On the partial orthogonality of faithful characters. Gregory M. Constantine 1,2

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1 On the partal orthogonalty of fathful characters by Gregory M. Constantne 1,2 ABSTRACT For conjugacy classes C and D we obtan an expresson for χ(c) χ(d), where the sum extends only over the fathful rreducble characters of a fnte group. Ths expresson s shown to be zero whenever the two conjugacy classes are dstnct modulo the socle of the group. When C and D are the same class, and the class preserves ts cardnalty modulo the socle, we express the above sum n terms of certan nvarants of the group. Key words and phrases: Möbus functon, socle, smple module, rreducble representaton Proposed runnng head: Fathful characters 1 Department of Mathematcs, Unversty of Pttsburgh, Pttsburgh, PA 15260; gmc@euler.math.ptt.edu 2 Funded n part by a Scafe Famly Foundaton grant 1

2 1. The lattce of normal subgroups All groups consdered n ths paper are fnte. For a subgroup H of G the set of subgroups of H that are normal n G, wth ncluson on subgroups as partal order, forms a lattce L G (H) wth meet T K and jon T K. We wrte L(G) for L G (G). In accordance wth the theory of lnear representatons over a feld of coprme characterstc to the order of G, t helps to vsualze ths lattce as the ntersecton of kernels of rreducble representatons of G; cf. Alpern and Bell (1995, p 150). Makng use of well-known results of Wesner (1935), Hall (1933, 1935) and Rota (1964), and the structure of the mnmal normal subgroups vewed as rreducble G modules over a prme-order feld, results of Kratzer and Thévenaz (1984, 1985) yeld an explct expresson for the Möbus functon of L(G). Palfy (1979) obtaned an explct formula for the sum of squares of degrees of the fathful rreducble representatons. Ths paper evaluates the nner product of the fathful characters on classes other than the dentty, and t hghlghts an orthogonalty modulo the socle that exsts for fathful characters; Palfy s result corresponds to the case when the class n queston s the dentty. Bascs on Möbus functons on partally ordered sets are found n Constantne (1987, Chapter 9). Let µ be the Möbus functon of L(G). Denote by S the subgroup of G generated by the set of all mnmal normal subgroups of G. The subgroup S s called the socle of G. In the context of ths paper Hall s result, whch states that µ(1, K) = 0 whenever K s not a jon of atoms of L(G), nforms us that µ(1, K) = 0 unless K s a subgroup of S. 2. Fathful rreducble representatons Denote by χ the character of the rreducble representaton ρ of G, over the feld of complex numbers. Fx conjugacy classes C and D of G. Let f(c, D, N) = χ(c) χ(d), where the sum s over all rreducble characters of G whose kernels (of the correspondng ρs) are equal to subgroup N exactly. We am to fnd an explct form for f n terms of some nvarants of G. The ndcator functon δ AB (H) s by defnton equal to 0 f conjugacy classes A and B of group H are dstnct n H, and s equal to 1 f they are the same n 2

3 H. To smplfy notaton we wrte δ AB (G/K) for δ Ā B(G/K), where A and B are classes of G and Ā and B are ther epmorphc mages n the factor group G/K. By the column orthogonalty of the character table of G, we have f(c, D, T ) = ( G / CN )δ CD (G/N), N T where the sum s over all rreducble representatons of G whose kernels contan N. Interpretng ths equalty on the lattce L(G), we use Möbus nverson to obtan f(c, D, N) = N T( G / CT )δ CD (G/T )µ(n, T ). By workng n the group G/N t suffces to evaluate f(c, D, 1). Usng Hall s result we may further smplfy by restrctng the sum only over normal subgroups T of G that are n the socle S of G. These observatons yeld f(c, D, 1) = )δ CD (G/T )µ(1, T ). (1) T S( G / CT Expresson (1) smplfes to closed form n certan nstances. Indeed, t s easy to see that f δ CD (G/S) = 0, then δ CD (G/T ) = 0 for all T S. We therefore conclude that when classes C and D are dstnct modulo the socle S, then every term n (1) s, n fact, zero. We call two classes A and B of G fathfully orthogonal f χ(a) χ(b) = 0, where the sum s taken only over the fathful characters of G. Fathful orthogonalty s a symmetrc relaton; therefore, f A and B are fathfully orthogonal we sometmes say that A s fathfully orthogonal to B. We now summarze our observatons as follows Theorem 1 If two conjugacy classes are dstnct modulo the socle, then they are fathfully orthogonal. An mmedate consequence s Corollary 1 Let G be a fnte group and S be the socle of G. Then: (a) Any conjugacy class nsde the socle s fathfully orthogonal to any conjugacy class outsde the socle. (b) There are at least m conjugacy classes of group G any two of whch are fathfully orthogonal, where m s the number of conjugacy classes of G/S. 3

4 We examne next the case n (1) when classes C and D are the same. A few nvarants of the socle of G need to be ntroduced. Vew an elementary Abelan mnmal normal subgroup M of G as a smple G module wth conjugaton as G module acton. Call two such mnmal normal subgroups G somorphc f there exsts a group somorphsm f between them that preserves the G module acton; that s, f(x g ) = f(x) g, for all g G. Partton the set of elementary Abelan mnmal normal subgroups of G nto G somorphsm classes. Denote these G classes by A 1,..., A a. Let A be representatves from the G classes A. The group generated by the elements of A s wrtten as S. It follows that S = A d, a drect product of d copes of A. Denote by Hom G (A, A ) the feld of G homomorphsms from A to tself; see Gorensten (1968, p 79). Let S a+1,..., S a+b be the set of non-abelan mnmal normal subgroups of G. Each S, a + 1 a + b, s a drect product of somorphc non-abelan smple groups. We can therefore express the socle S of G as the followng drect product: S = a+b =1 S = a =1 A d a+b j=a+1 S j. We now call on propertes of the Möbus functon, see Kratzer and Thévenaz (1984), to obtan the followng generalzed verson of Lemma 2 of Pálfy (1979): Theorem 2 (a) The Möbus functon on the lattce of normal subgroups of G s µ(h, T ) = 0, unless T/H s n the socle S of G/H. (b) If S = a =1 A d where q = Hom G (A, A ). a+b j=a+1 S j, and T/H = a =1 A α ck=1 S jk, then µ(h, T ) = ( 1) c a =1 ( 1) α q (α 2 ), To smplfy notaton we wrte f(c, N) for f(c, C, N). When C = D equaton (1) becomes f(c, 1) = T S( G / CT )µ(1, T ). Under certan condtons ths expresson can be wrtten as a product. Assume that the class C s such that CS = C S, where S s the socle of G. Such classes generally exst; for nstance any element of the center of G s such a class. We shall also gve an example 4

5 of a noncentral class n SL(2, 3) that satsfes ths condton. For any T S, T normal n G, we then obvously have CT = C T, snce each coset of S contans at most one element of C, and therefore the same must be true of cosets of T. Usng Theorem 2 we thus obtan, = G C f(c, 1) = T S( G / C T )µ(1, T ) (1/ T )µ(1, T ) T 1 S 1 T a+b S a+b = G a+b ( (1/ T )µ(1, T )). C =1 T S We can turn the above sums over the elements of a lattce nto ordnary sums. For a + 1 a + b, we have T S (1/ T )µ(1, T ) = 1 1 S, snce n ths case S s nonabelan mnmal normal n G, and therefore T s ether 1 or S. As to the Abelan case, for 1 a we have T S T 1 µ(1, T ) = d A k (qd 1) (q d q k 1 ) (q k 1) (q k q k 1 ) ( 1)k q 2) (k Usng the q dentty = d A k ( 1) k (qd 1) (q d (q k 1) (q 1) q k 1 ). we obtan (q m 1) (q m q k 1 ) (q k 1) (q 1) = (qm 1) (q m q k 2 ) q k 1 (q k 1 1) (q 1) (qm q k 1 + q m (qm 1) (q m q k 2 ) (q k 1 1) (q 1) = (qm+1 1) (q m+1 q k 1 ), (q k 1) (q 1) m ( A k ( 1) k (qm 1) (q m q k 1 (q k 1) (q 1) = and hence, by nducton, m+1 + qm+k q m ) q k 1 ) A k ( 1) k (qm+1 1) (q m+1 (q k 1) (q 1) )(1 qm A ) q k 1 ), 5

6 Therefore, d A k ( 1) k (qd f(c, 1) = G C We proved the followng: 1) (q d (q k 1) (q 1) a d 1 =1 j=0 q k 1 ) (1 qj a+b A ) Theorem 3 Let C be a conjugacy class of group G. = k=a+1 d 1 j=0 (1 qj A ). (1 1 S k ). sum extendng over the fathful rreducble characters of G. We have (a) Consder f χ(c) χ(c), wth the χ(c) χ(c) = G (1/ C T ) µ(1, T ) f T 1 S 1 T a+b S a+b (b) If class C s such that CS = C S, where S s the socle of G, then f χ(c) χ(c) = G C a d 1 =1 j=0 (1 qj a+b A ) k=a+1 (1 1 S k ). Theorem 1 and part (b) of Theorem 3 may be vewed as a form of column orthogonalty for the character table of any fnte group, n whch the nner product of columns s only taken over the fathful characters. The rght hand sde of the equaton can be nterpreted as a correcton factor to the usual ndex of the centralzer when the nner product extends over all rreducble characters. To examne a few specal cases, let C be a class n the center of G. Then C = 1, and part (b) of Theorem 3 nforms us that, rrespectve of C, f χ(c) χ(c) s equal to f χ(1) 2, whch represents the sum of squares of the degrees of the fathful rreducble representatons of G. Wth C = 1, ths s the man result of Pálfy (1979) who further explores the condtons under whch G has a fathful rreducble representaton. Corollary 2 (Pálfy) The sum of squares of degrees of the fathful rreducble representatons s a d χ(1) 2 1 = G (1 qj a+b f =1 j=0 A ) k=a+1 (1 1 S k ). 6

7 Interestngly, f χ(1) 2 s dvsble by C, where C s any conjugacy class of G such that CS = C S. Indeed, Theorem 3 (b) can be wrten as f χ(c) χ(c) = 1 C f χ(1) 2. The rght hand sde s a ratonal number, whereas the left hand sde s an algebrac nteger. It follows that the rght hand sde s n fact an nteger, possbly zero. We summarze: Corollary 3 The sum f χ(1) 2 s dvsble by the cardnalty of any conjugacy class C of G that satsfes CS = C S, where S s the socle of G. As an example, we examne the group SL(2, 3). There s a sngle mnmal normal subgroup, namely S =< 1 0 >, whch s the socle of SL(2, 3). Corollary 3 yelds 0 1 f χ(1) 2 = 24(1 1) = 12, snce n ths case a = 1, b = 0, and d 2 1 = 1. Consder the conjugacy class C of the element 1 1. Class C contans 4 elements, and t satsfes 0 1 CS = C S. Indeed, SL(2, 3) has three fathful rreducble representatons of degree 2, and 12 = f χ(1) 2 = s dvsble by 4 = C, as Corollary 3 ntmates. REFERENCES 1. Alpern, J. L. and Bell, R. B. (1995) Groups and representatons, Sprnger, New York. 2. Constantne, G. (1987). Combnatoral Theory and Statstcal Desgn, Wley, New York. 3. Fulton, W. and Harrs, J. (1991) Representaton Theory, Sprnger, New York. 4. Gorensten, D. (1968) Fnte Groups, Harper and Row, New York. 5. Hall, P. (1933) A contrbuton to the theory of groups of prme-power order, Proc. London Math. Soc., 36, Hall, P. (1935) The Euleran functon of a group, Quart. J. Math., 7,

8 7. Kratzer, C. and Thévenaz, J. (1984) Foncton de Möbus d un groupe fn et anneau de Burnsde, Comment. Math. Helv., 59, Kratzer, C. and Thévenaz, J. (1985) Type d homotope de trells et trells des sous-groupes d un groupe fnt, Comment. Math. Helv., 60, Pálfy, P. P. (1979) On fathful rreducble representatons of fnte groups, Studa Sc. Math. Hungar. 14, Rota, G-C. (1964) On the foundatons of combnatoral theory, I. Theory of Möbus functons, Z. Wahrsch. Verw. Gebete, 2, Wesner, L. (1935) Abstract theory of nverson of fnte seres, Trans. Amer. Math. Soc., 38,