Spectral Measures for G 2

Abstract

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the rank two Lie group G 2 , including the McKay graphs for the irreducible representations of G 2 and its maximal torus, and fusion modules associated to all known G 2 modular invariants.

Spectral measures for G2, II: Finite subgroups

“…Thus there is an infinite family of pullback measures ε µ over T 2 for any measure ν µ on I µ , that is, any ε µ such that ε µ (I −1 µ [x]) = ν µ (x) for all x ∈ I µ will yield the probability measure ν µ on I µ as a pushforward measure by (4). As in [9], we instead work with an intermediate probability measure ν λ,µ which lives over the joint spectrum D λ,µ ⊂ I λ × I µ ⊂ R 2 , for λ, µ ∈ P + , where there is no loss of dimension.…”

“…Here we review the results, determined in [9], for the spectral measures for G 2 . Let ρ 1 , ρ 2 denote the fundamental representations of G 2 of dimensions 7, 14 respectively.…”

“…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 the representation graphs for the irreducible representations of the exceptional compact Lie group G 2 and its maximal torus T 2 , and for nimrep graphs associated to the G 2 modular invariants, were studied in [9].…”

“…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 the representation graphs for the irreducible representations of the exceptional compact Lie group G 2 and its maximal torus T 2 , and for nimrep graphs associated to the G 2 modular invariants, were studied in [9]. The spectral measure for G 2 itself appears in [26] as the reduced Haar measure, and is useful for the study of finite-temperature G 2 gluodynamics [29].…”

“…The paper is organised as follows. In Section 2 we present some preliminary material, including a discussion on the relation between spectral measures over certain different domains in Section 2.1 and a summary of relevant results for G 2 and its nimreps from [9].…”