Conjugacy Classes in Finite Orthogonal Groups

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Conjugacy Classes in Finite Orthogonal Groups Conjugacy Classes in Finite Orthogonal Groups D. E. Taylor Version of 17 March, 2016 TEXed: 10 October, 2018 The definitive and very general treatment of the conjugacy classes of the unitary, symplectic and orthogonal groups was given by Wall [17] in 1963, building on the work of Williamson [18] for perfect fields of characteristic other than two. The following sections describe Magma code [1, 2] implementing the construction of conjugacy classes for the special case of the finite orthogonal groups defined over a Galois field GF.q/, where q is odd. The approach given here follows Milnor [12] combined with the work of Wall [17] as interpreted by Fulman [4, Chapter 6]. For other approaches with more emphasis on the theory of algebraic groups see Springer-Steinberg [15], Humphreys [8] and the recent book of Liebeck and Seitz [10]. For fields of characteristic two see Hesselink [7] and Xue [20]. The conjugacy classes are obtained by first computing a complete collection of invariants and then determining a representative matrix for each invariant. A partial analysis of similar algorithms for finite unitary groups can be found in [6]. There are some remarks about the orthogonal groups in the unpublished draft [13]. 1 Orthogonal groups In MAGMA the general orthogonal group in dimension n over the field k GF.q/, where q is D odd, consists of n n matrices A over k such that AJAtr J , where Atr denotes the transpose D of A and J is a ‘standard’ non-degenerate symmetric matrix. For all n there are two isometry classes of symmetric bilinear forms. However if n is odd, up to isomorphism, there is just one orthogonal group, whereas if n is even there are two orthogonal groups. If V kn, the matrix J defines a symmetric bilinear form on V given by ˇ.u; v/ uJ vtr. D D Odd dimensional orthogonal groups, q odd In n 2m 1, every non-degenerate symmetric bilinear form is congruent to a form with D C matrix 00 0 0 11 0 1 0 0 ƒm B0 0 1 0C B . C J @ 0 a 0 A where ƒm is the m m matrix B .. C D B C ƒm 0 0 @0 1 0 0A 1 0 0 0 1 and where a is a non-zero element of k. There are two congruence classes of these forms determined by whether or not a is a square in k. In the following code we use the forms with a 1 and a ı, where ı is a fixed non-square in k. In MAGMA, the default return value of D D STANDARDSYMMETRICFORM(n, q) has a 2 whereas the symmetric bilinear form preserved by D GO(n, q) has a 1=2; this is the form returned when the Variant := “Revised” option is D used. Even dimensional orthogonal groups, q odd tr The group GOC.2m; q/ consists of the matrices A such that AJA J , where J  à D D 0 ƒm is the matrix returned by STANDARDSYMMETRICFORM(n, q). ƒm 0 tr The group GO.2m 2; q/ consists of the matrices A such that AJA J , where J C D D 0 0 0 ƒ 1 m Â1 0 à @ 0 J2 0 A, J2 and, as above, ı is a non-square in k. This is returned by D 0 ı ƒm 0 0 the MAGMA command STANDARDSYMMETRICFORM(2 m+2, q : VARIANT := REVISED, MINUS). Conjugacy and modules GO".V / denotes an orthogonal group on V kq, where " 0; ; and where 0 D 2" f C g GO .V / GO.V /. The description of the conjugacy classes of GO .V / closely parallels D the description of the conjugacy classes of GL.V /. For g GL.V /, the space V becomes a kŒt-module Vg by defining vf .t/ vf .g/ for all 2 D v V and all f .t/ kŒt. 2 2 The symmetric form ˇ defines an isomorphism V V v ˇ. ; v/. An element W !1 W 7! g GL.V / acts on V according to the rule v. g/ .vg / for all v V and all V ; 2 D 2 2" that is, g g 1 . With this action V becomes a kŒt-module V and g belongs to GO .V / D g if and only if Vg V is an isomorphism of kŒt-modules. W ! g If g; h GL.V /, then T Vg Vh is a kŒt-isomorphism if and only if gT T h if and 12 1 1 1 W ! D " only if T g h T if and only if T Vg Vh is an isomorphism. Since T GO .V / D W !" " 2 if and only if T T it follows that g; h GO .V / are conjugate in GO .V / if and only if D 2 there is a kŒt-isomorphism T Vg V such that the diagram W ! h T Vg Vh ? ! ? ? ? Ây y Vg V !T h commutes. As shown in Macdonald [11, Chap. IV], if P is the set of all partitions and ˆ is the set of all monic irreducible polynomials (other than t), then for g GL.V / there is a function 2 ˆ P such that W ! M i .f / Vg kŒt=.f / (1.1) D f ˆ;i 2 and .f / .1.f /; 2.f /; : : : ; / is a partition such that D X deg.f / .f / n: j j D f ˆ 2 2 For g GO".V / there are restrictions—to be determined in the sections which follow—on 2 the polynomials and partitions that can occur in this decomposition. The functions listed in the following import statement were defined in SpConjugacy.tex and written to the file common.m. For clarity of exposition some of this the code will be repeated below coloured red, but not written to the primary MAGMA file GOConjugacy.m. import “common.m” : convert , allPartitions, primaryParts, stdJordanBlock , centralJoin, getSubIndices, restriction, homocyclicSplit , type1Companion ; Polynomials Definition 1.1. (i) If f .t/ kŒt is a monic polynomial of degree d such that f .0/ 0, the dual of f .t/ is 2 ¤ the polynomial f .t/ f .0/ 1t d f .t 1/: D (ii) The polynomial f .t/ t is -symmetric if f .t/ f .t/. ¤ D (iii) A polynomial f .t/ is -irreducible if it is -symmetric and has no proper -symmetric factors. d 1 d The monic polynomial f .t/ a0 a1t ad 1t t is -symmetric if and only D C C C C if 2 a0 1 and ad i a0ai for 0 < i < d. (1.2) D D Hence a0 1. Furthermore, an element a in an extension field of k is a root of a -symmetric D ˙ 1 polynomial f .t/ if and only if a is also a root. If g preserves the symmetric form ˇ introduced above, then for all u; v V we have 2 ˇ.ug; v/ ˇ.u; vg 1/ D and thus for f .t/ kŒt we have 2 ˇ.uf.g/; v/ ˇ.u; vf .g 1//: (1.3) D In particular, if m.t/ is the minimal polynomial of g, then vm.g 1/ 0 for all v and therefore D gd m.g 1/ 0, where d is the degree of m.t/. Thus m .g/ 0 and it follows that m .t/ D D D m.t/; that is, the minimal polynomial of g is -symmetric. Lemma 1.2. Let f .t/ be a monic -irreducible polynomial. (i) If f .t/ is reducible, there exists an irreducible polynomial g.t/ such that f .t/ g.t/g .t/ and D g.t/ g .t/. ¤ (ii) If the degree of f .t/ is even, then f .0/ 1. D (iii) If f .t/ is irreducible and of odd degree, then f .t/ is either t 1 or t 1. C (iv) If f .t/ is irreducible of even degree 2d, there is an irreducible polynomial g.t/ of degree d such that f .t/ t d g.t t 1/. D C 3 Proof. (i) Suppose that g.t/ is an irreducible factor of f .t/. Then g .t/ divides f .t/ f .t/ D and since f .t/ is -irreducible f .t/ g.t/g .t/ or f .t/ g.t/. D D (ii) Suppose that the degree of f .t/ is 2d. We may suppose that the characteristic of the field is not 2. If a0 1 it follows from (1.2) that ad 0 and that a2d i ai for 1 i < d. D D D Ä Thus f .1/ 0 and so t 1 divides f .t/. If f .t/ g.t/g .t/ and g.t/ is irreducible, as in (i), D D then t 1 divides g.t/, whence g.t/ t 1 and thus g.t/ g .t/, contradicting (i). If f .t/ D D is irreducible, then f .t/ t 1 contradicting the assumption that the degree of f .t/ is even. D (iii) Suppose that f .t/ is irreducible and that its degree is odd. It follows from (1.2) that f . a0/ 0 where a0 1 is the constant term of f .t/. Thus f .t/ t a0, proving (iii). D D ˙ D C (iv) Suppose that f .t/ is irreducible of degree 2d. Then from (ii) we have a0 1 and D it follows by induction (successively subtracting multiples of .t t 1/i from t d f .t/) that C there exists a polynomial g.t/ such that f .t/ t d g.t t 1/.
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