Reality properties of conjugacy classes in algebraic groups

Singh, Anupam; Thakur, Maneesh

doi:10.1007/s11856-008-1001-6

Cited by 17 publications

(18 citation statements)

References 14 publications

(24 reference statements)

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“…al. [7], Singh-Thakur [10,11], Tiep-Zalesski [12]. It is an important problem to classify real elements in a group G. Wonenburger [14] offered a characterization of real elements in GL(n, F) as a product of two involutions.…”

Section: Introductionmentioning

confidence: 99%

On the existence of an invariant non-degenerate bilinear form under a linear map

Gongopadhyay

Kulkarni

2011

Linear Algebra and its Applications

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Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F) > dim V. Let T : V → V be an invertible linear map. We answer the following question in this paper. When does V admit a T -invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit-Zuckerman [2], an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V, F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V, F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I (V, B). A non-degenerate S-invariant subspace W of (V, B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is nontrivial for the unipotent elements in I (V, B). The level of a unipotent T is the least integer k such that (T − I) k = 0. We also classify the levels of unipotents in I(V, B).

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Section: Introductionmentioning

confidence: 99%

On the existence of an invariant non-degenerate bilinear form under a linear map

Gongopadhyay

Kulkarni

2011

Linear Algebra and its Applications

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“…For the groups G 2 (q), Singh and Thakur [40,Corollary A.1.6] proved that when q is not a power of 2 or 3, then all real elements of G 2 (q) are strongly real. We take care of the remaining cases with a computational proof, and conclude this section with the following.…”

Section: Examples and Related Resultsmentioning

confidence: 99%

A computational approach to the Frobenius–Schur indicators of finite exceptional groups

Trefethen

Vinroot

2019

Int. J. Algebra Comput.

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We prove that the finite exceptional groups F 4 (q), E 7 (q) ad , and E 8 (q) have no irreducible complex characters with Frobenius-Schur indicator −1, and we list exactly which irreducible characters of these groups are not real-valued. We also give an exact list of complex irreducible characters of the Ree groups 2 F 4 (q 2 ) which are not real-valued, and we show the only character of this group which has Frobenius-Schur indicator −1 is the cuspidal unipotent character χ 21 found by M. Geck.

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“…The following theorem characterizes reality in SO(n). It may be considered as a special case of the results of Knüppel-Nielsen [10] and Singh-Thakur [18]. The arguments we have used in the proof is motivated by Singh-Thakur [18, section 3.4].…”

Section: Reality Properties Of Conjugacy Classes Inmentioning

confidence: 96%

“…[13], Singh-Thakur [18,19], Tiep-Zalesski [20]. The reality properties of the elements in SO(n, 1) follow from recent results of Singh-Thakur [18,Theorem 3.4.6] combining it with earlier results of Knüppel-Nielsen [10] and Wonenburger [22]. However, the results of these authors do not carry over to the identity component SO o (n, 1), and none of these authors have addressed this issue either.…”

Section: Introductionmentioning

confidence: 99%

Conjugacy classes in Möbius groups

Gongopadhyay

2010

Geom Dedicata

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Abstract. Let H n+1 denote the n + 1-dimensional (real) hyperbolic space. Let S n denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of S n is denoted by M (n). Let Mo(n) be its identity component which consists of all orientation-preserving elements in M (n). The conjugacy classification of isometries in Mo(n) depends on the conjugacy ofFor an element T in M (n), T and T −1 are conjugate in M (n), but they may not be conjugate in Mo(n). In the literature, T is called real if T is conjugate in Mo(n) to T −1 . In this paper we classify real elements in Mo(n).Let T be an element in Mo(n). Corresponding to T there is an associated element To in SO(n+1). If the complex conjugate eigenvalues of To are given by {e iθ j , e −iθ j }, 0 < θj ≤ π, j = 1, ..., k, then {θ1, ..., θ k } are called the rotation angles of T . If the rotation angles of T are distinct from eachother, then T is called a regular element. After classifying the real elements in Mo(n) we have parametrized the conjugacy classes of regular elements in Mo(n). In the parametrization, when T is not conjugate to T −1 , we have enlarged the group and have considered the conjugacy class of T in M (n). We prove that each such conjugacy class can be induced with a fibration structure.

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Reality properties of conjugacy classes in algebraic groups

Cited by 17 publications

References 14 publications

On the existence of an invariant non-degenerate bilinear form under a linear map

On the existence of an invariant non-degenerate bilinear form under a linear map

A computational approach to the Frobenius–Schur indicators of finite exceptional groups

Conjugacy classes in Möbius groups

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