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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 \(q=2^{2m+1}>2\).

20C33 Representations of finite groups of Lie type
20C15 Ordinary representations and characters
Full Text: DOI Link
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