Corrigenda: Low-dimensional Representations of Quasi-simple Groups

Abstract

This paper contains corrections to the tables of low-dimensional representations of quasi-simple groups published in the paper, 'Lowdimensional representations of quasi-simple groups', LMS Journal of Computation and Mathematics 4 (2001) 22-63.In our paper 'Low-dimensional representations of quasi-simple groups', we determine all the absolutely irreducible representations of quasi-simple groups of dimension at most 250, excluding those of groups of Lie type in their defining characteristic.Martin Liebeck has kindly pointed out to us three omissions in our tables: the 12-and 13-dimensional representations of the group L 3 (3), and the 248-dimensional representations of L 4 (5) in characteristic 2.When checking our arguments and calculations we realized that in fact all the representations of L 3 (3) were missing, as well as the representations of L 4 (5) of dimension exceeding 247.The absolutely irreducible representations of L 3 (3) can be found in the modular Atlas [7]. This leads to the first part of Table 1 below. The absolutely irreducible representations of L 4 (5) of degree up to 247 were classified by Guralnick and Tiep [3], and are contained in the original table. From the proofs given by Tiep

The non-coprime k(GV) problem

“…The bound on Q m implies that V = V 0 = F 128 q . For g ∈ G \ Z, by [13, Theorem 4.3] we have α(g) 10 and so e(g) 115 by Lemma 3.2, unlessḡ is a transvection, for which α(g) 15 and so e(g) 119. Proposition 3.8 now yields k(GV)/|V | < 0.27.…”

“…Then α = 5, P = 98280, and k(G/Z) = 101, so (7) implies |V | < 3.4 · 10 11 . Using the results of [16] and [15] we get |V | = Q 24 with Q = 2 or 3. If Q = 3, 2.…”

“…Then α = 5, P = 1782, and k(G/Z) 80, so (7) implies |V | < 1.06 · 10 11 . Using the results of [15,16] we get |V | = Q 12 with Q = 3, 4, 5, or 7. If Q = 7, Lemma 3.2 yields |[g, V ]| 7 3 for any g ∈ G \ Z(G), so k(GV) |V | in G/Z, then direct calculation using [17] shows that e(g) 6, except possibly for one class of involutions, for which we can take e(g) 8.…”

Fixed point ratios in actions of finite classical groups, IV

“…Then Theorems 2.2 and 2.4 apply and we use work of Lübeck [25] and Hiss-Malle [14] to quickly reduce to a small collection of irreducible embeddings which we can deal with on a case-by-case basis. This is the collection D (see Tables 2.10 Proof.…”

“…For the remainder let us assume H 0 = PSL 2 (l). In [14], Hiss and Malle list all the absolutely irreducible representations of degree at most 250 of quasisimple finite groups, excluding groups of Lie type in their defining characteristic. Frobenius-Schur indicators are also recorded and further information is given which allows one to calculate the smallest field over which each representation can be written.…”