G₂quivers

Abstract

This is the unspecified version of the paper.This version of the publication may differ from the final published version. Abstract: We present, in explicit matrix representation and a modernity befitting the community, the classification of the finite discrete subgroups of G 2 and compute the McKay quivers arising therefrom. Of physical interest are the classes of N = 1 gauge theories descending from M-theory and of mathematical interest are possible steps toward a systematic study of crepant resolutions to smooth G 2 manifolds as well as generalised McKay Correspondences. This writing is a companion monograph to hep-th/9811183 and hep-th/9905212, wherein the analogues for Calabi-Yau three-and four-folds were considered. Permanent

Spectral measures for G2, II: Finite subgroups

“…The classification of finite subgroups of G 2 is due to [27,3] (see also [12,13]). The reducible (i.e.…”

“…As a subgroup of GL (7) it is generated by the permutations (1234567) and (124)(365), and the signed permutation δ − {1,2,4,7} (12)(36), where for a set S the notation δ − S means that the entries in the rows indexed by elements of S are multiplied by −1 [6,3]. Its character table, generated by the GAP computational discrete algebra system, is given in Table 5 (see also [13]) where elements in C 4 , C 4 , C 8 , C 8 , C 7 , C 7 , C 6 , C 3 are of cycle type (C 4 , C 3 4 ), (C 4 , C 3 4 ), (C 8 , C 3 8 , C 5 8 , C 7 8 ), (C 8 , C 3 8 , C 5 8 , C 7 8 ), (C 7 , C 2 7 , C 4 7 ), (C 3 7 , C 5 7 , C 6 7 ), (C 6 , C 5 6 ), (C 3 , C 2 3 ) respectively.…”

“…The subgroup P GL(2; 7) of G 2 is an irreducible primitive group of order 336. It has nine irreducible representations, all real, and its character table is given in Table 9 [4,13].…”

“…1 are given in Figure 15, where we use the same notation as previously. Note that the graph given in [13, Figure 2] is not the McKay graph for the restriction 1 of the fundamental seven-dimensional representation of G 2 as claimed in [13], but is rather the McKay graph for the sevendimensional representation Σ 7 , which as shown above does not define an embedding of P GL(2; 7) in G 2 .…”

“…In Section 4 we discuss each finite subgroup of G 2 individually, including determining all non-conjugate embeddings into the fundamental representations of G 2 . For each such embedding we construct their McKay graphs, some of which have appeared before in [13] whilst the rest are new, and we determine their joint spectral measures.…”