Self-dual modules in characteristic two and normal subgroups
aa r X i v : . [ m a t h . R T ] J u l SELF-DUAL MODULES IN CHARACTERISTIC TWO AND NORMALSUBGROUPS
ROD GOW AND JOHN MURRAY
Abstract.
We prove Clifford theoretic results on the representations of finite groupswhich only hold in characteristic 2.Let G be a finite group, let N be a normal subgroup of G and let ϕ be an irreducible2-Brauer character of N which is self-dual. We prove that there is a unique self-dualirreducible Brauer character θ of G such that ϕ occurs with odd multiplicity in therestriction of θ to N . Moreover this multiplicity is 1.Conversely if θ is an irreducible 2-Brauer character of G which is self-dual but not ofquadratic type, the restriction of θ to N is a sum of distinct self-dual irreducible Brauercharacter of N , none of which have quadratic type.Let b be a real 2-block of N . We show that there is a unique real 2-block of G covering b which is weakly regular. Statement of results
Throughout the paper G is a finite group and N is a normal subgroup of G . We fix a2-modular system ( K, R, F ) for G . So R is a complete discrete valuation ring which hasfield of fractions K of characteristic 0 and residue field F = R/J ( R ) of characteristic 2.We will assume that K and F are splitting fields for all subgroups of G . For example,this holds if K contains a primitive | G | -th root of unity. We use r ∗ to denote the imageof r ∈ R in F . Each integer m can be factored as m = m m ′ , where m is a power of 2and m ′ is odd.We use Irr( G ) to denote the irreducible K -characters of G . These have values in acyclotomic subfield of K which can be identified with a subfield of C . So Irr( G ) canbe identified with the irreducible complex characters of G . Next recall that the Brauercharacter of an F G -module is an R -valued class function defined on the 2-regular (oddorder) elements of G . The Brauer characters of the irreducible F G -modules are called theirreducible 2-Brauer characters of G . We use IBr( G ) to denote all such characters. Thedual of a character θ is the character θ defined by θ ( g ) := θ ( g − ), for all g ∈ G . We saythat θ is self-dual if θ = θ . This holds if and only if θ is the character of some self-dualmodule.Let θ be an irreducible Brauer character of G and let ϕ be an irreducible Brauercharacter of a subgroup H of G . We say that θ lies over ϕ if θ is constituent of the inducedcharacter ϕ ↑ G . Likewise we say that ϕ lies under θ if ϕ is a constituent of the restricted Date : July 6, 2020. character θ ↓ H . There is no analogue of Frobenius reciprocity for Brauer characters. Sothe fact that θ lies over ϕ does not imply that ϕ lies under θ , and conversely. Howeverthese implications do hold if H is a normal subgroup of G . See [N, p155 and (8.7)]. Hereis our first result: Theorem 1.
Let ϕ be an irreducible -Brauer character of N . Then ϕ lies under someself-dual irreducible -Brauer character θ of G if only if ϕ is G -conjugate to ϕ . If sucha self-dual θ exists, it can be chosen so that ϕ occurs with odd multiplicity in θ ↓ N . Example:
Let G = h a, b | a = b = 1 , a b = a i be the semi-dihedral group of order16. The two faithful irreducible K -characters µ = µ of h a i are G -conjugate. Now theirreducible K -characters of G lying over µ consist of an irreducible K -character and itsdual. So Theorem 1 does not generalize to irreducible K -characters nor to irreducible p -Brauer characters, for primes p = 2.Our second result is: Theorem 2.
Let ϕ be a self-dual irreducible -Brauer character of N . Then (i) ϕ extends to its stabilizer in G , and exactly one such extension is self-dual. (ii) G has a unique self-dual irreducible -Brauer character θ such that ϕ occurs withodd multiplicity in θ ↓ N . (iii) ϕ occurs with multiplicity in θ ↓ N . We will refer to θ as the canonical irreducible Brauer character of G lying over ϕ . Themodule form of this theorem is stated and proved in Theorem 9 below.Theorem 2 is similar in flavour to a result of I. M. Richards. In [R] he proved that when G/N has odd order, each self-dual irreducible K -character of N extends to its stabilizerin G , and has a unique self-dual extension. Example:
Let G = h a, b | a = b = 1 , a b = a − i be the dihedral group of order 8. Thenthe non-trivial irreducible K -character of h a i is self-dual and G -invariant, but it does notextend to D . So Theorem 2 does not generalize to self-dual irreducible K -characters norto irreducible p -Brauer characters, for primes p = 2.Next recall that a non-trivial irreducible 2-Brauer character of G is said to have qua-dratic type if the corresponding F G -module affords a non-zero G -invariant quadraticform. Our first application is: Theorem 3.
Let θ be a non-quadratic type self-dual irreducible -Brauer character of G which does not lie over the trivial character of N . Then θ ↓ N is a sum of non-quadratictype self-dual irreducible Brauer characters of N , each occurring with multiplicity . The second application is to blocks. For undefined notation, see Section 6 below andfor a full exposition of block theory, see Chapter 5 of [NT].Let B be a 2-block of irreducible K -characters of G . Then the duals of the charactersin B form another 2-block B o , called the contragredient of B . We say that B is real if B = B o . Recall that B is said to cover a 2-block b of N if the restriction of an irreducible ELF-DUAL BRAUER 3 character in B contains an irreducible character in b . Also we say that B is weakly regular(with respect to N ) if it has maximal defect among the blocks of G which cover b . Theorem 4.
Let b be a -block of N . Then (i) G has a real weakly regular block covering b if and only if b and b o are G -conjugate. (ii) if b = b o , then G has a unique real weakly regular block of G covering b . We prove (i) in Lemma 18 and (ii) in Lemma 20.Recall that corresponding to each irreducible 2-Brauer character θ , G has a principalindecomposable character Φ θ . Then Φ θ is a K -character of G which vanishes off the 2-regular elements of G . We use the following result, which is implicit in [GW93, 1.4], toprove Theorem 4. As it may be of independent interest, we include a short proof here: Lemma 5.
Let B be a -block of G . Then B has an odd number of height irreducibleBrauer characters θ such that Φ θ (1) = | G | .Proof. Let D be a defect group of B . Then Brauer showed that dim( B ) | G || G : D | is a unit in R .See [NT, 5.10.1]. Now dim( B ) = P θ ∈ IBr( b ) Φ θ (1) θ (1). It is known that Φ θ (1) / | G | and θ (1) / | G : D | belong to R , for all θ ∈ IBr( B ). So Brauer’s result gives us an identity in F : X θ ∈ IBr( B ) (cid:18) Φ θ (1) | G | (cid:19) ∗ (cid:18) θ (1) | G : D | (cid:19) ∗ = (cid:18) dim( B ) | G || G : D | (cid:19) ∗ = 1 F . The contribution of θ to the left hand side is 1 F , if θ has height 0 and Φ θ (1) = | G | .Otherwise the contribution is 0 F . So the lemma follows directly from the above equality. (cid:3) Many 2-blocks have an odd number of height 0 irreducible Brauer characters. Forexample, the main result of [KOW] is that each 2-block of a symmetric group has aunique height 0 irreducible Brauer character. Furthermore, it is known that each principalindecomposable character Φ of a finite solvable group satisfies Φ(1) = | G | . So Lemma5 implies that each 2-block of a finite solvable group has an odd number of height 0irreducible Brauer characters. However, as B. Sambale has pointed out, the faithful 2-block of 3 . Suz . Real orbits of irreducible Brauer characters
Recall that g ∈ G is said to be real in G if xgx − = g − , for some x ∈ G . Similarlya conjugacy class of G is real if its elements are real, and 2-regular if its elements haveodd order. Now G acts on the conjugacy classes, the irreducible K -characters and theBrauer characters of its normal subgroup N . We say that a G -orbit of conjugacy classesof N is real if its union is a real conjugacy class of G . Likewise we say that a G -orbit ofirreducible Brauer characters of N is real if it contains the duals of its characters. ROD GOW AND JOHN MURRAY
Proof of Theorem 1.
It is clear that each self-dual irreducible Brauer character of G liesover a real G -orbit of irreducible Brauer characters of N .Suppose that G has ℓ conjugacy classes of 2-regular elements, with representatives g , . . . , g ℓ . Let θ , . . . , θ ℓ be the irreducible Brauer characters of G and let Φ , . . . Φ ℓ bethe corresponding principal indecomposable characters of G . The second orthogonalityrelations give X χ ∈ Irr( G ) χ ( g − i ) χ ( g j ) = δ i,j | C G ( g i ) | , for all i, j ∈ { , . . . , ℓ } .Now for all χ ∈ Irr( G ), we have χ ( g j ) = P ℓu =1 d χ,θ u θ u ( g j ), where the d χ,θ u are non-negative integers, called decomposition numbers. Then for all u = 1 , . . . , ℓ , we haveΦ u ( g i ) = P χ ∈ Irr( G ) d χ,θ u χ ( g i ). It is known that Φ u ( g − i ) | C G ( g i ) | ∈ R , for all u, i . So the abovedisplayed equation can be rewritten in R as(1) ℓ X u =1 Φ u ( g − i ) | C G ( g i ) | θ u ( g j ) = δ i,j , for all i, j ∈ { , . . . , ℓ } .In particular the Brauer character table [ θ i ( g j )] of G is a non-singular ℓ × ℓ matrix.(We note that the proof of [N, (2.18)] shows that det[ θ i ( g j )] = ± Q ℓj =1 | C G ( g j ) | ′ .)Suppose that G has r real conjugacy classes of 2-regular elements, which we may assumehave representatives g , . . . , g r . We choose notation so that θ , . . . , θ r are the self-dualirreducible Brauer characters of G . Then the self-dual Brauer character table of G isthe r × r submatrix T := [ θ i ( g j )] of the Brauer character table. Suppose that u ∈{ r + 1 , . . . , ℓ } . Then there is a unique u ∈ { r + 1 , . . . , ℓ } , with u = u , such that θ u = θ u .So Φ u ( g − i ) | C G ( g i ) | θ u ( g j ) = Φ u ( g − i ) | C G ( g i ) | θ u ( g j ), for all i, j . So the contribution of the summands indexedby u and u to (1) is 0 mod J ( R ), and we deduce that r X u =1 Φ u ( g − i ) | C G ( g i ) | θ u ( g j ) ≡ δ i,j mod J ( R ) , for all i, j ∈ { , . . . , r } .As R is a local ring, it follows that T is invertible, with inverse congruent to the r × r -matrix (cid:20) Φ i ( g − j ) | C G ( g j ) | (cid:21) mod J ( R ). In particular det( T ) J ( R ).Now suppose that G has t real conjugacy classes of 2-regular elements which are con-tained in N , with representatives n , . . . , n t . We relabel the θ , . . . , θ r so that the t × t submatrix S := [ θ i ( n j )] of T satisfies det( S ) J ( R ).For i = 1 , . . . , t , let ϕ i be an irreducible Brauer character of N which is a constituentof θ i ↓ N , and set ˆ ϕ i as the sum of the distinct G -conjugates of ϕ i . Then θ i ↓ N = e i ˆ ϕ i ,for some positive integer e i . The non-singularity of S implies that all the multiplicities e , . . . , e t are odd and ϕ , . . . , ϕ t lie in distinct G -orbits. Moreover each of these orbits isreal, as each θ i is self-dual. ELF-DUAL BRAUER 5
By the non-singularity of the Brauer character table of N and Brauer’s permutationlemma, G has t real orbits on the irreducible Brauer characters of N . So ϕ , . . . , ϕ t represent all real G -orbits of irreducible Brauer characters of N .Our work above shows that if ϕ is an irreducible Brauer character of N which is G -conjugate to ϕ , then G has a self-dual irreducible Brauer character θ such that ϕ occurswith odd multiplicity in θ ↓ N . This concludes the proof of Theorem 1. (cid:3) Clifford Theory for self-dual irreducible modules
We prove Theorem 2 in this section. Recall that the dual of an
F G -module V is the F G -module V ∗ := Hom F ( V, F ). We say that V is self-dual if V ∼ = V ∗ as F G -modules.For the reader’s convenience, we begin by stating a module version of Clifford’s Theorem:
Lemma 6 (Clifford 1937) . Let F be an arbitary field, let V be an irreducible F G -moduleand let W be an irreducible submodule of V ↓ N . Set T as the stabilizer of W in G . Then (i) V ↓ N ∼ = e ( W ⊕ · · ·⊕ W n ) for some integer e > , where W , . . . , W n are the distinct G -conjugates of W . (ii) Let U be the sum of all submodules of V ↓ N which are isomorphic to W . Then U is an irreducible F T -module, U ↓ N = eW and U ↑ G = V . (iii) If W is absolutely irreducible, it extends to a projective F T -module ˆ W and U ∼ = P ⊗ ˆ W , for some projective F ( T /N ) -module P . (iv) If W extends to an F T -module ˆ W , then the distinct irreducible F G -modules lyingover W are ( S ⊗ ˆ W ) ↑ G , . . . , ( S t ⊗ ˆ W ) ↑ G where S , . . . , S t are the distinct irreducible F ( T /N ) -modules.Proof. See for example Huppert and Blackburn, Finite Groups II, VII, 9.12. (cid:3)
We would next like to point out that Fong’s Lemma holds for all self-dual irreducible
F G -modules V over all fields F of characteristic 2. In particular F need not be perfect: Lemma 7 (Fong’s Lemma) . Let G be a finite group, let F be an arbitrary field of char-acteristic and let V be a non-trivial self-dual irreducible F G -module. Then V affords anon-degenerate G -invariant alternating bilinear form. In particular dim( S ) is even.Proof. As V is self-dual, it affords a non-degenerate G -invariant bilinear form B . Itmay happen that B is symmetric. If not, set ˆ B ( v , v ) = B ( v , v ) + B ( v , v ), for all v , v ∈ V . Then ˆ B is a non-zero G -invariant symmetric bilinear form. Now the radicalof ˆ B is a submodule of V . As V is irreducible, it follows that the radical is zero. So ˆ B isnon-degenerate.By the previous paragraph V affords a G -invariant non-degenerate symmetric bilinearform, henceforth denoted B . We claim that B is alternating, meaning B ( v, v ) = 0, for all v ∈ V . For suppose otherwise. Set Q ( v ) := B ( v, v ), for all v ∈ V . Then Q is a non-zero ROD GOW AND JOHN MURRAY G -invariant quadratic form on V . Now Q is additive, as for all v , v ∈ V , we have Q ( v + v ) = B ( v + v , v + v )= B ( v , v ) + B ( v , v ) + B ( v , v ) + B ( v , v )= Q ( v ) + Q ( v ) , using B ( v , v ) + B ( v , v ) = 0.Moreover Q ( λv ) = λ Q ( v ), for all λ ∈ F and v ∈ V . Define U := { v ∈ V | Q ( v ) = 0 } .Then our work shows that U is a submodule of V . But U = V , as Q = 0. So U = 0, byirreducibility of V . Let v ∈ V and g ∈ G . Then Q ( gv + v ) = Q ( gv ) + Q ( v ) = 0, as Q isadditive and G -invariant. So gv + v ∈ U , whence gv = v . But then G acts trivially on V ,contrary to hypothesis. This proves our claim.The final statement follows as every symplectic vector space has even dimension. (cid:3) Corollary 8.
Let G be a finite group and let F be an arbitrary field of characteristic .Then the radical of F G has odd codimension in
F G .Proof.
We use rad(
F G ) to denote the radical of
F G , which is the intersection of theannihilators of all irreducible
F G -modules. Suppose first that F is a splitting field for G . Let θ , . . . , θ ℓ be the irreducible Brauer characters of G , with θ , . . . , θ r precisely theself-dual characters, and θ the trivial Brauer character. We havedim( F G ) − dim(rad( F G )) = θ (1) + · · · + θ ℓ (1) . Now θ (1) = 1 and θ i (1) is even for 2 ≤ i ≤ r , by Fong’s Lemma. If i > r , we may pair θ i with its dual, and these two characters have the same degree. It is now clear that θ (1) + · · · + θ ℓ (1) is odd, and the result follows in this case.Now suppose that F is any field of characteristic 2. Set E = F ( ω ), where ω is a primitive | G | ′ -th root of unity in an extension field of F . Then E is a splitting field for G and a finiteseparable extension of F . As F G contains an E -basis of EG , it is a standard fact thatrad( EG ) is the E -span of rad( F G ). In particular dim E (rad( EG )) = dim F (rad( F G )). Thefirst part shows that dim E ( EG ) − dim E (rad( EG )) is odd. So dim F ( F G ) − dim F (rad( F G ))is odd in this case also. (cid:3)
For the rest of this section F is a perfect field of characteristic 2. Here is the moduleversion of Theorem 2: Theorem 9.
Let W be a self-dual irreducible F N -module. Then W extends to its stabi-lizer in G , and there is a unique extension ˆ W which is self-dual.Set V := ˆ W ↑ G . Then V is a self-dual irreducible F G -module, and V ↓ N ∼ = W ⊕· · ·⊕ W n ,where W , . . . , W n are the distinct G -conjugates of W . Moreover V is the unique self-dualirreducible F G -module such that W occurs with odd multiplicity in V ↓ N .Proof. We may assume that W is non-trivial and G -invariant. As W is a self-dual F N -module, it affords a non-degenerate N -invariant bilinear form B : W × W → F . Anapplication of Schur’s Lemma shows that B is unique up to scalars. ELF-DUAL BRAUER 7
Let X : N → GL( W ) be the F -representation given by W . For each g ∈ G , we definethe conjugate representation X g of N by X g ( n ) = X ( gng − ) , for all n ∈ N .Then X and X g are equivalent representations, as W is G -invariant. So there is Y ( g ) ∈ GL( W ) such that Y ( g ) X ( n ) = X g ( n ) Y ( g ) , for all n ∈ N .We choose Y ( g ) = X ( g ), whenever g ∈ N . Now for all g, h ∈ G , we have[ Y ( gh ) − Y ( g ) Y ( h )] X ( n ) = X ( n ) [ Y ( gh ) − Y ( g ) Y ( h )] . So by Schur’s Lemma there is a non-zero α ( g, h ) ∈ F such that(2) Y ( gh ) = α ( g, h ) Y ( g ) Y ( h ) . Then α : G × G → F × is an F -valued cocycle and Y is a projective representation of G which extends X .Next, for all g ∈ G , we define the bilinear form B g : W × W → F by B g ( u, v ) = B ( Y ( g ) u, Y ( g ) v ) , for all u, v ∈ W .Then for all n ∈ N we have B g ( X ( n ) u, X ( n ) v ) = B ( Y ( g ) X ( n ) u, Y ( g ) X ( n ) v )= B ( X ( gng − ) Y ( g ) u, X ( gng − ) Y ( g ) v )= B ( Y ( g ) u, Y ( g ) v ) , as B is X -invariant= B g ( u, v ) . This shows that B g is X -invariant. As B is unique up to scalars(3) B g = λ ( g ) B, for some λ ( g ) ∈ F × .As B is N -invariant, we have λ ( n ) = 1, for all n ∈ N .Now for all g, h ∈ G we have B gh = λ ( gh ) B . On the other hand B gh ( u, v ) = B ( Y ( gh ) u, Y ( gh ) v )= B ( α ( g, h ) Y ( g ) Y ( h ) u, α ( g, h ) Y ( g ) Y ( h ) v ) , by (2),= α ( g, h ) λ ( g ) λ ( h ) B ( u, v ) , by (3).Comparing these expressions, we see that(4) λ ( gh ) = α ( g, h ) λ ( g ) λ ( h ) . Since F is perfect, for each g in G , there exists µ ( g ) ∈ F such that µ ( g ) = λ ( g ). Setˆ Y ( g ) = µ ( g ) − Y ( g ) for all g ∈ G . Then ˆ Y is a projective representation of G whichextends X . Moreover ˆ Y corresponds to the cocycle β ( g, h ) := µ ( g ) µ ( h ) µ ( gh ) − α ( g, h ).Now β ( g, h ) = λ ( g ) λ ( h ) λ ( gh ) − α ( g, h ) = 1 and char( F ) = 2. So β ( g, h ) = 1, for all g, h ∈ G . This means that ˆ Y is an F -representation of G which extends X . ROD GOW AND JOHN MURRAY
If we now consider the action of the elements ˆ Y ( g ) on the bilinear form B , a repetitionof an earlier argument shows that for all u and v in W , and all g ∈ G , B ( ˆ Y ( g ) u, ˆ Y ( g ) v ) = ǫ ( g ) B ( u, v )for some nonzero scalar ǫ ( g ). The fact that ˆ Y is a representation of G , and B is N -invariant now implies that ǫ is a homomorphism G/N → F × .Finally, as F has characteristic 2, ǫ has odd order in the character group of G/N . Soas F is perfect, ǫ = δ for a unique homomorphism δ : G/N → F × . Then if we setˆ X ( g ) = δ ( g ) − ˆ Y ( g ), we find that ˆ X is also an F -representation of G which extends X .Moreover ˆ X is self-dual, as we can easily check that it leaves B invariant.Let ˆ W be the irreducible self-dual F G -module corresponding to ˆ X . So ˆ W extends W .Then S ⊗ ˆ W give all irreducible F G -modules lying over W , as S ranges over all irreducible F G/N -modules. Recall that S ⊗ ˆ W ∼ = S ′ ⊗ ˆ W if and only if S ∼ = S ′ . So S ⊗ ˆ W is self-dualif and only if S is self-dual. Fong’s Lemma implies that dim( S ⊗ ˆ W ) is an even multiple ofdim( ˆ W ), if S is non-trivial and self-dual. So ˆ W is the unique extension of W to G whichis self-dual. The statements about V are now consequences of Lemma 6. (cid:3) We will refer to V as the canonical self-dual irreducible F G -module lying over W .Our Corollary, which is probably known, is an analogue of Richards’ Theorem [R] forirreducible 2-Brauer characters: Corollary 10.
Suppose that | G : N | is odd. Then (i) If W is a self-dual irreducible F N -module, then W ↑ G has a unique self-dual com-position factor, up to isomorphism. (ii) If V is a self-dual irreducible F G -module then V ↓ N is a sum of distinct self-dualirreducible F N -modules.In particular induction-restriction defines a natural correspondence between the self-dualirreducible
F G -modules and the G -orbits of self-dual irreducible F N -modules.Proof.
We may assume that W is G -invariant. Let ˆ W be the unique self-dual irreducible F G -module which extends W . Then all irreducible F G -modules lying over W have theform U ⊗ W , for some irreducible F G/N -module U . But G/N has odd order. So U isself-dual if and only if it is trivial. It follows that ˆ W is the unique self-dual irreducible F G -module lying over W . All composition factors of W ↑ G lie over W . So (i) holds.For (ii), write V ↓ N = e ( W ⊕ · · · ⊕ W t ), where e, t ≥ W , . . . , W t are distinctirreducible F N -modules. Now V ↓ N is a self-dual irreducible F N -module. So for each i = 1 , . . . , t , W ∗ i ∼ = W j , for some j = 1 , . . . , t . The map i j is an involutary permutationof { , . . . , t } . But t is odd, as by Clifford Theory it divides | G : N | . So there exists i suchthat W ∗ i ∼ = W i , whence all W , . . . , W t are self-dual. Now by part (i), V is the uniqueself-dual irreducible F G -module lying over W . It then follows from Theorem 9 that e = 1i.e. V ↓ N = W ⊕ · · · ⊕ W t .The last statement follows from (i) and (ii). (cid:3) ELF-DUAL BRAUER 9
We also have a fairly obvious extension of Theorem 2 to subnormal subgroups:
Corollary 11.
Let S be a subnormal subgroup of G and let U be a self-dual irreducible F S -module. Then there is a unique self-dual irreducible
F G -module V such that U occurswith odd multiplicity in V ↓ S .Proof. We use induction on | G : S | . By Theorem 9 we may assume that S is not a normalsubgroup of G . So there exists S ( N ( G such that S is subnormal in N and N isnormal in G . As | N : S | < | G : S | , our inductive assumption implies that there is aunique self-dual irreducible F N -module W such that U has odd multiplicity in W ↓ S . Let V be the canonical F G -module over W .Now V ↓ N = W ⊕ W ⊕ · · · ⊕ W t is the sum of the distinct G -conjugates of W . As W is self-dual, all the W i are self-dual. By choice of W , U appears with even multiplicity in W i ↓ S , for i = 2 , . . . , t . Since V ↓ S = W ↓ S ⊕ W ↓ S ⊕ · · · ⊕ W t ↓ S , we deduce that U appearswith odd multiplicity in V ↓ S .Now let V ′ be a self-dual irreducible F G -module such that U occurs with odd multi-plicity in V ′ ↓ S . Write V ′ ↓ N = e ( W ′ ⊕ · · · ⊕ W ′ s ), for some odd integer e , where W ′ , . . . , W ′ s are distinct irreducible F N -modules. We claim that one and hence all W ′ i are self-dual.For suppose otherwise. As V ′ ↓ N is self-dual, for each i there is a unique j = i such that W ′ ∗ i ∼ = W ′ j . Then U , being self-dual, occurs with the same multiplicity in W ′ i ↓ S as in W ′ j ↓ S . So U occurs with even multiplicity in e ( W ′ i ⊕ W ′ j ), and hence with even multi-plicity in V ′ ↓ S . This contradiction proved our claim. So e = 1 and V ′ is the canonical F G -module over W ′ . Now we may assume that U appears with odd multiplicity in W ′ ↓ S .So W ∼ = W ′ , by uniqueness of W over U , and then V ′ ∼ = V , by uniqueness of V over W . (cid:3) Remark:
In the context of the Corollary, suppose that S E N E G , and let U be anirreducible self-dual F S -module. Let W be the canonical F N -module over U and let V be the canonical F G -module over W . We claim that that U occurs with multiplicity 1 in V ↓ S . For suppose otherwise. Then V ↓ N has an irreducible submodule W ′ = W such that U occurs with non-zero multiplicity in W ′ ↓ S . As W ′ is G -conjugate to W , it is self-dualand dim( W ′ ) = dim( W ). On the other hand, U occurs with even multiplicity e > W ′ ↓ S . Setting t as the number of distinct N -conjugates of U , we get the contradictiondim( W ′ ) = et dim( U ) = e dim( W ) > dim( W ) . This proves our claim. In view of this, we ask:
Question: do there exist (
G, V ) and (
S, U ), where G is a finite group, S is a subnormalsubgroup of G , V is a self-dual irreducible F G -module and U is a self-dual irreducible F S -module, such that U occurs with multiplicity e in V ↓ S , where e is odd but not 1?4. Irreducible self-dual modules of non-quadratic type
Let V be a non-trivial self-dual irreducible F G -module. Then by Lemma 7, V affordsa non-degenerate G -invariant alternating form B . Let Q : V → F be a quadratic form which polarizes to B . This means that Q ( λv ) = λ Q ( v ) and Q ( v + v ) = Q ( v ) + B ( v , v ) + Q ( v ), for all λ ∈ F and v , v ∈ V . However, contrary to what happens whenchar( F ) = 2, Q is not uniquely determined by B . In particular Q need not be G -invariant.On the other hand, for many F G -modules each G -invariant quadratic form is uniquelydetermined by its polarization: Lemma 12.
Let G be a finite group and let F be an arbitrary field of characteristic .Suppose that V is an F G -module which affords a non-degenerate G -invariant alternatingbilinear form B but V has no trivial quotient. Then V affords at most one G -invariantquadratic form which polarizes to B .Proof. Let Q and Q be G -invariant quadratic forms on V which polarize to B . Then P := Q + Q is a G -invariant quadratic form which polarizes to 2 B = 0. Thus P ( λv ) = λ P ( v ), and P ( v + v ) = P ( v ) + P ( v ), for all λ ∈ F and v , v ∈ V .Set U := { m ∈ V | P ( v ) = 0 } . Then U is a submodule of V . Let g ∈ G and v ∈ V .Then gv + v ∈ U , as P ( gv + v ) = P ( gv ) + P ( v ) = 0. So G acts trivially on V /U , whence U = V by our hypothesis on V . We conclude that P = 0, or equivalently Q = Q . (cid:3) Recall the notion of a canonical irreducible
F G -module introduced after Theorem 9.
Proposition 13.
Let W be a non-trivial self-dual irreducible F N -module and let V bethe canonical self-dual irreducible F G -module lying over W . Then V has quadratic typeif and only if W has quadratic type.Proof. We adopt the notation of Theorem 9. So W has a unique self-dual extension ˆ W to its stabilizer T and V = ˆ W ↑ G . Also V ↓ N is the sum of the distinct G conjugates of W , each occurring with multiplicity 1. So we can identify W with an F -subspace of V .Suppose first that V affords a G -invariant quadratic form Q , and let B be its polariza-tion. So B is a non-degenerate G -invariant alternating form on V . Let W ⊥ = { v ∈ V | B ( v, w ) = 0 , for all w ∈ W } . Then W ⊥ is a submodule of V ↓ N and V ↓ N /W ⊥ ∼ = W ∗ ∼ = W as F N -modules. As W occurs with multiplicity 1 in V ↓ N , we deduce that W ∩ W ⊥ = 0.So the restriction B ↓ W to W is non-degenerate. Moreover the restriction Q ↓ W of Q to W is an N -invariant quadratic form on W which polarizes to B ↓ W . So W is of quadratictype.Conversely, suppose that W affords a non-degenerate N -invariant quadratic form q ,and let b be its polarization. Then b is a non-degenerate N -invariant alternating form on W . We identify W and ˆ W as F -vector spaces. As ˆ W is self-dual and irreducible, it affordsa T -invariant non-zero bilinear form, say b ′ . Now all N -invariant non-zero bilinear formson W are scalar multiples of each other, as W is irreducible. So b is a scalar multiple of b ′ , and in particular b is T -invariant.For t ∈ T , we define a quadratic form q t on W by setting q t ( w ) := q ( tw ) , for all w ∈ W . ELF-DUAL BRAUER 11
It is clear that q t is N -invariant, and also that q t polarizes to b , as b is T -invariant. So q t = q , according to Lemma 12. This establishes that q is T -invariant, and shows that ˆ W is of quadratic type.Next, we may decompose V = ˆ W ↑ G as F -vector space V = ( g ⊗ ˆ W ) ⊕ ( g ⊗ ˆ W ) ⊕ · · · ⊕ ( g n ⊗ ˆ W ) , where g , . . . , g n is a transversal to T in G . By a standard procedure, we may define theinduced forms b ↑ G and q ↑ G on V using b ↑ G n X i =1 g i ⊗ w i , n X j =1 g j ⊗ x j ! = n X i =1 b ( w i , x i ) ,q ↑ G n X i =1 g i ⊗ w i ! = n X i =1 q ( w i ) , for all w i , x j ∈ ˆ W . Then b ↑ G is a G -invariant alternating bilinear form on V and q ↑ G is a G -invariant quadratic form on V which polarizes to b ↑ G . So V is of quadratic type. (cid:3) Proposition 14.
Let V be a self-dual irreducible F G -module and suppose that some self-dual irreducible
F N -module W occurs with multiplicity e > in V ↓ N . Then e is evenand V has quadratic type.Proof. We adopt the notation of Theorem 9. So W has a unique self-dual extension ˆ W toits stabilizer T and V = ( S ⊗ ˆ W ) ↑ G , where S is a self-dual irreducible F ( T /N )-module.Now W occurs with multiplicity 1 in ( ˆ W ↑ G ) ↓ N . So the multiplicity e of W in V ↓ N equalsdim( S ). As e >
1, we deduce that S is non-trivial. But then dim( S ) is even, according toLemma 7. Finally, since S and ˆ W are both non-trivial and self-dual, S ⊗ ˆ W has quadratictype, by the remark below. This in turn implies that V = ( S ⊗ ˆ W ) ↑ G has quadratictype. (cid:3) Remark:
Suppose that U and V are F G -modules which afford non-degenerate G -invariant alternating bilinear forms B U and B V , respectively. According to Sin andWillems [SW, Proposition 3.4] there is a quadratic form Q on U ⊗ F V , which polar-izes to B U ⊗ B V , such that Q ( u ⊗ v ) = 0, for all u ∈ U and v ∈ V . These propertiesuniquely specify Q . For, if u , . . . , u n and v , . . . , v m are bases for U and V , respectively,then for all λ ij ∈ FQ (cid:16)X λ ij u i ⊗ v j (cid:17) := X λ ij λ i ′ j ′ B U ( u i , u i ′ ) B V ( v j , v j ′ ) , where i, i ′ range over 1 , . . . , n and j, j ′ over 1 , . . . , m . Any basic tensor u ⊗ v can be writtenas P α i β j u i ⊗ v j , for scalars α i , β j . Then in the expression for Q ( u ⊗ v ), the term indexedby ( i, j ) , ( i ′ , j ′ ) can be cancelled with the term indexed by ( i ′ , j ) , ( i, j ′ ), for i = i ′ . Likewisepairs of terms with j = j ′ cancel. Finally, all terms with i = i ′ or j = j ′ are zero as B U and B V are alternating. It is now clear that Q is G -invariant. We turn our attention to those irreducible
F N -modules that are not self-dual but are G -conjugate to their duals. To investigate these, we require a familiar concept.Suppose that W is an irreducible F N -module, with stabilizer T in G . Then T ∗ = { g ∈ G | W g ∼ = W or W g ∼ = W ∗ } is a subgroup of G containing T , called the extended stabilizer of W . If W and W ∗ arenon-isomorphic and G -conjugate, then | T ∗ : T | = 2. Otherwise T = T ∗ . Proposition 15.
Let W be an irreducible F N -module which is not self-dual. Then allself-dual irreducible
F G -modules lying over W are of quadratic type.Proof. If W is not G -conjugate to W ∗ , there are no self-dual irreducible F G -modules lyingover W . So we may assume that W is G -conjugate to W ∗ and that | T ∗ : T | = 2.Let V be a self-dual irreducible F G -module lying over W . Then V = U ↑ G , where U isthe unique irreducible submodule of V ↓ T lying over W . Likewise V ∼ = V ∗ = ( U ∗ ) ↑ G . So U ∗ is the unique irreducible submodule of V ↓ T lying over W ∗ . Note that U ∗ = U , as W and W ∗ are not T -conjugate.Set X := U ↑ T ∗ . Then V = X ↑ G , and X is an irreducible submodule of V ↓ T ∗ . Now U is a submodule of X ↓ T . So by uniqueness of U , X is the unique irreducible submodule of V ↓ T ∗ lying over W . Likewise X ∗ is the unique irreducible submodule of V ↓ T ∗ lying over W ∗ . But X lies over W ∗ , as W and W ∗ are T ∗ -conjugate. So X ∼ = X ∗ .Let τ ∈ T ∗ \ T . Then X ↓ T = U ⊕ τ U . But U and U ∗ are non-isomorphic irreduciblesubmodules of X ↓ T . So X ↓ T = U ⊕ U ∗ , whence τ U ∼ = U ∗ .Let B be a G -invariant non-degenerate alternating bilinear form on V , and let X ⊥ bethe orthogonal complement of X in V ↓ T ∗ with respect to B . Then X ∼ = X ∗ ∼ = V /X ⊥ .So X ∩ X ⊥ = 0, by uniqueness of X . This shows that the restriction B X of B to X is a( T ∗ -invariant) non-degenerate alternating bilinear form on X . Since U is irreducible butnot self-dual, it is totally isotropic with respect to B X , and likewise, so is τ U . We define Q : X → F via Q ( u + τ u ) = B ( u , τ u ) , for all u , u ∈ U .Then Q is a quadratic form which polarizes to B X . As Q vanishes on the subspaces U and τ U , it is an example of a hyperbolic form.We now check that Q is T ∗ -invariant. It is certainly T -invariant, as T fixes U and τ U ,and preserves B . Suppose that τ ′ ∈ T ∗ \ T . Then τ ′ u ∈ τ U and τ ′ τ u ∈ U . So Q ( τ ′ ( u + τ u )) = B ( τ ′ τ u , τ ′ u ) = B ( τ u , u ) = B ( u , τ u ) = Q ( u + τ u ) , since τ ′ also leaves B invariant. So Q is indeed T ∗ -invariant.Finally, the induced form Q ↑ G is a G -invariant quadratic form on V which polarizesto the G -invariant non-degenerate alternating bilinear form B X ↑ G . So V is of quadratictype, as required. (cid:3) Proof of Theorem 3.
Let V be a self-dual irreducible F G -module which is not of quadratictype, such that N does not act trivially on V . Write V ↓ N = e ( W ⊕ · · · ⊕ W t ), where ELF-DUAL BRAUER 13 e > W , . . . , W t is a G -orbit of irreducible F N -modules, each of which is irreducibleand non-trivial.It follows from Proposition 15 that each W i is self-dual, and then from Proposition 14that e = 1. So V is the canonical self-dual irreducible F G -module over each W i . ThenProposition 13 implies that each W i has non-quadratic type. (cid:3) We now show how the techniques we have developed above can be employed to obtaina criterion for all self-dual irreducible
F G -modules to be of quadratic type.
Corollary 16.
For each normal subgroup N of a finite group G , all non-trivial self-dualirreducible F G -modules are of quadratic type if and only if the same is true for both N and G/N .Proof.
Suppose first that all non-trivial self-dual irreducible
F G -modules are of quadratictype. Then the same is obviously true for
G/N and we must show that it is true for N . Let W be a non-trivial self-dual irreducible F N -module and let V be the canonicalself-dual irreducible F G -module over W . Then V is irreducible and of quadratic type,from the hypothesis. So W is of quadratic type, according to Proposition 13.Conversely, suppose that all non-trivial self-dual irreducible modules of both N and G/N are of quadratic type. Then for the sake of contradiction, suppose that V is a self-dual irreducible F G -module which has non-quadratic type. Theorem 3 implies that N acts trivially on V . So V is a self-dual irreducible F ( G/N )-module of non-quadratic type,contrary to hypothesis. (cid:3) Irreducible self-dual modules of non-abelian finite simple groups
The proof of Corollary 16 shows that in an inductive approach to deciding whether aself-dual irreducible
F G -module is of quadratic type, the main difficulty lies in solvingthe problem for non-abelian simple groups, and as far as we know, this is an unsolved(and difficult) question. See [W, Remark 3.4(a)].At a simpler, but by no means straightforward, level, we can ask if every non-abeliansimple group has a non-trivial irreducible module of quadratic type. The answer is no, foraccording to [HM], the Mathieu simple group M has no such modules. We were unableto find an explicit reference to the calculations needed to verify this in the literature. Sowe outline a proof here, which only assumes some knowledge of the irreducible Brauercharacters of certain groups.We use the notation and decomposition matrices from [ModAtlas] and character tablesfrom [GAP]. So M has exactly two non-trivial self-dual irreducible Brauer characters φ and φ , with degrees 34 and 98, respectively.Now M has two conjugacy classes of maximal subgroups isomorphic to the alternatinggroup A . The restriction of φ to any A is the sum of an irreducible character ψ of degree20 plus another of degree 14. In turn, the restriction of ψ to A is the sum of an irreduciblecharacter µ of degree 4 plus two irreducible characters of degree 8. Examining the valuesof µ , we see that it is the Brauer character of a representation defined over F . So µ cannot be of quadratic type. For the order of A is greater than the order of each of thetwo orthogonal groups O + (4 , ∼ = S ≀ C and O − (4 , ∼ = S . It now follows from [GW95,Lemma 1.2] that φ is not of quadratic type.Next we observe that φ φ = 2 φ + φ . Here φ and φ = φ are of degree 10, and φ is the trivial Brauer character. Now φ φ is the Brauer character of the ring E of F -endomorphisms of a module affording φ . Let Tr : E → F be the trace map and let W = { A ∈ E | Tr( A ) = 0 F } . Then W is a submodule of E and E/W is the trivialmodule. So W has Brauer character φ + φ . Clearly the identity map I ∈ E spans theunique trivial submodule of W . Now for each A ∈ W , set Q ( A ) as the coefficient of x n − in the F -characteristic polynomial of A . Then Q is a G -invariant quadratic form, withpolarization B ( A, B ) = Tr( AB ), for all A, B ∈ W . In particular I spans Rad( Q ). As I has characteristic polynomial ( x − = x + x + x + 1, we see that Q ( I ) = 1 F . Sothe singular radical Rad ( Q ) is 0. Now [GW95, Theorem 1.3] implies that φ is not ofquadratic type.On the other hand, each of the remaining 25 sporadic finite simple groups does haveirreducible F G -modules of quadratic type. To show this, we need the following knownresult. We prove it here, as we could not locate a self-contained proof in the literature:
Lemma 17.
Let G be a finite group and let F be a perfect field of characteristic .Then each self-dual irreducible F G -module which is not in the principal -block of G hasquadratic type.Proof. Let M be a self-dual irreducible F G -module which is not in the principal 2-block.In particular there is no module W such that soc( W ) is trivial and W soc( W ) ∼ = M .Let B be a G -invariant non-degenerate symplectic bilinear form on M , and let Q be aquadratic form on M which polarizes to B . For all g ∈ G , define Q g ( m ) := Q ( gm ). Then Q g is a quadratic form which polarizes to B , as Q g ( m + m ) = Q ( gm ) + B ( gm , gm ) + Q ( gm ) = Q g ( m ) + B ( m , m ) + Q g ( m ) . Consider the quadratic form Q + Q g . This is additive and satisfies ( Q + Q g )( λm ) = λ ( Q + Q g )( m ), for all λ ∈ F and m ∈ M . As F is perfect, there exists φ g ∈ M ∗ suchthat Q ( gm ) = Q ( m ) + φ g ( m ) , for all m ∈ M . Now for g, h ∈ G , and m ∈ M we have Q ( m ) + φ gh ( m ) = Q ( ghm )= Q ( hm ) + φ g ( hm ) = Q ( m ) + φ h ( m ) + h − φ g ( m ) So φ : G → M ∗ satisfies the cocycle condition φ gh = φ h + h − φ g , for all g, h ∈ G .Now take W to be the Cartesian product M × F , endowed with the obvious F -vectorspace structure. Define an F G -module structure on W via g ( m, λ ) = ( gm, φ g ( m ) + λ ) , for all m ∈ M, λ ∈ F and g ∈ G . ELF-DUAL BRAUER 15
This is an action because for all g, h ∈ G , we have( gh )( m, λ ) = ( ghm, φ gh ( m ) + λ )= ( ghm, φ g ( hm ) + φ h ( m ) + λ )= g ( hm, φ h ( m ) + λ )= g ( h ( m, λ )) . Clearly W has a submodule 0 × F isomorphic to the trivial F G -module F , and W modulothis submodule is isomorphic to M . Our assumption on M forces W ∼ = M ⊕ F as F G -modules. So there is ψ ∈ M ∗ such that m ( m, ψ ( m )), for m ∈ M , is an injective F G -module map M → W .Now on the one hand g ( m, ψ ( m )) = ( gm, ψ ( gm )). On the other hand g ( m, ψ ( m )) =( gm, φ g ( m ) + ψ ( m )). Comparing these expressions, we see that φ g ( m ) + ψ ( gm ) = ψ ( m ),for all g ∈ G . Finally, define the quadratic form ˆ Q on M viaˆ Q ( m ) = Q ( m ) + ψ ( m ) , for all m ∈ M .Then it is clear that ˆ Q polarizes to B . Furthermore, for all g ∈ G we haveˆ Q ( gm ) = Q ( gm ) + ψ ( gm ) = Q ( m ) + φ g ( m ) + ψ ( gm ) = Q ( m ) + ψ ( m ) = ˆ Q ( m ) . So ˆ Q is G -invariant. (cid:3) Using [GAP] and [ModAtlas], the only sporadic finite simple groups which do not havea real non-principal 2-block are M , M , M and M . Now M has an orthogonalirreducible K -character χ , of degree 10, whose restriction to 2-regular elements is aself-dual irreducible Brauer character φ . So φ has quadratic type. Similarly M hasan orthogonal irreducible K -character χ , of degree 252, whose restriction to 2-regularelements contains the self-dual irreducible Brauer character φ with multiplicity 1, butdoes not contain the trivial Brauer character. So φ has quadratic type. Finally, φ restricts to an irreducible Brauer character of a maximal subgroup M . So M also hasa quadratic type irreducible Brauer character.All other simple group whose modular representations are tabulated in the modular[Atlas] have quadratic type irreducible Brauer characters, and we suspect that M maybe unique among all non-abelian finite simple groups in not having such a character. Wenote that the automorphism group of M does have irreducible modules of quadratictype, since Proposition 15 applies to certain irreducible modules of the automorphismgroup that are induced from irreducible modules of M that are not self-dual.6. Real weakly regular -blocks We continue to assume that G is a finite group and N is a normal subgroup of G . Theresults in this section include real refinements of [M, Theorem 4.4, Corollary 4.5]. If C is a conjugacy class of G , then C + is the sum of its elements in RG . Also C o is theclass consisting of the inverses of the elements of C . Each z ∈ Z( F G ) can be written z = P β ( z, C ) C + , where C ranges over the conjugacy classes of G and β ( z, C ) ∈ F . We use standard notation and results on blocks. In particular, corresponding to each 2-block B of G , there is a primitive idempotent e B of the centre Z( F G ) of
F G , an F -algebrahomomorphism ω B : Z( F G ) → F , called the central character of B , and a 2-subgroup D of G called a defect group of B . Then D is only determined up to G -conjugacy, and | D | = 2 d , where d ≥ B . We use Irr( B ) and IBr( B ) to denote theirreducible K -characters and irreducible Brauer characters in B , respectively.Let χ ∈ Irr( B ), let ψ be an irreducible constituent of χ ↓ N and let b be the 2-block of N containing ψ . Then B is said to cover b , and the 2-blocks of N covered by B form a single G -orbit. Set e Gb as the sum of the distinct G -conjugates of e b . Then e Gb is an idempotentin Z( F G ) which is the sum of the block idempotents of all blocks of G which cover b .Recall that B is said to be weakly regular (with respect to N ) if it has maximal defectamong the set of blocks of G which cover b . This happens if and only if B has a defectgroup D such that DN/N is a Sylow 2-subgroup of the stabilizer of b in G .Let χ be a K -character or Brauer character belonging to B . Then χ (1) ≥ | G : D | . Ifequality occurs, we say that χ has height 0. Recall that if χ is an irreducible K -character,its central character is defined by ω χ ( C + ) := χ ( C + ) /χ (1), for all conjugacy classes C of G . It is classical that ω χ ( C + ) ∈ R , and indeed ω B ( C + ) = ω χ ( C + ) ∗ is independent of χ ∈ Irr( B ). Suppose now that θ is an irreducible Brauer character in B which has height0. We claim that for all 2-regular conjugacy classes C of G (5) θ ( C + ) θ (1) ∈ R and (cid:18) θ ( C + ) θ (1) (cid:19) ∗ = ω B ( C + ) . For, it is known that there are integers n χ such that θ ≡ P χ ∈ Irr( B ) n χ χ on the 2-regularelements of G . As χ ( C + ) /χ (1) and χ (1) /θ (1) belong to R , we get θ ( C + ) θ (1) = X χ ∈ Irr( B ) (cid:18) χ ( C + ) χ (1) (cid:19) (cid:18) n χ χ (1) θ (1) (cid:19) belongs to R .Moreover (cid:16) θ ( C + ) θ (1) (cid:17) ∗ = ω B ( C + ) (cid:16) P χ ∈ Irr( B ) n χ χ (1) θ (1) (cid:17) ∗ = ω B ( C + ).Our first result includes a proof of Theorem 4(i): Lemma 18.
Let b be a -block of N . Then the number of weakly regular -blocks of G which cover b is odd. So G has a real weakly regular -block which covers b if and only if b is G -conjugate to b o .Let B be a weakly regular -block of G which covers b . Then β ( e B , C ) ω B ( C + ) = β ( e Gb , C ) ω b ( C + ) , for all conjugacy class C of G contained in N .Proof. The first statement is proved in Lemma 5.1 of [GM], so we merely summarize theargument here. There is a defect preserving bijection between the blocks of G covering b and the blocks of the G -stabilizer of b covering b . So we may assume that b is G -invariant.Let B be as in the statement. In particular e B = e B e b . So 1 F = ω B ( e B ) = ω B ( e b ).Thus there is a conjugacy class L of G contained in N such that β ( e b , L ) ω B ( L + ) = 0 F .Now L is 2-regular, as it is in the support of the block idempotent e b . As e b is a sum ELF-DUAL BRAUER 17 of block idempotents of blocks of G with a defect group contained in D , L has a defectgroup contained in D . But ω B ( L + ) = 0 F . So L has a defect group containing the defectgroup D of B . We deduce that D is a defect group of L .Corollary 3.2 of [GM] implies that β ( e B , L ) = ω B ( L o + ). But ω B ( L o + ) = ω B ′ ( L o + ),for each block B ′ of G which covers b , as L ⊆ N . So, again by Corollary 3.2 of [GM] β ( e B , L ) = β ( e B ′ , L ), if B ′ is in addition weakly regular. On the other hand β ( e B ′ , L ) = 0 F ,if B ′ is not weakly regular. As e b is the sum of the block idempotents of all blocks of G covering b , we see that β ( e b , L ) = β ( e B , L ) ρ , where ρ is the number of weakly regular2-blocks of G covering b . It follows from this that ρ is odd.Suppose that there is a real weakly regular 2-block B of G which covers b . Then B = B o also covers b o . So b is G -conjugate to b o . Conversely, suppose that b is G -conjugate to b o . Then taking contragredients of blocks is an involution on the set of weakly regular2-block of G covering b . As this set has odd size ρ , we deduce that there is a real weaklyregular 2-block of G which covers b .For the last statement, let C be a conjugacy class of G which is contained in N for which β ( e B , C ) ω B ( C + ) = 0 F or β ( e b , C ) ω b ( C + ) = 0 F . As ω B ( C + ) = ω b ( C + ), the argumentabove implies that D is a defect group of C . But then β ( e b , C ) = β ( e B , C ) ρ = β ( e B , C ),as char( F ) = 2. We conclude that β ( e B , C ) ω B ( C + ) = β ( e b , C ) ω b ( C + ). (cid:3) We need one more result before proving part (ii) of Theorem 4:
Lemma 19.
Let b be a real G -invariant -block of N . Then G has a self-dual Brauercharacter φ such that φ vanishes off N and φ ↓ N = e ( θ + · · · + θ t ) where both e and t areodd and θ , . . . , θ t are distinct self-dual height irreducible Brauer characters in b .Proof. Note that we are not claiming that φ is irreducible.Consider the G -set X := { θ ∈ IBr( b ) | θ has height zero and Φ θ (1) = | N | } . Then X has odd size, according to Lemma 5. Also duality is an involution on X . So there is a G -orbit θ , . . . , θ t in X , with t odd and all θ i self-dual and of height 0.Let T be the inertial group of θ in G . Then T contains a Sylow 2-subgroup S of G .As SN/N is a 2-group, θ has a unique extension ˆ θ to an irreducible Brauer character of SN . Notice that ˆ θ vanishes off N , as N contains all 2-regular elements of SN .Set φ := ˆ θ ↑ G . Then φ is self-dual and φ ↓ N = SN : N ] ( θ ↑ G ) ↓ N = e ( θ + · · · + θ t ), where e = [ T : SN ] is odd. Finally φ vanishes off N as ˆ θ vanishes off N . (cid:3) We now prove the uniqueness part (ii) of Theorem 4:
Lemma 20.
Let b be a real -block of N . Then G has a unique real -block which covers b and which is weakly regular with respect to N .Proof. We may assume that b is G -invariant, and we let B be any real weakly regular2-block of G covering b . Let φ be the Brauer character of G defined in Lemma 19. So φ ↓ N = e ( θ · · · + θ t ) where et is odd and θ , . . . , θ t are distinct self-dual height 0 irreducibleBrauer characters in b . Write φ = P µ ∈ IBr( G ) m µ µ , where m µ are non-negative integers.Then φ B := P µ ∈ IBr( B ) m µ µ is the B -part of φ . Let C be a 2-regular conjugacy class of G which is contained in N . Then θ i ( C + ) = θ ( C + ), for i = 1 , . . . , t , as θ i is G -conjugate to θ . So (cid:18) φ ( C + ) θ (1) (cid:19) ∗ = (cid:18) etθ ( C + ) θ (1) (cid:19) ∗ = ω b ( C + ) = ω B ( C + ) , where we have used (5).Next let ˆ e B be the unique idempotent in Z( RG ) with ˆ e B ∗ = e B . Then for all µ ∈ IBr( G )we have µ ( ˆ e B ) = µ (1) or 0 R , as µ does or does not belong to B , respectively. So (cid:18) φ B (1) θ (1) (cid:19) ∗ = (cid:18) φ ( ˆ e B ) θ (1) (cid:19) ∗ = X β ( e B , C + ) ω B ( C + ) = X β ( e b , C + ) ω b ( C + ) = ω b ( e b ) = 1 F . Here in both sums, C ranges over the conjugacy classes of G which are contained in N ,as φ vanishes off N . Also the middle equality arises from the last assertion in Lemma 18.Now for each µ ∈ IBr( B ) with m µ = 0, we have µ ↓ N = e µ ( θ · · · + θ t ), for some integer e µ >
0. Then by the previous displayed equation φ B (1) θ (1) = t X µ ∈ IBr( B ) m µ e µ is an odd integer.As m µ e µ = m µ e µ , it follows that there is a self-dual µ ∈ IBr( B ) such that m µ e µ is odd.Then µ is the canonical irreducible Brauer character of G lying over θ given by Theorem2. As θ determines µ , which in turn determines B , we conclude that B is the only realweakly regular 2-block of G which covers b , as we wished to show. (cid:3) References [Atlas] C. Jansen, K. Lux, R. Parker, R. Wilson,
An atlas of Brauer characters.
Lond. Math. Soc. Mono.New Series, . Oxford Science Publications. The Clarendon Press, Oxford University Press, NewYork, 1995.[GAP] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0 ; 2020, .[G] R. Gow, Real valued and 2-rational group characters,
J. Algebra (1979) 388–413.[GM] R. Gow, J. Murray, Real 2-regular classes and 2-blocks, J. Algebra (2) (2000) 455–473.[GW93] R. Gow, W. Willems, Quadratic geometries, projective modules and idempotents,
J. Algebra (1993) 257–272.[GW95] R. Gow, W. Willems, Methods to decide if simple self-dual modules over fields of characteristic2 are of quadratic type,
J. Algebra (1995) 1067–1081.[Gr] J. A. Green, On the indecomposable representations of a finite group,
Math. Zeit. (1959) 430–445.[HM] G. Hiss, G. Malle, Low-dimensional representations of quasi-simple groups, Lond. Math. Soc. J.Comput. Math. (2001) 22–63.[KOW] M. Kiyota, T. Okuyama, T. Wada, The heights of irreducible Brauer characters in 2-blocks ofthe symmetric groups, J. Algebra (2012) 329–344.[M] M. Murai, Block induction, normal subgroups and characters of height zero,
Osaka J. Math. (1994) 9–25.[ModAtlas] The Modular Atlas homepage. (2018). .[NT] H. Nagao, Y. Tsushima, Representations of finite groups , Academic Press, Inc. , 1989.
ELF-DUAL BRAUER 19 [N] G. Navarro,
Characters and blocks of finite groups,
London Math. Soc. Lecture Notes Series ,Cambridge Univ. Press (1998), 287pp.[R] I. M. Richards, Characters of groups with quotients of odd order,
J. Algebra (1985) 45–47.[SW] P. Sin, W. Willems, G -invariant quadratic forms, J. reine angew. Math. (1991) 45–59.[W] Wolfgang Willems, Duality and forms in representation theory,
Representation theory of finite groupsand finite-dimensional algebras (Bielefeld, 1991) , Progr. Math., , Birkhuser, Basel, 1991, 509–520. School of Mathematical Sciences,, University College Dublin, IRELAND
E-mail address : [email protected]
Department of Mathematics and Statistics, Maynooth University, IRELAND
E-mail address ::