Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory

Abstract

We determine spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with SU (3) modular invariants and subgroups of SU (3). Our methods also give an alternative approach to deriving the results of Banica and Bisch for ADE graphs and subgroups of SU (2) and explain the connection between their results for affine ADE graphs and the Kostant polynomials. We also look at the Hilbert generating series of associated pre-projective algebras.

The Nakayama Automorphism of the Almost Calabi-Yau Algebras Associated to SU(3) Modular Invariants

“…The permutation matrix P corresponds precisely to the Nakayama permutation for A. This result was mentioned without proof in [29]. In [13,Prop.…”

“…where ψ = (ψ i ) is the isomorphism given in (29) and d is the Koszul differential defined in Sect. 2.6.…”

Spectral measures for G2, II: Finite subgroups

“…A fundamental domain C of T 2 under the action of the dihedral group D 12 is illustrated in Figure 2, where the axes are labelled by the parameters θ 1 , θ 2 in (e 2πiθ 1 , e 2πiθ 2 ) ∈ T 2 , which is a quotient of the fundamental domain of T 2 /S 3 illustrated in Figure 1 (see [7]) by the Z 2 -action given by the matrix -1. Note that in Figure 2, the lines θ 1 = 0 and θ 2 = 0 are also boundaries of copies of the fundamental domain C under the action of D 12 , whereas in Figure 1 they are not boundaries of copies of the fundamental domain under the action of S 3 .…”

“…Spectral measures associated to the compact Lie groups SU (2) and SU (3) and their maximal tori, nimrep graphs associated to the SU (2) and SU (3) modular invariants, and the McKay graphs (or representation graphs) for finite subgroups of SU (2) and SU (3) were studied in [1,7,8]. Spectral measures associated to the compact Lie group Sp(2) are studied in [10], whilst spectral measures associated to other compact rank two Lie groups and their maximal tori are studied in [11].…”

Spectral Measures for G 2

“…Spectral measures associated to the compact Lie groups SU(2), SU(3) and G 2 , their maximal tori, nimrep graphs associated to the SU(2), SU(3) and G 2 modular invariants, and the McKay graphs for finite subgroups of SU(2), SU(3) and G 2 were studied in [1,20,21,23,24]. Spectral measures associated to other compact rank two Lie groups and their maximal tori are studied in [25].…”