Spectral measures for G2, II: Finite subgroups

Abstract

Joint spectral measures associated to the rank two Lie group G 2 , including the representation graphs for the irreducible representations of G 2 and its maximal torus, nimrep graphs associated to the G 2 modular invariants have been studied. In this paper we study the joint spectral measures for the McKay graphs (or representation graphs) of finite subgroups of G 2 . Using character theoretic methods we classify all non-conjugate embeddings of each subgroup into the fundamental representation of G 2 and present their McKay graphs, some of which are new. In the process we also collect together in one paper much information about the finite subgroups of G 2 and their representation theory, drawn from scattered sources.

M-theory cosmology, octonions, error correcting codes

“…The group G 2 has some important finite subgroups, in particular, G 2 (2) of order 12096 and [Z 2 ] 3 • PSL 2 (7), of order 1344. We refer the reader to relatively recent papers on the finite subgroups of G 2 [58][59][60][61][62][63][64] which include more details on the discrete finite subgroups of G 2 (R). In particular in [62] in table 1 one can find the list of all the important finite subgroups Γ ⊂ G 2 .…”

Spectral measures for $Sp(2)$

“…x,y is invariant under the action of D 8 . An expression for J 2 x,y in terms of the D 8 -invariant variables x, y may be obtained as a product of the roots appearing as the equations of the boundary of D x,y in (26), and is given as (see also [41]) The factorizations of J x,y in ( 30) and the equations for the boundaries of D x,y given in ( 26) will be used in Sections 2.3, 3.2 to determine explicit expressions for the weights which appear in the spectral measures µ v u Z over I u in terms of elliptic integrals. Similarly, under the change of variables y = χ ρy (ω 1 , ω 2 ), z = χ ρz (ω 1 , ω 2 ), the Jacobian J y,z = det(∂(y, z)/∂(θ 1 , θ 2 )) is given by…”

“…Similarly, setting ψ(x, y) = y m in (34), the measure µ y,Sp(2) Z is obtained by integrating with respect to x. More explicitly, using the expressions for the boundaries of D given in (26), the spectral measure µ…”

“…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 [1,22,23,25,26]. Spectral measures associated to other compact rank two Lie groups and their maximal tori are studied in [27].…”

“…Thus the joint spectral measure over D x,y is |J x,y (x, y)|dx dy/16π4 , which is the reduced Haar measure on Sp(2) [41, §6.2]. The measure µ x,Sp(2) Nover I x is obtained by integrating with respect to y in(40), whilst the measure µ y,Sp(2) N over I y is obtained by integrating with respect to x in(40).More explicitly, using the expressions for the boundaries of D x,y given in(26), the spectral measure µx,Sp(2) N (over [−4, 4]) for the graph A ρx ∞ (Sp(2)) is dµ x,y (x, y)| dy for x ∈ [−4, 0], (x 2 +4)/4 2x−3 |J x,y (x, y)| dy for x ∈ [0, 4]. The spectral measure µ y,Sp(2) N (over [−3, 5]) for the graph A ρy ∞ (Sp(2)) is dµ…”