mckay correspondence iku nakamura

MCKAY CORRESPONDENCE 5ρ Tr ρ 1 σ τρ 0 χ 0 1 1 1ρ 1 χ 1 1 1 −1ρ 2 χ 2 2 1 0ρ 3 χ 3 2 −1 0ρ 4 χ 4 1 −1 iρ 5 χ 5 1 −1 −iTable 2. Character table of D 3 (of type D 5 )˜Γ GROUP (G) consisting of the vertices v(ρ) for ρ ∈ Irr G and all the edgesbetween them.For example, let us look at the D 5 case. Let χ j := Tr(ρ j ) be the characterof ρ j . Then from Table 2 we see thatχ 2 (g)χ nat (g) =χ 2 (g)χ 2 (g) =χ 0 (g)+χ 1 (g)+χ 3 (g), for g =1,σ or τ.Hence χ 2 χ nat = χ 0 + χ 1 + χ 3 . General representation theory says that anirreducible representation of G is uniquely determined up to equivalence byits character. Therefore ρ 2 ⊗ρ nat = ρ 0 +ρ 1 +ρ 3 . Hence a ρ2 ,ρ j= 1 for j =0, 1, 3and a ρ2 ,ρ j= 0 for j =2, 4, 5. Similarly, we see thatχ 0 χ nat = χ 2 , χ 1 χ nat = χ 2 ,χ 3 χ nat = χ 2 + χ 4 + χ 5 ,χ 4 χ nat = χ 3 and χ 5 χ nat = χ 3 .In this way we obtain a graph – the extended Dynkin diagram ˜D 5 of Figure 3.Thus we see that there are two completely different ways to obtain the sameextended Dynkin diagram ˜D 5 as ˜Γ SING (A 2 /G, 0) and ˜Γ GROUP (G), while D 5as Γ SING (A 2 /G, 0) and Γ GROUP (G).˜D 5ρ 0 ✈ ρ 2❅ ✈ρ ✈ 3ρ 4✈ ρ1 ✈❅ ❅✈ ρ 5Figure 3. ˜Γ GROUP (D 3 )The same is true in the other cases. Namely the two graphs Γ SING (A 2 /G, 0)and Γ GROUP (G) turn out to be one of the Dynkin diagrams ADE and coincidewith each other, while both ˜Γ SING (A 2 /G, 0) and ˜Γ GROUP (G) are thecorresponding extended Dynkin diagram (See Figure 4). It is also interestingto note that the degrees of the characters deg ρ j = χ j (1) are equal to themultiplicities of the fundamental cycle we computed in section 1.3.