Fleischmann, P., and Janiszczak, I., The semisimple conjugacy classes of finite groups of Lie type E 6 and E 7 , Comm. Algebra 21 (1993), 93-161.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[4] showed that a genus of semisimple elements of G F corresponds to a pair (J; w] where J 6= Delta is a proper subset of the vertex set Delta of the extended Dynkin diagram (up to W conjugacy) and [w] is a conjugacy class representative of NW (W J ) W J . Many authors ( 3] 6] [7], 8] 13] have considered the problem of counting semisimple conjugacy classes of G F according to genus. As emerges from their work, the number of semisimple classes belonging to the genus (J; w] is equal to f(J; w] jC NW (W J ) W J (w)j where f(J; w] is the number of t in a maximal ....

Fleischmann, P., and Janiszczak, I., The semisimple conjugacy classes of finite groups of Lie type E 6 and E 7 , Comm. Algebra 21 (1993), 93-161.

....G s i are conjugate in G if and only if [s 1 ] and [s 2 ] are of the same genus. We are especially interested in generic genus numbers , i.e. formulae depending on q, which give these numbers uniformly for a whole series of groups with the same Dynkin type (e.g. 24] 23] for groups of type F 4 , [11], 12] for E n ; n = 6; 7; 8 and [8] for general linear and unitary groups) Since stabilizers of semisimple elements are subgroups of G of maximal Lie rank, they can be described by closed subsystems of the root system Phi of G. This yields combinatorial interpretations of genus numbers in ....

....Lusztig s parametrization of irreducible characters (section 1 ( Since jE(G s ; 1] j is constant (and known) for all semisimple classes of given genus in G, one can use genus numbers to compute the total number of irreducible characters or, equivalently, the generic class number of G . In [11] and [12] all generic semisimple genus numbers f T ( Phi 1 ; w] have been calculated for all simply connected groups of type E 6 ; E 7 and E 8 (see also [6] for E 6 ) The computations were done by implementing the formulae in section 2 on a computer. The following example shows the generic ....

P. Fleischmann, I. Janiszczak, The semisimple conjugacy classes of finite groups of Lie type E 6 and E 7 , Comm. in Alg. 21(1),(1993), 93-161

....) q 2 = 2 2n 1 (see [Shi2] The order of W is 2 7 Delta 3 2 and so p = 3 since q is even. We choose the representative of a semisimple regular class t 6 in [Shi2] Then p divides jZ(t 6 )j = q 4 Gamma 1 and N(t 6 ) 1=4) Delta q 2 Delta (q 2 Gamma 2) G = E 6 (q) sc (see [FJ1]) The order of W is 2 7 Delta 3 4 Delta 5 and so p 2 f2; 3; 5g. Let w = s(ff 2 ) Delta s(ff 1 ff 2 ff 3 ) Delta s(ff 1 ) Delta s(ff 1 2ff 2 2ff 3 ff 4 ff 5 ) Delta s(ff 3 ff 4 ) This is w20 for J = in [FJ1] Then t w = q Gamma 1) Delta (q 3 1) Delta (q 2 1) and c ....

....) 1=4) Delta q 2 Delta (q 2 Gamma 2) G = E 6 (q) sc (see [FJ1] The order of W is 2 7 Delta 3 4 Delta 5 and so p 2 f2; 3; 5g. Let w = s(ff 2 ) Delta s(ff 1 ff 2 ff 3 ) Delta s(ff 1 ) Delta s(ff 1 2ff 2 2ff 3 ff 4 ff 5 ) Delta s(ff 3 ff 4 ) This is w20 for J = in [FJ1]) Then t w = q Gamma 1) Delta (q 3 1) Delta (q 2 1) and c w = 1=12) Delta q 3 Delta (q 1) Delta (q Gamma 1) 2 . G = 2 E 6 (q) sc (see [FJ1] As in the nontwisted case p 2 f2; 3; 5g but we have to replace q by Gammaq, i.e t w = q 1) Delta (q 3 Gamma 1) Delta (q ....

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P.Fleischmann, I.Janiszczak, The semisimple conjugacy classes of finite groups of Lie type E 6 and E 7 , Comm. in Alg., 21(1993), 93-161;

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