities, and finite groups ↩