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Principal blocks and the Steinberg character. (English) Zbl 1194.20012
From the text: We determine the finite simple groups of Lie type of characteristic \(p\), for which the Steinberg character lies in the principal \(\ell\)-block for every prime \(\ell\neq p\) dividing the order of the group.
Theorem. Let \(G\) be a finite simple group of Lie type of characteristic \(p\). Then the Steinberg character of \(G\) lies in the principal \(\ell\)-block of \(G\) for all primes \(\ell\neq p\) dividing the order of \(G\), if and only if \(G\) is one of the groups in the following list:
(i) \(\text{PSL}_n(q)\) with \(2\leq n\leq 4\) and \((n,q)\neq(2,2),(2,3)\).
(ii) \(\text{PSU}_n(q)\) with \(3\leq n\leq 4\) and \((n,q)\neq(3,2)\).
(iii) \(\text{PSp}_4(q)\) with \(q\neq 2\).
(iv) \(\text{P}\Omega^+_8(q)\).
(v) \(G_2(q)\) with \(q\neq 2\).
(vi) \(F_4(q)\).
(vii) \(^3D_4(q)\).
(viii) \(^2B_2(q)\) with \(q=2^{2m+1}>2\).
(ix) \(^2G_2(q)\) with \(q=3^{2m+1}>3\).
(x) \(^2F_4(q)\) with \(q=2^{2m+1}>2\).

MSC:
20C33 Representations of finite groups of Lie type
20C15 Ordinary representations and characters
Software:
GAP; CHEVIE
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References:
[1] DOI: 10.1112/S0024610705022556 · Zbl 1096.20010
[2] DOI: 10.1016/j.aim.2007.12.010 · Zbl 1185.20009
[3] Broué M., Astérisque 212 pp 7–
[4] B. Chang and R. Ree, Symposia Mathematica, The characters of G2(q) 13 (Academic Press, London, 1974) pp. 395–413.
[5] Conway J. H., Atlas of Finite Groups (1985)
[6] Deriziotis D. I., Trans. Amer. Math. Soc. 303 pp 39–
[7] Enomoto H., Japan. J. Math. 2 pp 191–
[8] Enomoto H., Japan. J. Math. 12 pp 325–
[9] DOI: 10.1007/BF01389188 · Zbl 0507.20007
[10] Fong P., J. reine angew. Math. 396 pp 122–
[11] DOI: 10.1007/BF01190329 · Zbl 0847.20006
[12] DOI: 10.1007/BF01188519 · Zbl 0669.20036
[13] DOI: 10.1007/978-3-642-67994-0
[14] DOI: 10.1080/00927879008824026 · Zbl 0721.20008
[15] Shinoda K., J. Fac. Sci. Univ. Tokyo (Sect. I A Math.) 21 pp 133–
[16] Shoji T., J. Fac. Sci. Univ. Tokyo (Sect. I A Math.) 21 pp 1–
[17] DOI: 10.2307/1970423 · Zbl 0106.24702
[18] Taylor D. E., The Geometry of the Classical Groups (1992) · Zbl 0767.20001
[19] Ward H. N., Trans. Amer. Math. Soc. 121 pp 62–
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