Small Degree Representations of Finite Chevalley Groups in Defining Characteristic

Abstract

The author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l 3 , and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

The probability of generating a finite simple group

“…S = F 4 (3): Note that |F 4 (3)| = 2 15 · 3 24 · 5 2 · 7 2 · 13 2 · 41 · 73, and that F 4 (3) = Aut F 4 (3), so we only need determine P F4(3) (2). Using the fact that F 4 (3) has a 25-dimensional faithful representation over a field of characteristic 3 (see [28]), whereas O + 8 (2) does not [16], and consideration of divisors of the group order to further refine the lists in [25, Table 1], we conclude that if M ∈ U (as defined earlier) and the socle S is not of Lie type of untwisted rank at most 2 and characteristic 3, then S is one of:…”

Fuchsian groups, finite simple groups and representation varieties

“…Inspection of the orders of the simple groups G = Cl n (q) now shows that n ≤ 12. For n ≤ 12 we may assume that F * (M ) ∈ Lie(p) where q = p e (otherwise |M | is bounded), and it is simple to list the possible such groups having irreducible representations of dimension n ≤ 12 (see [34] for example). In all cases (18) holds.…”

A lower bound for the dimension of a highest weight module

“…Thus n ≤ 5 and dim V (ω) ≤ 64. An inspection of the Tables 6.6-6.9 of [11] shows that the only possibilities (up to graph automorphism) for (n, ω) are (n, 3ω 1 ), 2 ≤ n ≤ 5, (n, 4ω 1 ), n = 2, 3, and (n, 5ω 1 ), n = 2, 3.…”