Real conjugacy classes in algebraic groups and finite groups of Lie type

Tiep, Pham Huu; Zalesski, Alexandre

doi:10.1515/jgth.2005.8.3.291

Cited by 58 publications

(53 citation statements)

References 12 publications

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“…, K m and K i = Spin(W i ). Recall (see [TZ05,§6] for instance) that the nondegenerate quadratic space W gives rise to the special Clifford group Γ + (W ) with…”

Section: 3(ii)mentioning

confidence: 99%

The Waring problem for finite simple groups

2011

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The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and simple groups in particular. Here the k-th power word can be replaced by an arbitrary group word w = 1, and the goal is to express group elements as short products of values of w.We give a best possible and somewhat surprising solution for this Waring type problem for (non-abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements.Along the way we also obtain new results, possibly of independent interest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes.Our methods involve algebraic geometry and representation theory, especially Lusztig's theory of representations of groups of Lie type.

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“…, K m and K i = Spin(W i ). Recall (see [TZ05,§6] for instance) that the nondegenerate quadratic space W gives rise to the special Clifford group Γ + (W ) with…”

Section: 3(ii)mentioning

confidence: 99%

The Waring problem for finite simple groups

2011

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“…The group PSL 2 ð9Þ G A 6 has seven conjugacy classes (with elements of order 1, 2, 3, 3, 4, 5 and 5), all of which are real (see Theorem 3.6 or [16]). Lemma 7.1 implies that 3:PSL 2 ð9Þ has seven real conjugacy classes with elements of the same orders.…”

Section: Covers Of Psl 2 (9)mentioning

confidence: 99%

Real and strongly real classes in PGL n (q) and quasi-simple covers of PSL n (q)

Gill¹,

Singh²

2011

Journal of Group Theory

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“…If q ≡ 1 mod 4, then all characters, and so all blocks, of PSp 2n (q) are real by [43], whereas if q ≡ 3 mod 4, then this is no longer true. However, the group H, got by extending the group PSp 2n (q) by the outer diagonal automorphism, only has real-valued characters by [44,Theorem 4].…”

Section: Classical Groups In Odd Characteristicmentioning

confidence: 99%

“…If N is the Steinberg triality group 3 D 4 (q), then all characters of N are real by [43], and so certainly all 2-blocks of N are real.…”

Section: Classical Groups In Odd Characteristicmentioning

confidence: 99%