Conjugacy Classes in Finite Unitary Groups D. E. Taylor Version of 26 June, 2021 TEXed: 3 August, 2021 This treatment of the general unitary groups is modelled on the reports ‘Conjugacy Classes in Finite Special and General Linear Groups’ and ‘Conjugacy Classes in Finite Symplectic Groups’. It also uses ideas from the preprint [2] and the C code of Sergei Haller implementing conjugacy classes for the conformal unitary groups (see the file classes_classical.c). The conjugacy classes are obtained by first computing a complete collection of invariants, determining a representative matrix for each invariant and then computing the order of its centraliser directly from the invariant. This construction is based on the work of Wall [4]. 1 Finite unitary groups For a prime power q, let k GF.q2/ and let k k x x denote the automorphism of q D W !n W 7! N n k defined by x x . If f .t/ a0 a1t ant kŒt, let f .t/ a0 a1t ant . N D D C C C 2 DN CN C C N The general unitary group The general unitary group GU.n; q/ considered here is the set of n n matrices A over the tr field k such that AƒA ƒ, where Atr is the transpose of A and ƒ is the ‘standard’ hermitian D form 00 0 0 11 B0 0 1 0C B . C ƒn B .. C : D B C @0 1 0 0A 1 0 0 0 The description of the conjugacy classes of GU.n; q/ closely parallels the description of the conjugacy classes of GL.n; q/ and Sp.2n; q/. The group GU.n; q/ acts on V kn and for g GU.n; q/, the space V becomes a kŒt- D 2 module Vg by defining vf .t/ vf .g/ for all v V and all f .t/ kŒt. The hermitian form D 2 2 on V preserved by GU.n; q/ is defined by ˇ.u; v/ uƒvtr. D The conformal unitary group The conformal unitary group CGU.n; q/ is the group of linear transformations g such that ˇ.ug; vg/ .g/ˇ.u; v/ for some .g/ k and all u; v V . In matrix terms this is equivalent tr D 2 2 to AƒA .g/ƒ. On applying the automorphism we see that .g/ k0, the fixed field D 2 of . 1 If we identify k with the corresponding scalar matrix we have ˇ.u; v/ and 2 D N since the norm map is onto we see that CGU.n; q/ Z GU.n; q/, where Z is the centre of D CGU.n; q/. Therefore, if g1, g2,..., gr represent the conjugacy classes of GU.n; q/, the conjugacy i classes of CGU.n; q/ are represented by the elements gj , where is a primitive element of k, 1 i < q and 1 j r. Thus from now on we restrict our attention to the general unitary Ä Ä Ä group. import “common.m” : centralJoin, convert ; import “Classes/translate/translateU.m” : tagToNameGU, tagToNameSU ; 2 Polynomials and partitions 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.n; q2/ there is a function ˆ P such that 2 W ! M i .f / Vg kŒt=.f / (2.1) 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 If g GU.n; q/ there are restrictions—to be determined in the sections which follow—on 2 the polynomials that can occur in this decomposition. Polynomials Definition 2.1. (i) Let f .t/ kŒt be a monic polynomial of degree d such that f .0/ 0. The twisted dual 2 ¤ of f .t/ is the polynomial 1 f .t/ f .0/ t d f .t 1/: Q D (ii) The polynomial f .t/ is symmetric if f .t/ f .t/. Q 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 if D C C C C a0a0 1 and ad i a0ai for 0 < i < d. (2.2) D D It is clear that for monic polynomials f and g we have f f and .fg/ f g. Q D D Q Q intrinsic TILDEDUALPOLYNOMIAL( f :: RNGUPOLELT ) RNGUPOLELT { The twisted dual of the polynomial f !} eseq := COEFFICIENTS(f ); a0 := eseq[1]; e 2 require a0 ne 0 : “Polynomial must have non-zero constant term”; K := COEFFICIENTRING(f ); flag, q := ISSQUARE(#K ); require flag : “Field size must be a square”; cseq := [e q : e in eseq ]; return cseq[1] 1 PARENT(f ) !REVERSE(cseq); end intrinsic; If g preserves the hermitian form ˇ, then for all u; v V we have 2 ˇ.ug; v/ ˇ.u; vg 1/ D and thus for f .t/ kŒt 2 ˇ.uf.g/; v/ ˇ.u; vf .g 1//: (2.3) 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/ m.t/; D Q D Q D that is, the minimal polynomial of g is symmetric. Lemma 2.2. Let f .t/ be a monic irreducible polynomial. (i) If f .t/ is reducible, there exists an irreducible polynomial h.t/ such that f .t/ h.t/h.t/ and D Q h.t/ h.t/. ¤ Q d (ii) If f .t/ is irreducible, the degree d of f .t/ is odd and q 1 1 for all such that f ./ 0. C D D Proof. (i) Suppose that h.t/ is an irreducible factor of f .t/. Then g.t/ divides f .t/ f .t/ and Q Q D since f .t/ is irreducible f .t/ h.t/h.t/ and h.t/ h.t/. D Q Q ¤ (ii) (Ennola [1, Lemma 2]) Let be a root of f and let e be the order of . Then f f D Q implies f . 1/ 0 and so f . q/ 0. The Galois group Gal.kŒ; k/ is generated by D D 2 x xq W 7! q i q2i 2d and therefore ./ for some i such that 0 i < d. Since kŒ q we also 2d D D Ä j j D have q . Thus q2i 1 1 .mod e/ and q2d 1 .mod e/. D Á Á Let s gcd.2i 1; 2d/. There exist integers a and b such that D s a.2i 1/ 2bd D C and it follows that qs qa.2i 1/ 2bd . 1/a .mod e/. But s is odd, hence s divides d and D C Á since a is odd, qs 1 .mod e/. Thus if d is even, qd 1 .mod e/. But then d=2./ , Á Á d D which is a contradiction. Therefore d is odd, qd 1 and so q 1 1. Á C D intrinsic TILDEIRREDUCIBLEPOLYNOMIALS( q :: RNGINTELT, d :: RNGINTELT ) SEQENUM {All monic polynomials of degree d with no proper ! tilde-symmetric factors} K := GF(q 2); P := POLYNOMIALRING(K ); is a generator of the group of field elements of norm 1. (q 1) := PRIMITIVEELEMENT(K ) ; 3 if d eq 1 then return [P P:1 i : i in [0:: q]]; end if; j m := d div 2; if ISEVEN(d ) then pols := (m eq 1) select [P P:1 a : a in K a ne 0 ] j j else ALLIRREDUCIBLEPOLYNOMIALS(K , m); pols := SETSEQ( h hh : h in pols h ne hh where hh is TILDEDUALPOLYNOMIAL(h) ); f j g else // d is odd pols := [P ]; j V := VECTORSPACE(K , m); Construct all symmetric polynomials of degree d. for i := 0 to q do i a0 := ; for v in V do eseq := [a0] cat ELTSEQ(v ); q eseq cat:= [a0 eseq[m+1 j ] : j in [0:: m 1]] cat [K ! 1]; f := P ! eseq ; if ISIRREDUCIBLE(f ) then APPEND( pols, f ); end if; end for; end for; end if; return pols ; end intrinsic; Conjugacy class invariants Proof We shall see that elements g1; g2 GU.n; q/ are conjugate in GU.n; q/ if and only if 2 required! they are conjugate in GL.n; q2/ and the conjugacy classes in GU.n; q/ are parametrised by sequences Œ f; f ˆ;e P , where ˆe is the set of irreducible polynomials. h i j 2 2 The class invariants can be constructed in several steps. Firstly, choose a partition D Œn1; n2; : : : ; nk of d. If has m parts of size n, choose m polynomials of degree n (with repetition) represented as a list of pairs, where f; r indicates that the polynomial f of h i degree n has been chosen r times. Secondly, refine by replacing each pair f; r by f; , where is a partition of r. This h i h i refinement step is carried out by the following function. refine := function() ƒ := [ @@ ]; f g for in do := []; f , r := EXPLODE(); for in PARTITIONS(r ) do ˇ := convert (); for in ƒ do APPEND( ,INCLUDE(, <f , ˇ> )); end for; end for; ƒ := ; end for; 4 return ƒ ; end function; intrinsic INTERNALCLASSINVARIANTSGU( d :: RNGINTELT, q :: RNGINTELT : SUBSET := “All”) SEQENUM {The! sequence of conjugacy class invariants for the general unitary group GU(d,q)} if SUBSET eq “Unipotent” then t := POLYNOMIALRING(GF(q 2)):1; return [ @ <t 1, convert (part )> @ : part in PARTITIONS(d )]; f g end if; pols := [TILDEIRREDUCIBLEPOLYNOMIALS(q, k ): k in [1:: d ]]; polsz := [ 1::#pols[k ] : k in [1:: d ]]; f g ptnz := [convert (): in PARTITIONS(d )]; inv := []; for ı in ptnz do prev := [ @@ ]; f g for term in ı do ss := []; n, m := EXPLODE(term); pp := pols[n]; for S in MULTISETS(polsz[n], m) do if SUBSET eq “Semisimple” then „ := [ @@ ]; f g for i r in S do ! „ := [ INCLUDE(, <pp[i ], [<1, r >]>): in „]; end for; else := [ < pp[i ], r > : i r in S ]; ! „ := refine(); end if; for stub in prev do for in „ do APPEND( ss, stub join ); end for; end for; end for; prev := ss ; end for; inv cat:= ss ; end for; return inv ; end intrinsic; Given the sequence pif of primary invariant factors f; m of a unitary matrix, for each h i polynomial f combine the multiplicities m into a partition and replace f by f f if f f .