RESTRICTION OF ODD DEGREE CHARACTERS AND NATURAL CORRESPONDENCES 6 EUGENIO GIANNELLI,ALEXANDERKLESHCHEV,GABRIEL NAVARRO,ANDPHAMHUUTIEP 1 0 2 Abstract. Let q be an odd prime power, n>1, and let P denote a maximal parabolic n subgroup of GLn(q) with Levisubgroup GLn−1(q)×GL1(q). Werestrict theodd-degree a irreducible characters of GLn(q) toP todiscover anatural correspondence of characters, J both for GLn(q) and SLn(q). A similar result is established for certain finite groups 6 withself-normalizingSylowp-subgroups. Wealsoconstructacanonicalbijectionbetween 2 the odd-degree irreducible characters of Sn and those of M, where M is any maximal subgroup of Sn of odd index; as well as between the odd-degree irreducible characters of T] G=GLn(q) or GUn(q) with q odd and those of NG(P), where P is a Sylow 2-subgroup of G. Sinceourbijections commutewith theaction of theabsolute Galois group overthe R rationals, we conclude that the fields of values of character correspondents are the same. h. We usethis to answer some questions of R. Gow. t a m [ 1. Introduction 2 v It is not often the case that a natural correspondence of characters between a group G 3 and a subgroupH of G is found. Even morerarely this correspondencecan bedescribedby 2 inspecting the restriction of characters from G to H. The paradigmatic example of this is 4 the Glauberman correspondence between the P-invariant irreducible characters of a finite 4 0 group G of order not divisible by a prime p, acted by the p-group P, and the irreducible 1. characters of the fixed point subgroup CG(P). 0 If G is a finite group and p is a prime, the McKay conjecture (cf. [M , M ]) asserts that 1 2 16 |Irrp′(G)| = |Irrp′(NG(P))| for P ∈ Sylp(G), where Irrp′(G) is the set of the irreducible complex characters of G of degree not divisible by p. : v i The focus of this paper is on the existence of canonical correspondences between Irrp′(G) X and Irrp′(H) for certain pairs (G,H) of finite groups with G > H ≥ NG(P). Even in the r caseswheretheMcKayconjectureisknowntoholdforGandH (cf. [O1],[MS])andthereby a the existence of a bijection between Irrp′(G) and Irrp′(H) is guaranteed, this resulting bijection may not give a canonical correspondence. For instance, one expects a canonical correspondenceto commute with the action of the absolute Galois group over the rationals, 2010 Mathematics Subject Classification. Primary 20C15. Key words and phrases. Restriction, Odd Degree Characters, Natural Correspondences. The first author gratefully acknowledges financial support by the ERC Advanced Grant 291512. The secondauthorwassupportedbytheNSF(grantDMS-1161094) andtheMax-Planck-Institut. Theresearch of the third author is supported by the Prometeo/Generalitat Valenciana, Proyectos MTM2013-40464-P. The fourth author was supported by theNSF(grant DMS-1201374). We thank Rod Gow for communicating his observation concerning GLn(q) in Corollary F to us, as well as for suggesting that thesame statement should hold for GUn(q). 1 2 E.GIANNELLI, A.KLESHCHEV,G.NAVARRO,ANDP.H.TIEP and in this case, the fields of values of character correspondents must be the same. This does not happen often. Also, one expects that canonical correspondences between Irrp′(G) and Irrp′(H) will commute with every automorphism of G that stabilizes H, and provide essential information on cohomological character theoretical questions. Furthermore, there is some hope that certain canonical correspondences will play an important role in proving various refinements of the McKay conjecture (eg. [N ]). 2 For any character χ of G we denote by χ its restriction to a subgroup H. H Definition 1.1. An arbitrary subgroup H ≤ G is called p-restriction good if for every χ ∈ Irrp′(G), there exists a unique χ∗ ∈ Irrp′(H) such that χH = χ∗ + ∆ and either ∆ = 0 or all irreducible constituents of ∆ have degrees divisible by p. A p-restriction good subgroup H ≤ G is called p-restriction canonical if the map χ 7→ χ∗ yields a bijection between Irrp′(G) and Irrp′(H). Very recently, the following result∗ has been proved: Theorem 1.2. [APS] Let n ∈ Z . Then S is a 2-restriction good subgroup in S . >1 n−1 n Moreover, if n is odd, then S is a 2-restriction canonical subgroup in S . n−1 n In this paper, we prove: Theorem A. Let n ∈ Z , q be an odd prime power, and P be a maximal parabolic >1 subgroup of GL (q) with Levi subgroup GL (q)×GL (q). Then P is a 2-restriction good n n−1 1 subgroup in GL (q). Moreover, if n is odd, then P is a 2-restriction canonical subgroup in n GL (q). n Theorem B. Let n ∈ Z be odd, q an odd prime power, and Q be a maximal parabolic >1 subgroup of SL (q) with Levi subgroup (GL (q) × GL (q)) ∩ SL (q). Then Q is a 2- n n−1 1 n restriction canonical subgroup in SL (q). n Theorem C. Let G be a finite group, p be a prime, and P ∈ Syl (G). Suppose that p P = N (P), and in addition that G is a solvable group if p = 2. Let P ≤H ≤ G. Then H G is a p-restriction canonical subgroup in G. All these theorems might suggest that further results of this type can hold true for arbitrary finite groups with self-normalizing Sylow 2-subgroups. However, the group G = SL (2) has a self-normalizing Sylow 2-subgroups, and an irreducible character χ ∈ Irr(G) 3 of degree 7 such that the restriction of χ to every odd-index proper subgroup H of G has exactly three irreducible constituents of odd degree. This example also shows that Theo- rem A does not hold when 2|q. Other examples also show that analogues of Theorem 1.2 and Theorems A, B do not seem to hold when p > 2. However, canonical character correspondences, although not necessarily defined by restriction, can be obtained for symmetric groups and finite general linear and unitary groups, again for p = 2. The meaning of the wordcanonicalinthefollowingtheoremswillbecomeclearinSections4,5;inparticular,the constructed character correspondences commute with the action of Galois automorphisms and group automorphisms. ∗Infact, weindependentlyconjectured thisstatement. Butwhile wewereworkingontheproof ofit, we learned of thepreprint [APS] in which theconjecture was proved. RESTRICTION OF ODD DEGREE CHARACTERS AND NATURAL CORRESPONDENCES 3 Theorem D. Let n ∈ Z and let M be a maximal subgroup of S of odd index. Then >1 n there is a canonical bijection between Irr2′(Sn) and Irr2′(M). Theorem E. Let n ∈ Z , q be an odd prime power, G = GL (q) or GU (q), and P ∈ ≥1 n n Syl2(G). Then there is a canonical bijection χ 7→ χ♯ between Irr2′(G) and Irr2′(NG(P)). We will prove (see Theorem 5.3) that our canonical bijection in Theorem E commutes with Galois action, and this will imply, for instance, that the fields of values of the odd- degree irreducible characters of GL (q) and GU (q), if q is odd, are in bijection with n n the fields of values of the odd-degree characters of N (P) for P ∈ Syl (G). This does not G 2 happeninGL (4)orGL (8). Twoothercaseswherethereexistacanonicalcorrespondence 2 2 for the McKay conjecture for p = 2 are in solvable groups [Is ] and symmetric groups [G], 1 see also Theorem 4.3. To illustrate the power of canonical maps, we can answer a question of Gow, which was privately communicated to us. Corollary F. The number of real-valued, irreducible characters of odd degree of G = GL (q)andGU (q), withq anyoddprime power, isequaltothatofN (P)forP ∈ Syl (G), n n G 2 which is 2n1+n2+...+nr+r if n = 2n1 + 2n2 + ... + 2nr is the 2-adic decomposition of n. Furthermore, all such characters are rational-valued. 2. Restriction to a maximal parabolic subgroup Unless otherwise stated, we always assume that p is a prime and H is a subgroup of a finite group G. We begin with some simple observations. Note that if the p-restriction good subgroup H of G satisfies |Irrp′(G)| = |Irrp′(H)|, then it is p-restriction canonical, by the following lemma. (The lemma also indicates a possible weakening of the notion of p-restriction good subgroupswhen one allows a multiplicity >1 of the p′-degree irreducible constituent). Lemma 2.1. Let H have p′-index in G and |Irrp′(G)| = |Irrp′(H)|. Suppose that for every χ ∈ Irrp′(G), among the irreducible constituents of χH there is only one (but possibly with multiplicity > 1), denoted by χ∗, that has p′-degree. Then the map ∗ : χ 7→ χ∗ is a bijection between Irrp′(G) and Irrp′(H). Proof. For any ρ ∈ Irrp′(H), the induced character ρG has p′-degree, so it contains a constituent χ ∈ Irrp′(G). By assumption, ρ= χ∗. Thus ∗ is surjective, and so it is injective as well. (cid:3) The following result is well known, see for example [J , 22.4]. 1 Lemma 2.2. Let a,b ∈ Z , n = a+b, and consider 2-adic decompositions ≥0 t t t n= 2in , a = 2ia ,b = 2ib . i i i Xi=1 Xi=1 Xi=1 Then the following statements are equivalent: (a) The binomial coefficient n is odd. a (b) ai +bi = ni for all i. (cid:0) (cid:1) (c) 0 ≤ a ≤ n for all i. i i 4 E.GIANNELLI, A.KLESHCHEV,G.NAVARRO,ANDP.H.TIEP For r ∈ Z we denote by [r] the largest p-power that divides r; we also set [0] := ∞. >0 p p Corollary 2.3. Let a ,...,a ∈ Z and n = m a . Suppose that n!/ m a ! is odd. 1 m >0 i=1 i i=1 i Then, by relabeling a1,...,am suitably, we may Passume that Q [n] = [a ] < [a ] < ... < [a ] . 2 1 2 2 2 m 2 Proof. There is nothing to prove for m = 1. Assume m = 2, and consider the 2-adic decompositions t t t n = 2in , a = 2ib , a = 2ic , i 1 i 2 i Xi=k Xi=0 Xi=0 with k ≥ 0 and n = 1. By Lemma 2.2 we have b = c = 0 for 0 ≤ i < k and, k i i moreover, relabeling a and a if necessary, we may assume that (b ,c ) = (1,0). Thus 1 2 k k [n] = [a ] < [a ] as needed. Now the case m > 2 follows by an easy induction on m 2 1 2 2 2 using the case m = 2. (cid:3) Corollary 2.4. Suppose that a,b ∈ Z , n = a+b, and n is odd. Then there is a unique >0 a c ∈ {a−1,a} such that n−1 is odd. Moreover, if we as(cid:0)su(cid:1)me additionally that [a] ≤ [b] , c 2 2 then n−1 is odd. (cid:0) (cid:1) a−1 (cid:0) (cid:1) Proof. As n = n−1 + n−1 ,thefirstclaimfollows. Forthesecondclaim,letc ∈ {a−1,a} a a−1 a be such th(cid:0)at(cid:1) n−(cid:0)1 i(cid:1)s o(cid:0)dd.(cid:1)By Corollary 2.3, the assumption [a] ≤ [b] implies that c 2 2 [n]2 = [a]2 < [(cid:0)b]2. (cid:1)If n is odd, then a is odd and b is even, and, by Lemma 2.2, c is even, whence c= a−1. If n is even, we consider the 2-adic decompositions t t t n= 2in , a = 2ia , b = 2ib , i i i Xi=k Xi=0 Xi=0 with k ≥ 1 and n = 1. By Lemma 2.2, k (a ,...,a )= (0,...,0,1), (b ,...,b ) = (0,...,0,0), 0 k 0 k and so n−1 is even. Hence again we must have that c= a−1. (cid:3) a (cid:0) (cid:1) Recall thatcomplex irreduciblecharacters of S arelabelled bypartitions λ ⊢ n: χ = χλ. n By Theorem 1.2, thereis a canonical map λ 7→ λ∗ such that, if χλ ∈ Irr(S )is of odddegree n then χ∗ = χλ∗ is the unique odd-degree irreducible constituent of χSn−1. From now on, we fix an odd prime power q. For any n ≥ 1, let G = GL (q), with n a natural module V = Fn = he ,...,e i . As in [KT ], it is convenient for us to use q 1 n Fq 2 the Dipper-James classification of complex irreducible characters of G, as described in [J ]. Namely, every χ ∈ Irr(G) can be written uniquely, up to a permutation of the pairs 2 {(s ,λ ),...,(s ,λ )}, in the form 1 1 m m (2.1) χ = S(s ,λ )◦S(s ,λ )◦...◦S(s ,λ ). 1 1 2 2 m m Here, s ∈ F¯× has degree d over F , λ ⊢ k , m k d = n, and the m elements s have i q i q i i i=1 i i i pairwise distinct minimal polynomials over FPq. In particular, S(si,λi) is an irreducible character of GL (q). kidi RESTRICTION OF ODD DEGREE CHARACTERS AND NATURAL CORRESPONDENCES 5 Furthermore, there is a parabolic subgroup P = U L of G with Levi subgroup L = χ χ χ χ GL (q)×...GL (q) and unipotent radical U . The (outer) tensor product k1d1 kmdm χ ψ := S(s ,λ )⊗S(s ,λ )⊗...⊗S(s ,λ ) 1 1 2 2 m m is an L -character, and χ is obtained from ψ via the Harish-Chandra induction RG , i.e. χ Lχ we first inflate ψ to a P -character and then induce it to G. The adjoint operation of χ Harish-Chandra restriction ∗RG takes any character ρ of G, afforded by a CG-module W Lχ to the L -character afforded by WUχ, the fixed point subspace for U on W. χ χ Let P = UL be a maximal parabolic subgroup of G with Levi subgroup L = GL (q)× 1 GL (q) and the unipotent radical U. Conjugating suitably in G and applying the n−1 transpose-inverse automorphism if necessary, we may assume that P = Stab (he i ) and G 1 Fq the second factor GL (q) of L fixes both e and he ,...,e i . n−1 1 2 n Fq Given the above notation, we can now prove the following theorem which implies Theo- rem A: Theorem 2.5. Let q be an odd prime power, n ≥ 2, G = GL (q), P = Stab (he i ). n G 1 Fq Suppose that χ ∈Irr2′(G). Then the following statements hold: (i) One can choose a label (2.1) for χ such that si ∈ F×q (so that di = 1) and χλi ∈ Irr2′(Ski) for all i = 1,...,m, and [n] =[k ] < [k ] < ... < [k ] ; 2 1 2 2 2 m 2 (ii) χP = χ∗ + ∆, where χ∗ ∈ Irr2′(P) and either ∆ = 0 or ∆ is a P-character all irreducible constituents of which are of even degree; (iii) χ∗ is trivial on U, and equal to S(s ,(1)) ⊗ (S(s ,λ∗) ◦S(s ,λ ) ◦... ◦ S(s ,λ )) 1 1 1 2 2 m m when viewed as a character of GL (q)×GL (q); 1 n−1 (iv) If n is odd, then the map χ 7→ χ∗ is a bijection between Irr2′(G) and Irr2′(P). Note that in 2.5(iii), the symbol S(s ,λ∗) is considered void if k = 1. We proceed in a 1 1 1 series of lemmas. Lemma 2.6. Statement (i) of Theorem 2.5 holds. Proof. Since the degree of χ is odd, so are the degrees of each S(s ,λ ), which implies that i i eachd = deg(s ) = 1,forexampleby[KT ,Lemma5.7(ii)], i.e. s ∈ F× foralli= 1,...,m. i i 2 i q Next, we also must have that |G : P | is odd, which implies by a repeated application of χ m [NT , Lemma 4.4(i)] that n!/ k ! is odd. So we may assume by Corollary 2.3 that 1 i=1 i [n]2 = [k1]2 < [k2]2 < ... < [kQm]2. Finally, it is well known (and follows from the hook formula for the degree of unipotent characters of G, see [FS, (1.15)]) that (2.2) χλ(1) ≡ deg(S(s,λ))(mod2) if s ∈F×q , and so we conclude that χλi ∈ Irr2′(Ski). (cid:3) Lemma 2.7. Let X = X ×X , where X ∼= GL (q), X ∼= GL (q), P = UL a parabolic 1 2 1 m 2 n 1 1 subgroup of X with unipotent radical U and Levi subgroup L , and let L = L ×X . 1 1 1 2 (i) If α is a character of X and β is a character of X , then ∗RX(α⊗β) = ∗RX1(α)⊗β. 1 2 L L1 (ii) If γ is a character of L and δ is a character of X , then RX(γ ⊗δ) = RX1(γ)⊗δ. 1 2 L L1 6 E.GIANNELLI, A.KLESHCHEV,G.NAVARRO,ANDP.H.TIEP Proof. (i)Letα,respectivelyβ,beaffordedbyaCX -moduleA,respectivelyaCX -module 1 2 B. Then ∗RX(α⊗β) is afforded by the L-module (A⊗B)U = AU ⊗B and so equal to L ∗RX1(α)⊗β. L1 (ii) Inflate γ to the character γ˜ of P using P /U ∼= L , and inflate γ⊗δ to the character 1 1 1 ρ = γ˜⊗δ of P = P ×X using P/U ∼= L. Then 1 2 RX(γ⊗δ) = ρX = ((γ˜⊗1 )·(1 ⊗δ) )X = L X2 X1 P = (γ˜⊗1 )X ·(1 ⊗δ) = (RX1(γ)⊗1 )·(1 ⊗δ) = RX1(γ)⊗δ. X2 X1 L1 X2 X1 L1 (cid:3) Proposition 2.8. Statements (ii) and (iii) of Theorem 2.5 hold. Proof. (a) First we note that, since L acts transitively on the qn−1 −1 non-principal irreducible characters of U and q is odd, any irreducible character of P which is nontrivial on U has even degree. Hence all the odd-degree irreducible constituents of χ are contained P in ∗RG(χ). Next, by Lemma 2.6, we already know that s ∈ F× for all i = 1,...,m. L i q Suppose that m = 1. Then χ is a unipotent character of G tensored with a linear character; in particular, χ belongs to the principal series. By the Comparison Theorem [HL, Theorem 5.9] (see also [C, Theorem 5.1] for the case of the principal series), the computation of ∗RGL(χ)can bereplaced by the computation of (χλ1)Sn−1, whereweidentify S ,respectivelyS ,withtheWeylgroupofG,respectivelyofL. (Seealso[FS,Proposition n n−1 (1C)] for the explicit formula in the case of G.) Applying Theorem 1.2 and (2.2), we are done in this case. (b) Now wewill assumem ≥ 2 and seta = k , b = n−k , wherek ,...,k satisfy 2.5(i); 1 1 1 m in particular, (2.3) [a] < [b] , 2 2 and so a 6= b. Let M = GL (q)×GL (q) so that χ = RG(α⊗β), where a b M α = S(s ,λ ), β = S(s ,λ )◦...◦S(s ,λ ). 1 1 2 2 m m This follows by the transitivity of the Harish-Chandrainduction [DM, Proposition 4.7]. By the Mackey formula for Harish-Chandra induction and restriction (see e.g. [DF, Theorem 1.14]), ∗RGχ = ∗RG(RG(α⊗β)) =RL ∗RM (α⊗β) L L M L∩M L∩M (2.4) (cid:16) (cid:17) ⊕RLL∩wMw−1 conjw(∗RMM∩w−1Lw(α⊗β)) , (cid:16) (cid:17) where w is the permutation matrix corresponding to the cycle (1,2,...,a+1) and conj w denotes the conjugation by w. (c) Herewe consider thefirstsummandin theright hand sideof (2.4). By Lemma 2.7(i), ∗RM (α⊗β)= ∗RGLa(q) (α)⊗β. L∩M T1×GLa−1(q) It is well known (see e.g. [L, p. 70]) that the Harish-Chandra induction and restriction respect the Lusztig series, which in our case is labeled by the semisimple element in the RESTRICTION OF ODD DEGREE CHARACTERS AND NATURAL CORRESPONDENCES 7 dual group G∗ ∼=G that has each s ∈ F× as eigenvalue with multiplicity k . Hence we can i q i write r (2.5) ∗RGLa(q) (α) = S(s ,(1))⊗S(s ,µ ) T1×GLa−1(q) 1 1 j Xj=1 for some r ≥ 1 and some partitions µ ⊢ (a−1). It then follows by Lemma 2.7(ii) and the j transitivity of the Harish-Chandra induction that r RLL∩M(∗RML∩M(α⊗β)) = RTL1×GLa−1(q)×GLb(q) S(s1,(1))⊗S(s1,µj)⊗β Xj=1    r r = S(s ,(1))⊗RGLn−1(q) (S(s ,µ )⊗β) = S(s ,(1))⊗γ 1 GLa−1(q)×GLb(q) 1 j 1 j Xj=1 Xj=1 whereγ := S(s ,µ )◦S(s ,λ )◦...◦S(s ,λ )∈ Irr(GL (q)). Certainly, theirreducible j 1 j 2 2 m m n−1 constituent S(s ,(1))⊗γ can be of odd degree only when deg(S(s ,µ )) and |GL (q) : 1 j 1 j n−1 (GL (q) × GL (q))| are both odd. The former implies, by applying (a) to (2.5) that a−1 b µ = λ∗. The latter implies by [NT , Lemma 4.4(i)] that n−1 is odd. j 1 1 a−1 We have shown that the first summand in (2.4) conta(cid:0)ins a(cid:1)t most one irreducible constituent of odd degree, namely S(s ,(1))⊗(S(s ,λ∗)◦S(s ,λ )◦...◦S(s ,λ )), 1 1 1 2 2 m m which can occur with multiplicity ≤ 1 in the summand only when n−1 is odd. a−1 (d) Arguing as in (c), we can show that each irreducible constitu(cid:0)ent(cid:1)in the second summand of (2.4) is of form S(s′,(1))⊗(S(s′,ν )◦S(s′,ν )◦...◦S(s′ ,ν )◦S(s ,λ )), 1 2 1 3 2 m m−1 1 1 where s′ ∈ {s′,...,s′ } = {s ,...,s } and ν ,...,ν are some partitions with the 1 2 m 2 m 1 m−1 lengths totalling to b−1. By [NT , Lemma4.4(i)], such an irreducibleconstituent can have 1 odd degree only when n−1 = n−1 is odd. b−1 a Given the condition(cid:0)(2.3(cid:1)), C(cid:0)orol(cid:1)lary 2.4 implies that n−1 is even. Thus the second a summand in (2.4) contains no odd-degree irreducible const(cid:0)ituen(cid:1)t. Since χ(1) is odd, we are done by the result of (c). (cid:3) Completion of the proof of Theorem 2.5. InviewofLemma2.6,Proposition2.8,andLemma 2.1,itremainstoprovethat|Irr2′(G)| = |Irr2′(P)|whenn = 2k+1. WecanchooseaSylow 2-subgroupS = S ×S < LofG, whereS = O (GL (q)), andS ∈ Syl (GL (q))hasthe 1 2 1 2 1 2 2 2k form S = A≀B, where A∈ Syl (GL (q)) and B ∈ Syl (S ). Then the S-module V decom- 2 2 2 2 k poses as V ⊕V′, where V = he i and all irreducible constituents of V′ = he ,...,e i 0 0 1 Fq 2 n Fq are of even dimension. It follows that N (S) is contained in Stab (V )∩Stab (V′) = L G G 0 G and so N (S) = N (S). By the main result of [O1], G L (2.6) |Irr2′(G)| = |Irr2′(NG(S))| = |Irr2′(NL(S))| = |Irr2′(L)|. As mentioned in part (a) of the proof of Proposition 2.8, |Irr2′(L)| = |Irr2′(P)|, and so we are done. (cid:3) 8 E.GIANNELLI, A.KLESHCHEV,G.NAVARRO,ANDP.H.TIEP Proof of Theorem B. Let G := GL (q), and consider normal subgroups S := SL (q) and n n H := Z(G)S of G. (i) Consider any θ ∈ Irr2′(S). Since H = Z(G)∗S is a central product, θ extends to H which has odd index gcd(n,q−1) in G. Hence θ lies under some χ ∈ Irr2′(G). Now we can find a label (2.1) for χ that satisfies 2.5(i). Applying [KT , Theorem 1.1] to χ, we then see 1 that χS is irreducible, i.e. χS = θ. We have shown that every θ ∈ Irr2′(S) extends to G, when (2.7) |Irr2′(S)| = |Irr2′(G)|/(q −1) as G/S ∼= Cq−1. (ii) We may assume that Q = P ∩S, where P is the maximal parabolic subgroup of G considered in Theorem 2.5. Then Q = UL1 with L1 := L∩S ∼= GLn−1(q). Note that the normal subgroup K := StabS(e1)∩StabS(he2,...,eniFq)∼= SLn−1(q) of L acts transitively on Irr(U) r {1 } since n ≥ 3. It follows that every irreducible 1 U character of Q that is nontrivial on U has even degree. In particular, (2.8) |Irr2′(Q)| = |Irr2′(L1)|. (iii) Now we keep the notation of (i) and consider any odd-degree irreducible constituent ζ of θ (which exists since θ has odd degree). We have shown that ζ is trivial on U and so Q is a constituent of ∗RG(χ). We can beview ζ as an L -character. Any irreduciblecharacter L 1 of Z(G)L = Z(G)∗L lying above ζ then also has degree equal to ζ(1). Next, Z(G)L has 1 1 1 odd index gcd(n,q−1) in L. Thus any irreducible character of L lying above ζ must have odd degree. Applying Theorem 2.5, we now see that ζ is a constituent of (χ∗) , and L1 (χ∗) = S(s ,λ∗)◦S(s ,λ )◦...◦S(s ,λ ) K˜ 1 1 2 2 m m for K˜ := StabG(e1)∩StabG(he2,...,eniFq) ∼= GLn−1(q). Since χ∗(1) is odd, by [NT , Lemma 4.4(i)] and Corollary 2.3 we have that [k −1] , [k ] , 1 1 2 2 2 ..., [k ] are pairwise different; in particular, k −1,k ,...,k are pairwise different. It m 2 1 2 m then follows by [KT , Theorem 1.1] that (χ∗) is irreducible. Hence, (χ∗) is irreducible 1 K L1 and ζ = (χ∗) . L1 (iv) To complete the proof of Theorem B, by Lemma 2.1 it remains to show that |Irr2′(S)| = |Irr2′(Q)|. Since |Irr2′(L)| = |Irr2′(GL1(q)×GLn−1(q))| = (q−1)|Irr2′(GLn−1(q))| = (q−1)|Irr2′(L1)|, we are done by combining (2.6), (2.7), and (2.8). (cid:3) 3. Finite groups with self-normalizing Sylow subgroups In this section we prove Theorem C. In fact, in this case the key hypothesis is the self- normalizing Sylow and not the nature of the prime p. The next result was proved in [N ] 1 for P = H (with a more complicated proof). RESTRICTION OF ODD DEGREE CHARACTERS AND NATURAL CORRESPONDENCES 9 Theorem 3.1. Suppose that G is a p-solvable group, P ∈ Syl (G), and assume that P = p N (P). Let P ≤ H ≤ G. If χ ∈ Irr(G) has p′-degree, then the restriction of χ to H G contains a unique p′-degree irreducible constituent ψ ∈ Irr(H). This establishes a natural bijection between the p′-degree irreducible characters of G and H. Proof. We proceed by induction on |G|. Let χ ∈ Irr(G) be of p′-degree. Let N be a normal subgroup of p′-order. Since χ(1) is not divisible by p, using the Clifford correspondence (Theorem (6.11) of [Is ]), let θ ∈ Irr(N) be P-invariant under χ. Then C (P) = 1 by 2 N hypothesis, and θ = 1 , by the Glauberman correspondence (Theorem (13.1) of [Is ]). N 2 Therefore, N ≤ ker(χ). Now consider χ¯ ∈ Irr(G/N) the corresponding character in the factor group. By induction χ¯ contains a unique p′-irreducible constituent ψ. Now, all NH the constituents of this character restrict irreducibly to H, so we are done in this case. So we may assume that N = 1. Let now N be a normal p-subgroup of G. Then χ contains P a linear constituent λ and λ = ν ∈ Irr(N). Let P ≤ T be the stabilizer of ν in G, and let N ψ ∈ Irr(T) be the Clifford correspondent of χ over ν. It is well-known by Clifford theory (see [S, Lemma 2.5]) that χ = (ψ )H +∆ H T∩H where no irreducible constituent of ∆ lies over ν. We claim that no irreducible constituent of ∆ has p′-degree. If ξ is an irreducible constituent of ∆ of p′-degree, then ξ has some N linear P-invariant constituent ǫ. But then χ has P-invariant irreducible constituents ν N and ǫ, and by a standard argument they are N (P)-conjugate. However, N (P) = P, and G G since both are P-invariant, they are equal. This contradicts the nature of ∆. Suppose now that T < G. Notice that T has a self-normalizing Sylow p-subgroup. By induction, ψ = µ+ρ, T∩H where µ has p′-degree and every irreducible constituent of ρ has degree divisible by p. All these constituents lie over ν, so all of them induce irreducibly to H by the Clifford correspondence. We conclude that χ = (ψ )H +∆ = µH +ρH +∆, H T∩H and the theorem is again proved in this case. Hence the last case is that T = G. We claim that ν extends to G. This is because ν extends to P, by coprimeness it extends to every Q/N ∈ Syl (G/N) if q 6= p (by Corollary q (6.27) of [Is ]), and therefore it extends to G by Theorem (6.26) of [Is ]. Let λ ∈ Irr(G) 2 2 be a linear extension of ν. By Gallagher’s Corollary (6.17) of [Is ], we can write χ = βλ, 2 where β ∈ Irr(G/N), has p′-degree. By induction, β = τ +Ξ, H where every irreducible constituent of Ξ has degree divisible by p. Now χ = λ β = λ τ +λ Ξ, H H H H H and this proves half of the theorem. The bijectivity of the map is proved similarly. (cid:3) Thefact that N (P) = P is key. For instance, take the productof an extraspecial group G F = 31+2 of exponent 3 acted on by C in such a way that C acts trivially on the center + 4 4 10 E.GIANNELLI, A.KLESHCHEV,G.NAVARRO,ANDP.H.TIEP of F. The normalizer of a Sylow 2-subgroup in G is N = C ×C , is even maximal. Every 4 3 restriction of the irreducible characters of degree 3 has three irreduciblelinear constituents. Proof of Theorem C. In view of Theorem 3.1, we may assume p > 2. Decompose χ = H t ρ + s ψ , where t ≥ 1, s ≥ 0, ρ ,ψ ∈ Irr(H), p ∤ ρ (1), p|ψ (1). Now each ρ | i=1 i j=1 j i j i j i P cPontains aPlinear character, and by [NTV, Corollary B], χP = λ+δ contains a uniquelinear character λ (so that either δ = 0 or all irreducible constituents of δ have degree divisible by p). Hence t = 1, and we have a well-defined map ∗ : χ 7→ χ∗ := ρ 1 from X := Irrp′(G) to Y := Irrp′(H). Now, for any ρ ∈ Y, ρG contains a p′-degree χ of G and then χ contains ρ, so χ∗ = ρ. H Thus ∗ is onto. By [NTV, Corollary B], |X| = |P/P′|= |Y|. So ∗ is a bijection. (cid:3) 4. Canonical character correspondences in symmetric groups Note that the natural character correspondences described in [APS] (see Theorem 1.2) and in Theorems A and B are given by restriction. In the case of G = S , one may ask n whether the restriction to a Young subgroup H of odd index in G would also give such a correspondence between the odd-degree characters of G and those of H. Unfortunately, the answer is no: if G = S and H = S ×S , then the restriction to H of any irreducible 7 5 2 character of degree 35 of G have three odd-degree irreducible constituents. When G = Sn and P ∈ Syl2(G), a canonical bijection between Irr2′(G) and Irr2′(P) was defined in [G] (although given by restriction to P only when n is a 2-power). We will constructanewsuchcanonicalbijection,where,inaddition,thecorrespondentχ♯ ∈Irr2′(P) of χ ∈ Irr2′(G) is an irreducible constituent of χP (although not necessarily occurring with multiplicity one). Let H(n) denote the set of n hook partitions (n−i,1i), 0 ≤ i < n, of n. If λ ⊢ n, then Y(λ) is the Young diagram of λ. Furthermore, we will say that λ is an odd partition of n (and write λ ⊢ n) exactly when χλ ∈ Irr(S ) has odd degree. o n Lemma 4.1. Let n ∈Z and let γ be a partition of n with a hook H of length m in Y(γ). ≥1 Let β ∈ H(m) be the hook partition corresponding to H, and let α ⊢ (n−m) be such that Y(α) is obtained from Y(γ) by removing the rim m-hook R of Y(γ) corresponding to H. Then (χγ)Sn−m×Sm contains χα⊗χβ as an irreducible constituent with multiplicity one. Proof. We apply the Littlewood-Richardson rule as given in [JK, Corollary 2.8.14]. Write β = (k,1m−k) for some 1 ≤ k ≤ m and consider the symbols a where the (i,j)-node ij belongs to Y(β): a ,a ,...,a ,a ,a ,...,a . 11 12 1k 21 31 m−k+1,1 We need to count the numberof ways of addingsymbolsa to Y(α) to get Y(γ), that is, to ij fill up the rim m-hook R of Y(γ) with these symbols a in such a way that all conditions ij (i), (ii), (iii) of [JK, Corollary 2.8.14] are fulfilled. First we have to put the k symbols a ,a ,...,a in k different columns of R consecutively starting from the rightmost to 11 12 1k get a proper diagram. Since R has exactly k columns, we have to put these symbols at the top of these k columns, and there is exactly one way to do it. Then we need to put the m−k symbols a ,a ,...,a in the m−k remaining rows of R consecutively 21 31 m−k+1,1

Restriction of odd degree characters and natural correspondences

Similar Restriction of odd degree characters and natural correspondences