Finite subgroups of G₂,(c)

G2quivers

“…As a companion monograph to [21,24], we have transcribed some known results, collected from mathematical works seemingly obscure to the physics literature [17,18,19] and recast them into a compilation explicit in representation, feasible to computation and abundant in tabulation. These shall constitute Sections 2 and 3.…”

𝐿₂(𝑞) and the rank two Lie groups: their construction in light of Kostant’s conjecture

“…However, this case is easily checked using Theorem 4.1 and Theorem 2.2. This allows us to prove (recalling h from equation (2) and that A is the fixed generator for F*) : Theorem 6.4. Using the symbol F*2 to denote the set of squares in ¥* and recalling the notation from Section 2 so that n(X) = e2n,mlh , then the alternating invariant 3-forms ( , , ) on V are described explicitly as follows :…”

“…The main group under consideration will be L2(q) = PSL(2, q) = the group of 2 x 2 matrices of determinant 1 over the field of q elements, F9, all modulo its center. Since q is odd, we may write (2) q = 2h+l…”

“…However, most proofs have relied on a computer and this was the case for E% in particular [5]. In fact, the result of ¿2(61) sitting inside E% had been a major stumbling block in the classification of all finite simple subgroups occurring in complex simple Lie groups (only £2 (2,29) for Ej and ¿2(32) and Sz(S) for Es are still in doubt-see [4, Table 1]). Of course, it is desirable to have a proof that does not need a computer.…”

“…Cohen and Wales in [3] and [2]) and, in particular, on Kostant's conjecture ( [2], [3], [5], [10], [13], and [10] of Cohen, Griess, Kleidman, Lisser, Meurman, Ryba, and Wales). The conjecture is verified easily in the nonexceptional cases by using a character table and Schur indicators (see [10] or [4]).…”