Involutions and representations of the finite orthogonal groups

Transcription

1 Invoutions and representations of the finite orthogona groups Student: Juio Brau Advisors: Dr. Ryan Vinroot Dr. Kaus Lux Spring 2007 Introduction A inear representation of a group is a way of giving the group a structure of geoetric syetries. It is a very effective way to study groups because it aows us to reduce any of the probes of abstract groups to probes in inear agebra. The present research has the goa of understanding when it is possibe to construct an invoution ode for the finite orthogona groups. After soe work, we have discovered that an invoution ode does not aways exist for O(n, q) and we conjuecture that it does so ony for n 5. We coud effectivey write out the su of the degrees of the irreducibe characters of the orthogona groups as a poynoia in q. To do so we ade use of the gaussian binoia coefficients. Fro writing out the su of the degrees in this for we can say soe things about the poynoia such as its degree, but we woud ike to abe to say ore, such as wether or not it is irreducibe over Q, athough in specific cases it appears not to be. Throughout our research we ade use of the coputer agebra syste GAP. 1 Representation theory Definition. A representation of a group G is a group hooorphis π : G GL n (C) where GL n (C) is the group of a invertibe n n atrices with entries in C. The degree of the representation is n. When n = 1 we say that π is a inear representation. 1

2 Definition. Two representations π and π are equivaent if there exists and invertibe atrix T such that π(g) = T 1 π (g)t for a g G. Definition. Let π be a representation of the group G. A subrepresentation of π is the restriction of the action of π to a subspace U V = C n such that U is invariant under a representation operators π(g), i.e. π(g)(u) U g G. Definition. A representation is said to be irreducibe if there exists no nontrivia invariant subspace. We denote the set of a irreducibe representations of a group G by Irr(G). Definition. Let π be a representation of a group G. The character of π is a function χ : G C defined by χ(g) = tr π(g) where tr denoted the trace of the atrix π(g). Reca that the trace function is we-defined since for two equivaent representations π 1 and π 2 of G, we have that π 1 (g) = T 1 π 2 (g)t for a g G and so consequenty tr π 1 (g) = tr T 1 π 2 (g)t = tr π 2 (g) for a g G. Often we wi repace the ter representation with the ter character, for exape, we wi speak of a inear character, irreducibe character, etc. Note aso that χ(1) is just the trace of the identity atrix, so this equas the degree of the representation. A character is part of a ore genera cass of functions of C G caed cass functions, which are functions of G that are constant on congugacy casses. We now reca soe basic definitions and resuts in character theory that we wi often use. Definition. Let G be a group. The character tabe of G is a square array of copex nubers with rows indexed by the inequivaent irreducibe characters of G and the couns indexed by the congugacy casses. The entry in row χ and coun K is the vaue of χ on the congugacy cass K. K.. χ... χ(k). 2

3 Definition. Let χ and ψ be characters of a group G. The inner product of χ and ψ is [χ, φ] = 1 χ(g)ψ(g). G g G Theore 1 Let G be a group. The irreducibe characters of G for an orthonora basis for the vector space of a cass functions of G with respect to the inner product [, ]. Coroary 2 Let χ be a character of G. Then χ is irreducibe if and ony if [χ, χ] = 1. 2 Induced characters The foowing concept provides a hepfu ethod for copute the irreducibe characters of a group G. Definition. Let H G and et χ be a character of H. Then the induced character on G is given by χ G (g) = 1 H χ (xgx 1 ), x G where χ is defined by χ (h) = χ(h) if h H and χ (y) = 0 if y / H. Note that, in genera, an induced irreducibe character wi not be an irreducibe of G, however, we ay then decopose that character to find new irreducibes of G. Of great hep is the foowing Lea 3 (Frobenius Reciprocity) Let H G and et ψ be a character of H and χ be a character of G. Then [ψ, χ H ] = [ψ G, χ]. where χ H is the restriction of χ to the subgroup H. 3

4 3 Biinear and Quadratic fors Definition. Let V be a vector space over a fied F. A biinear for on V is a function B : V V F that is inear in each variabe, that is, that satisfies B(u + v, w) = B(u, w) + B(v, w); B(u, v + w) = B(u, v) + B(u, w); B(cu, v) = cb(u, v); B(u, cv) = cb(u, v) for a u, v, w V and a c F. We say then that B is a biinear for on V. If B is a biinear for on V and P = {v 1, v 2,..., v n } is a basis for V then the atrix A = [B(v i, v j )] is caed the atrix of B reative to P. We aso say that A is a representing atrix of B. If u, v V and u = i c iv i and v = i b iv i then ( B(u, v) = B c i v i, ) b i v i = c i B(v i, v j )b j = C t AB i i i,j where C = (c 1,..., c n ) t and B = (b 1,..., b n ) t. Two atrices A and C of a for B reative to different basis of V are congruent, that is, A = P t CP for soe invertibe atrix P. In particuar note that det A = (det P ) 2 det C. Since the deterinant of the atrix of the for B depends on the choice of basis for V, we ake the foowing definitions. Definition. If F is a fied, then we denote its utipicative group of nonzero eeents by F. Define F 2 = {a 2 : a F }. If A is a representing atrix of B then the discriinant of a biinear for B is { 0 if det A = 0 discr(a) = (det A)F 2 F /F 2 otherwise. Note that the discriinant function is we defined. Definition. A biinear for B is nondegenerate if discr(b) 0. 4

5 Definition. A biinear for B is syetric if B(u, v) = B(v, u) for a v, u V. In this paper we wi be interested ony in nondegenerate syetric biinear fors. Given a syetric biinear for B on V we ay define the quadratic for Q : V F by Q(v) = B(v, v). Note that the biinear for B is aso copetey deterined by the quadratic for Q. We wi show that, when the fied F is finite, there are exacty two inequivaent nondegenerate quadratic fors. To this end we ake the foowing definitions. Definition. Let B 1 and B 1 be biinear fors on spaces V 1 and V 2, respectivey. We say that B 1 and B 2 are equivaent if there exists an isoetry σ : V 1 V 2, that is, σ satisfies B 2 (σv, σw) = B 1 (v, w) for a v, w V 1. Proposition 4 biinear fors B 1, B 2 on spaces V 1 and V 2 are equivaent if and ony if there exists bases for V 1 and V 2 such that [B 1 (v i, v j )] = [B 2 (v i, v j )]. Given a set of vectors {v 1,..., v } we say that the set is orthogona reative to the for B if B(v i, v j ) = 0 i j. Theore 5 Let B be a syetric for on a vector space V. Then V has an orthogona basis {v 1,..., v n } reative to which B has representing atrix b A =. b r., where the Q(v i ) = b i 0. In the past theore one ay choose v 1 so that b 1 coud be any eeent in the iage of Q. Afterwards at each stage we ay choose v i so that b i can be any eeent in the iage of the restriction of Q to the orthogona copeent of v 1,..., v i 1. With this in ind we aso have that Theore 6 If B is a nondegenerate syetric biinear for on a space V over a finite fied F of diension n 2, then there is a basis for V reative to which the representing atrix A = diag(1,..., 1, d), 0 d F. Since in the above theore discr(b) = d F 2 and [F : F 2 ] = 2 then there are up to equivaence two nondegenerate syetric biinear (quadratic) fors over a space V, one with discriinant 1 (od F 2 ) and the other with discriinant 1 (od F 2 ). 5

6 4 Orthogona groups Let V be a vector space of diension n 2 over a fied F with char F 2 having a non-degenerate syetric biinear for. Definition. The orthogona group O(V ) consists of a isoetries of V, that is, O(V ) = {τ GL(V ) : B(τu, τv) = B(u, v), for a u, v V }. Here GL(V ) denotes the genera inear group of V, which consists of a invertibe inear transforations of V. Note that O(V ) GL(V ). Let τ O(V ). Suppose we choose a basis for V, and et T be the atrix representing τ reative to the basis. Then we have that T t AT = A where A is the atrix representing the for B reative to the basis. Then it foows that (det T ) 2 det A = det A so det T = ±1. When F is a finite fied with q eeents, the orthogona group on V is finite and we denote it by O(n, F q ). Theore 7 Let V be a vector space over a finite fied F. If n is even, there are exacty two non-isoorphic orthogona groups over V. When n is odd, there is exacty one orthogona group over V. Proof: First consider the case n = 2k. Reca that there are exacty two inequivaent nondegenerate quadratic fors, one with discriinant 1 (od F 2 ) and the other with discriinant 1 (od F 2 ). Let d be a nonsquare in F. We ay choose an orthogona basis reative to which the fors have representing atrix A given by 1. A = diag(1, 1, 1, 1,..., 1, 1), 2. A = diag(1, 1, 1, 1,..., 1, d). The discriinants of these fors are respectivey given by ( 1) k and ( 1) k d (od F 2 ). If k is even then ( 1) k = 1 is a square and ( 1) k d is a nonsquare. If k is odd then if 1 is a square then d is a nonsquare and if 1 is a nonsquare then d is a square. Then we aways have that the discriinants 6

7 of these fors are square and nonsquare in soe order, which ipies that these are the two inequivaent fors. Now suppose n = 2k + 1 is odd. Again we ay choose basis so that the representing atrices of the fors are given by (3) A = diag(1, 1, 1, 1,..., 1, 1, 1), (4) A = diag(1, 1, 1, 1,..., 1, 1, d). The discriinants of these fors are respectivey ( 1) k+1 and ( 1) k+1 d (od F 2 ) and by the sae anaysis as before these are a square and a nonsquare in soe order. However note that if we repace the quadratic for Q by Q a (v) = aq(v), then these two fors deterine the sae orthogona group. Then if we scae Q by d in case 4 we obtain A = diag(d, d, d, d,..., d, d, d 2 ), where discra = ( 1) k+1 d 2k+2 ( 1) k+1 od F 2 which is the discriinant for the for of case 3. Thus these two fors deterine the sae orthogona group and so when n is odd we have exacty one orthogona group. We sha denote the group of type 1 by O + (2n, q), of type 2 by O (2n, q) and of type 3 by O(2n + 1, q). Theore 8 Let F q be a finite fied with char F q 2. Then 1. O + (2k, q) = 2q k(k 1) (q k 1) k 1 1 (i 1), 2. O (2k, q) = 2q k(k 1) (q k + 1) k 1 1 (i 1), 3. O(2k + 1, q) = 2q k2 k 1 (q2i 1). 7

8 Definition. Let π be a representation of the group G. Then π is a rea representation if we can choose a basis for V = C n such that the corresponding atrix representation has iage in GL n (R). Definition. Let G be a group. A ode for G is the direct su of its irreducibe characters each appearing with utipicity one. That is, a ode for a group G is χ. We wish to construct a ode for the orthogona groups by inducing inear characters of the centraizers of invoutions in G. In other words, if x 1, x 2,..., x n are a set of representatives of conjugacy casses of order 2 or 1 in G, then we say that an invoution ode of G is a set of inear characters {χ i } where χ i is a inear character of C G (x i ) and n Ind G C G (x i )(χ i ) = i=1 Definition. Let G be a group and et χ be a character of G. The Frobenius- χ Schur indicator of χ is ε(χ) = 1 G χ(g 2 ). g G This indicator takes the vaues 1,0 and -1 (See [4]). The Frobenius-Schur indicator wi prove very usefu as the foowing is aso true. Theore 9 (Frobenius-Schur) Let χ Irr(G). Then ε(χ) 0 χ is rea vaued. Proposition 10 (Frobenius-Schur) Let G be a finite group such that ε(χ) = 1 for every irreducibe character χ of G. Then χ(1) = {g G g 2 = 1}. 8

9 Theore 11 (Gow, 1985) If χ is any irreducibe character of O(2n + 1, q) G = O + (2n, q) O (2n, q) with q odd, then ε(χ) = 1. These resuts ipy that in the orthogona groups the su of the degrees of the irreducibe characters is exacty equa to the nuber of eeents whose square is the identity. Note that since two conjugate eeents have the sae order, then {g G g 2 = 1} = a 2 =1 cc(a) = a 2 =1 G C G (a). In GL(n, q) there are exacty n + 1 conjugacy casses of eeents whose square is the identity, two of which are I and I, where I denotes the identity atrix. Each conjugacy cass has a representative that is a diagona atrix with a certain nuber of 1 s and -1 s on the ain diagona. If the representative has has k 1 s and n k -1 s then we wi say that the conjugacy cass represented by (k, n k). If we consider the n 1 conjugacy casses different fro I and I, then each of these casses spits into two distinct ones in the orthogona group, giving us a tota of 2n casses whose square is the identity. It foows fro [6] that the centraizers of invoutions in O ± (n, q) are products of saer orthogona groups. Suppose first that G = O + (n, q). Wa showed that if C is a centraizer in GL(n, q) of a conjugacy cass represented by (k, n k) and it spits into centraizers C 1 and C 2 in G, then C 1 = O + (k, q) O + (n k, q) and If G = O (n, q) then C 2 = O (n k, q) O (k, q). C 1 = O + (k, q) O (n k, q) and C 2 = O + (n k, q) O (k, q). 9

10 Since we know the order of these groups, we ay now copute for any pair of specific vaues of n and q the su of the degrees of the irreducibe characters. For exape if we et G = O + (2, q) then χ(1) = 2 + G O + (1, q) O + (1, q) + G O (1, q) O (1, q) 2(q 1) = = q (q 1) 4 Using this sae approach we can write a genera forua for the su of the degrees of the irreducibes. After doing soe agebra we have the foowing Definition. We define the q-binoia coefficients, aso caed Gaussian binoias, by ( ) = (q 1)(q 1 1) (q r+1 ). r q (q 1)( 1) (q r 1) Theore 12 Let γ(q) = χ(1). Then (i) If G = O(n, q) where n = then n 2 ( ) χ(1) = 2 q ( k/2 )(k+1) k. 2 k=0 (ii) If G = O + (n, q) where n = 2 and is even, then 1 ( ) 1 χ(1) = ( 1)+( 1) (q 1) =0 ( ) ( ) + ( ) + q =0. 10

11 (iii) If G = O + (n, q) where n = 2 and is odd, then 1 ( ) 1 χ(1) = ( 1)+( 1) (q 1) =0 ( ) ( ) + ( ) + q (q 1) =0 (iv) If G = O (n, q) where n = 2 and is even, then ( ) 1 χ(1) = ( 1)+( 1) (q + 1) =0 ( ) ( ) + ( ) + q =0 (v) If G = O (n, q) where n = 2 and is odd, then 1 ( ) 1 χ(1) = ( 1)+( 1) (q + 1) =0 ( ) ( ) + ( ) + q (q 1) =0 Using the fact that the degree of ( ) k foowing equas k(2 k + 1) we have the Coroary 13 Let γ(q) = χ(1). Then we have that 1. For O(2k + 1, q), deg(γ) = k(k + 1). 2. For O ± (2k, q), deg(γ) = k 2. The foowing GAP code defines a function which takes two inputs, a group G and a subgroup H G, and it outputs the utipicities of the induced inear characters of H to G. 11

12 indfunction:=function(g,h) a:=charactertabe(g); b:=charactertabe(h); in:=fitered(irr(b),x->x[1]=1); ind:=induced(b,a,irr(b)); return List(ind,x->List(Irr(a),y->ScaarProduct(y,x))); end; Using this function we found an counterexape to there being an invoution ode for every orthogona group. For O(5, q) there does not exist an invoution ode since a of the inear characters of one of the centraizers is never utipicity free when induced. However, for n 4 the induced inear characters appear to aways be utipicity free, so this eads us to beieve that O(n, q) does have an invoution ode for this case. We have then the foowing Conjecture. Let G = O ± (n, q) with q odd. 1. Let ψ be a inear character of a centraizer of a non-centra invoution of G. Then ψ G is utipicity free if and ony if n G has an invoution ode if and ony if n 4. Note that if part 1 is true then part 2 woud foow. One of the steps in proving this conjecture woud be to construct invoution odes for the orthogona groups with n 4. The foowing theores show this for n = 2. The proof of the foowing theore can be found in [2]. Theore 14 O + (2, q) is dihedra of order 2(q 1) and O (2, q) is dihedra of order 2(q + 1). The proof of the foowing theore can be found in [7]. Theore 15 The dihedra groups have invoution odes. References [1] Gow, R. Rea representations of the finite orthogona and sypectic groups of odd characteristic, J. Agebra 96 (1985), no. 1,

13 [2] Grove, L. Cassica Groups and Geoetric Agebra. Graduate Studies in Matheatics, Voue 39. Aerican Matheatica Society [3] Grove, L. Groups and Characters. Wiey Interscience [4] Isaacs, I.M. Character theory of finite groups. Acadeic Press. Pure and Appied Matheatics, No. 69. New York, [5] Sagan, E. Bruce The Syetric Group. Representations, Cobinatoria Agoriths, and Syetric Functions. Graduate Texts is Matheatics. Springer-Verag New York, [6] Wa, G.E. On the conjugacy casses in the unitary orthogona and sypectic groups. J. Asutraian. Math. Soc. 3, (1962), [7] Vinroot, Ryan C. Invoution odes of finite Coxeter groups. To appear in J. Group Theory. 13