Symmetric forms, idempotents and involutary antiisomorphisms

P. Landrock and 0. Manz Nagoya Math. J. Vol. 125 (1992), 33- 51 S YM M ETRIC FORM S , ID EM P OTEN TS AN D IN VOLU TARY AN TΠ S OM ORP H IS M S PETER LANDROCK AND OLAF MANZ Introduction Let G be a finite group, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA F a field and M an irreducible F[G ]- module. By we denote the F - linear involutary antiautomorphism of F[G], induced by inversion on group elements. Suppose t h at char (F) Φ 2. We then show th at M carries a non- singular G - invariant symmetric bilinear form with values in F if and only if there exists a A- invariant idempotent e e F[G] which generates the projective cover of M. This extends earlier results of W. Willems [Wi]. The assertion is not true if c h a r ( F ) = 2. We even consider this question in th e class of those finite- dimensional algebras which admit an .F- linear involutary antiautomorphism r and which are symmetric with respect to a r- invariant symmetric functional. Besides group algebras, also involutary semi- simple F - algebras belong to t h at class. Λ In th e final part of this paper, we let G be represented irreducibly and orthogonally on a real vector space M. We then show t h at there is a relationship between G - orbits on the un it sphere of M and idempotents e e R[G ] such t h at M ^ R[G ]e and e = e. This has some connection to a problem in Coding Theory, namely to find G - orbits on the un it sphere whose minimal Euclidian distance is considerably large. § 1. Involutary and symmetric algebras Let A be a finite- dimensional F - algebra over a field F. We set A = A/ J(A), where J(A) denotes th e Jacobson Radical of A. ΐn th e following, each A- module should be understood as a finitely generated A- Ze/ ΐ- module. 1.1 LEMMA. Let e, f e A be primitive d ί empotents such that ef Φ Then the following assertions hold. 0. Received Ju ly 9, 1990. 33 Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 34 PETER LAN D ROCK AN D OLAF M AN Z (a) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The map Ae —> Af, ae^- ^ae f, is an A- module isomorphism. (b) fAe = fAf (as F- vector spaces). (c) fAe is a local algebra sί omorphic to fAf, via the algebra- isomorphism fae H- > foe • / (a e A). Proof, (a) Since ~ef Φ 0 and both Ae and A/ are irreducible, the map Ae- +Af, ae- >ae- f, is an isomorphism. Consequently, Ae ^ Af via αβ *- >ae- f (cf. [H B; VII, 11.6]). (b) I t follows from (a) th at fAf = Hom A(A/ , A/ ) ^ Hom A(A/ , Ae) = / Ae (as F- vector spaces). (c) By (a), the map fae *- +fae- f (a e A) is a vector space monomorphism between fAe and fAf, and (b) implies th at it even is an isomorphism. The assertion now follows from (fae)(fbe)f = (fae)f. (fbe)f (a, be A). D If A admits an F - linear involutary antiautomorphism τ , we call (A, τ ) an involutary F- algebra. Observe th at τ leaves J(A) in varian t and th us τ induces an involutary antiautomorphism on A. Let V be an A- module over an involutary F - algebra (A, τ ) . An F - bilinear form < , >: Vx V- >F is called a τ - form if it is non- degenerate and if (av, w} = yζ , aτ w} for all ae A, v, w e V (i.e. th e adjoint mapping of a with respect to <( , ) is given by aT ). 1.2. LEMMA. Let (A, τ) be an involutary F- algebra and M an irreducible A- module. (a) / / there exists a primitive idempotent f e A such that M = Af and f f φ 0, then there even exists a primitive idempotent e e A such that M = Ae and eτ = e. Moreover, e can be chosen an the 1- element in fAfτ . (b) Let M carry a τ - form < , ) . If M contains a non- sί otropίc vector x, then there exists an idempotent feA which satisfies M— Af, fτ f φ θ and fx = x. τ Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYMMETRIC FORMS, IDEMPOTENTS 35 Prcof. (a) By Lemma 1.1 (c), th e mapping fAΓ- + fAf, faf*~faf*.f, is an algebra- isomorphism. Let eefAfτ be th e preimage of fefAf. Then τ e is a primitive idempotent and is th e 1- element of fAf\ Since fAf is τ r- invariant, we also have e = e. F inally ae ι- » αe • / = af (a e A) yields Ae = A/ , hence M ~ Ae. (b) We consider th e map A —> M, α (- > αx. Then there exists a primitive idempotent / e A such th at A(ϊ — f) is its kernel. Consequently, M = A/ and / x = x. Since 0φ <x, x> = </x, fx) = </</*, x> = f / ^ 0 follows. • We denote by P(V) th e projective cover of an A- module V. 1.3 TH EOREM. L et (A, τ) 6β an inυ olutary F- algebra. Suppose that the irreducible A- module M carries a symmetric τ - form < , >. If c h ar( F ) Φ 2, then there exists a primitive idempotent e e A swcΛ that eτ = e and P(M) s Ae. Proof. Since char (F ) ^ 2, th e symmetric form < , > is n ot symplectic. Therefore, Lemma 1.2(b) applies and Lemma 1.2(a) yields an idempotent e e A such th at eτ = e and M ^ Ae. H ence P(M) = Ae. Π We shall see in Example 3.1 th at th e hypothesis c h a r ( F ) Φ 2 is n ot superfluous. It is well- known t h at an idempotent d = d + J(A) can be lifted to an idempotent e e A which is a polynomial in d with integer coefficients. This observation applies to our question about τ- invariant idempotents as follows. 1.4 PROPOSITION . Suppose that (A, τ ) is an involutary F- algebra. If d = d + J(A) is a τ - invariant idempotent in A, then there exists a τ nί varίant idempotent e e A such that e — d. Proof. We may assume t h at d is τ- invariant. Otherwise namely d τ τ 2 can be replaced by dd% because dd = dd — d = d. Arguing by induc2 tion, we may as well assume th at J( A) = 0. We set e = 3d2 — 2d\ Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 36 PETER LAN D ROCK AN D OLAF M AN Z Then e2 = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA e + (d 2 - d)\ 2d + l)(2d - 3) = β, because d2 - de J(A). Since Γ e = e and e = d, th e assertion follows. • It is clear t h at in Proposition 1.4, J(A) can be replaced by any τin varian t nilpotent ideal / . A finite- dimensional F - algebra is called symmetric if there exists a functional ω e H om F (A, F) which satisfies φ ( ab) = ^>(6α) for all a, be A and which does not contain any non- zero right- (or left- ) ideal of A in its kernel. We call φ a symmetric functional for A. (Equivalently, A can be characterized by a non- degenerate symmetric associative F - bilinear form ( , ): A X A ~- > F. Observe t h at then (α, b) = φ ( ab). But we prefer to work with the functional .ψ ) Let (A, τ) be an involutary F - algebra which is symmetric with respect to ψ e H om F (A, F). We call (A, τ , φ ) a. symmetric involutary algebra if φ ( a τ ) = φ ( a) for all a e A. 1.5 LEMMA. Let (A, τ, φ ) be a symmetric involutary e e A is an d ί empotent satisfying eτ = e, then (υ , w} = φ ( vw τ ), F- algebra. If υ , w e Ae, defines a symmetric τ - form on Ae. Proof. Suppose t h at 0 = (v0, w} = φ {\ι w τ ) for all w = ae e Ae. Since eT = e, we obtain 0 = φ ( v ϋ eaτ ) — φ ( v Qaτ ) for all ae A, and £> contains the right- ideal v0A in its kernel. Thus vQ = 0 and < , > is non- degenerate. Since φ is τ- invariant, we have (v, w} = <p((vwT ) τ ) = 9?(α;ι;Γ) = <w;, u> and < , > is symmetric. Let ae A. Then <αu, w} = <p(avwτ ) = φ ( vw ta) = φ ( v(a T w) 7) — <u, aτ uf) and < , > is a τ- form. • The following is inspired by [CR; 9.17], where the element z is defined for semi- simple algebras over a splitting field. 1.6 L E M M A. Let A be a symmetric algebra with respect to ψ e H o m F ( A, F), and let M be an irreducible A- module with character β e H o m F ( A, F). For dual bases {α lr , un} and {bl9 *• - ,&«} of A (i.e. φ {a tb^ = δij), we set Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYMMETRIC FORMS, IDEMPOTENTS z = Σ βiaΰ bi We then have: (a) β( a) = φ ( za) for all (b) z e Z(A) Π soc (A). (c) 37 6A . aeA. Let e e A be a primitive idempotent such that M = Ae. Then the map Ae - > soc (Ae), ae >- > aez, is an A- module isomorphism. (d) Suppose t h at (A, τ , φ ) is a symmetric involutary F - algebra and T x assume t h at β is r- invariant (i.e. β( a ) — β( a) for all a e A). Then z = z. Proof, (a) Observe t h at φ ( za 3) = Σ?= i β( )βι Φψia ,/ = 1, 3) = β ( a ά ) for , n. Since {α 1? , α j is an F- basis of A, the assertion follows, (b) Let c e J(A). By (a), φ ( zc) = β( c) = 0 and φ contains the rightideal zJ(A) in its kernel. Thus z e ann (J(A)) = soc (A). If α, 6 e A, then Therefore, again (a) shows φ ( a zb) — <p(zb a) = /3(6α) = β( ab) = φ ( zab). (α^ — 2α)A < ker(9) and az = a α for all α e A. (c) Let ε e A be any lift of the Wedderburn idempotent ε e A, corresponding to M. F or each α e A, we th us obtain • (ψ za) = j9(α) = j8(αe) = ^(^α ε) = φ ( ε za) . This implies t h at ε z = z and β2 ^ 0. Since soc (Ae) is irreducible and z e Z(A) Π soc (A), the map Ae - » soc (Ae), αβ ι- > aez = aze, is an isomorphism. (d) Since φ is r- invariant, we have φ ( a\ b)) — φ ( (a,ibj) τ ) — φ ( a ί bj) — δ i} and {αί, , α^}, {bl, - - , bT n) are dual bases of A as well. Since also β is assumed to be r- invariant, we obtain We now apply (a) to both z and z\ for all α e A, i.e. z = zτ . Γ Consequently, (ψ za) — β( a) = ^(2 α) Π If (A, τ ) is an involutary F - algebra and M an A- module, then M * = H om F (M, F) becomes an A- left- module by (aa)(m) = a(aτ m), a e A, m e M, a e M *. The module M * is called the d^αZ module of M (with respect to τ). It is easy to see t h at M is self- dual (i.e. M * = M ) if and only if M carries a Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 38 PETER LANDROCK AN D OLAF MAN Z τ- form (cf. [H B; VII, 8.10]). Our next aim is to "lift" symmetric r- forms from zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ P(M) to M. 1.7 PROPOSITION . Let (A, τ , φ ) be a symmetric involutary F- algebra. For a primitive idempotent e e A, suppose that Ae carries a symmetric τ - form ( , >. Then M ^ Ae = soc(Ae) as well carries a symmetric τ - form. Proof module We may assume th at Ae is reducible and consider the subsoc (Ae) 1 = {veAe\ (v, w} = 0 for all w e soc (Ae)}. Since dim (soc (Ae) 1) = dim (Ae) — dim (soc (Ae)) = dim (J(A)e), we conclude soc(Ae) 1 = J(A)e. As Ae is assumed to carry a τ- form, Ae is self- dual and th us M= M *. If β denotes th e F - eharaeter of M, it follows t h at β( a τ ) — β( a) for all ae A. F or dual bases {α£} and {bj} of A, we consider z = ίΣ β( ai)°t a n ( i define a bilinear form { , ) ; on M by Since by Lemma 1.6(c), x H- > XZ is an isomorphism from Ae onto soc (Ae), and since soc (Ae) 1 = J(A)e, the form < , >' is well- defined. Suppose t h at 0 = (x, 3>o>' = (xz, y0} for all x e Ae. Thus xz run s through the whole of 1 soc (Ae) and y0 e soc (Ae) = J(A)e. Therefore, yQ = 0 and < , y is nondegenerate. Since obviously (ax, y}; = (x, aτ y}' for all ae A, itrem ains to show t h at ( , >' is symmetric. By part (b) and (d) of Lemma 1.6, T z = ze Z(A), and therefore (x>y)' = (xz, y} = (zx, y) = <x, zT^> = <x, 2:,y> = <y2, x> = (y, x}' for all x,y e Ae. This completes the proof. • We are now able to formulate our main result. 1.8 TH EOREM. Let (A, r, φ ) be a symmetric involutary F- algebra, and M an irreducible A- module. If char (F) Φ 2, then the following statements are equivalent. (1) P(M) carries a symmetric τ - form. (2) M carries a symmetric τ - form. (3) (4) There exists an idempotent ee A such that M = Ae and eτ = e. τ There exists an idempotent ee A such that P(M) ^ Ae and e = e. Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYMMETRIC FORMS, IDEMPOTENTS Proof. (1) (2)^(3): (3)= >(4): (4)= φ (l): 39 z=> (2): Proposition 1.7. Theorem 1.3. Proposition 1.4. Lemma 1.5. Π Recall t h at th e symmetry of A is not needed for (2) => (3) and (3) Φ (4), and t h at char (F) φ 2 is only relevant for (2) => (3). § 2. SomezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA applications As our main application, we consider the group algebra F[G] of a finite group G over the field F. Then = Σ a8g~ x α = Σ agg^ά geσ gβG is an F - linear involutary antiisomorphism of F[G]. Moreover, λ { e HomF(F[G], F) defined by λ ^ a) = ax is a symmetric functional on F[G]. A Since X0) = λ ^ a) for all aeF[G], (F[G], , λ t) is a symmetric involutary F - algebra. Let V be an F[G ]- module and < , ) a A- form on V. Then (gυ , gw} = (υ , w} for all υ , w e V, g e G. Thus th e Λ- forms on V" are just the G - invariant non- degenerate i^- bilinear forms on V. 2.1, COROLLARY. Theorem 1.8 Λo/rfs for (A, r, ^) = CF[G], A, ^ ) . The Corollary extends earlier results of W. Willems. H e showed in his (unpublished) dissertation [Wi; 2.19] t h at F[G]e (for a primitive idempotent e) carries a symmetric G - invariant non- degenerate F - bilinear form if and only if there exists deF[G] such t h at d = d and F[G]e ^ F[G]d. Observe t h at it is easy to see th at (F[G]e)* ^ F[G]e (cf. [OT; Lemma 1]) and hence e = e implies the existence of a G - invariant non- degenerate F - bilinear form on F[G]e. We generalize the approach above. Let H < G be a subgroup of G such th at c h a r ( F ) ||iff|. Then / = / * = (1/ |# |) Σ ΛG ITΛ is an idempotent of ^ [G ], and F[G]f is a transitive permutation module. Its endomorphismring EndFίG1 (F[G]f) ^ nΛ ti fF[G\ f = : hF(H, G) = 6 is called Hecke algebra. Observe th at H = 1 implies t h at 6 = JF [G ]. We Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 40 PETER LAN D ROCK AN D OLAF M AN Z choose representativeszyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA xt == 1, rc2, , xt for (£Γ, i7)- double cosets in G and Xi set in d fo ) = \ H: HC) H\ . Then 23 = [fxj\ i = 1, , ί} is an F- basis for b. Let a — 2ΐ= i QiifXif) e b (α* e F ) . Then b is a symmetric algebra with respect to ^ eH o m F ( b, F ) , defined by φ ( a) = α 1? Also, {i n d ^ ) - / ^ 1 / ^ = 1, , £} is a dual basis of 93, whence α^ = φ ( a n Λ d(x jj- fxj^), j = 1, , ί. (This paragraph is a special case of [CR; 11.30 (i) and (Hi)].) We define τ to be the restriction of A on b. Since / is A- invariant, τ is an involutary antiisomorphism on b. N ote t h at aτ = U Σ \ «*(/#*" V)We now expand aτ in terms of S3, say α τ = X^ = 1 &*(/#*/), ^ e F . Since indix^'fx^f = f = fxj, it follows from the previous paragraph t h at ^(α Γ) = 6i = 9>(Σί 6^ (/ x,/ ) • /) = ^ ( 2 , α, (fxΐ ι f)./ ) = α! = 9>(α) for α e b. Consequently, (b, r, 9) is a symmetric involutary F - algebra. 2.2. COROLLARY. Lei h be a Hecke algebra over F and τ the involutary antiisomorphism induced by A. Then Theorem 1.8 holds for (b,τ fφ ) . Theorem 1.3 clearly can be applied to any no m atter whether A is symmetric or not. symmetric with respect to φ , ψ e H om F (A, F ) . (A, τ , φ ) is a symmetric involutary F - algebra, involutary F - algebra (A, r), Suppose however th at A is It might then happen t h at but (A, r, )ψ is not. 2.3. EXAM PLE. Let q be an odd prime power. Set A = GF(q2) and consider A as an algebra over F = GF(q). Let τ be the F robenius automorphism of A over F . Then τ is an F - linear involutary (anti- ) isomorphism of A. F or ae A, we consider φ ( a) = trA/ F(a) = aτ + a an d ψ(α ) = aτ — a. Then φ ( A) = F , and ψ(α) = 0 if and only if aeF. Thus both φ and ψ are symmetric functionals for A. H owever φ ( a τ ) = ^(α ), but ψ(α τ) = — ψ(α) for all ae A. The situation of Example 2.3 is typical. N amely let (A, τ ) be an involutary F - algebra with center Z = Z(A). Suppose t h at Z is a field and consider the subfield i£" of Z, consisting of all r- invariant elements. Then Z\ K is a separable field extension of degree at most 2 (see [Al; X, Thm. 10]). We do not exploit this further on. If φ is a symmetric functional for A, th en it is easy to see t h at any other symmetric functional ψ is given by (ψ a) = φ ( za) for all ae A, Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYM M ETRIC F ORM S, I D EM P OTEN TS 41 wherezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA z is a central element of A. The following fact as well is easy and will be needed later on. 2.4 LEMMA. L et A be a symmetric F- algebra with respect to φ e H om ^A, F). If x is an inυ ertible element in Z(A), then φ x e H om F (A, F) defined by φ x(a) = φ ( xa) 9 ae A, as well is a symmetric functional for A. Proof. Since x e Z(A), we have φ x(ab) —φ ( xab) = φ ( bxa) = φ ( xbά) = φ x{ba) If φ x h as t h e right ideal cA in its kernel, then φ ( xcA) Since x is invertible, it follows t h at c = 0. for all a, be A. = 0, whence xc = 0. • We now consider a semi- simple algebra S. Recall t h at S then is symmetric (cf. [CR; 9.8]). By Wedderburn's Theorem, n S = 0 Mat m .(Z) ί) with finite- dimensional skew- fields Dt. i= l It M is an irreducible S- module, it belongs to a unique Wedderburn component of S. Thus in view of an application of § 1, we may assume th at S = M at w(D ) is simple. 2.5. PROPOSITION . L et D be a finite- dimensional skew- field over F and assume that (D, ή) is an involutary F- algebra. Then there exists X e Hom F (Z), F) such that (/ ), r, X) is a symmetric involutary F- algebra. Proof. Set Z = Z(D). Case 1: Suppose t h at η induces th e identity on Z. We then consider D as a Z- algebra and pick a symmetric functional φ e Hom^(Z), Z). Let L be a splitting field for D. Since D is centrally simple over Z, it follows th at L ®z D = M at w(L) for some n e N . Since both φ and η are Z- linear, we can define φ , r) e H om L (M at n (L), L) by a = id L (x) y> and ή = id L (x) 57. Then 37 is an involutary antiisomorphism on Mat7Z(L) and ψ satisfies iψ iXiMyij)) - 9((Jΰ)(^;)) for all ( Xij), (ytj) e M at n (L). It follows t h at φ is (up to some F - scalar factor; th e trace on M at n (L). Since (x3ί ) >-> (JC^)' is an L- algebra automorphism on M at n (L), an elementary version of th e Skolem- N oether Theorem implies t h at there exists an invertible matrix (ckl) e Matn(L) such t h at ( Xijy = (ckl)'\ xjt)(ckl) for all ( Xij) e M at n (L). Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 42 PETER LANDROCK AN D OLAF M AN Z In particular, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ^(xtjy) = φ ( (Xij)) for all (xtj) e M at n (L). Consequently, we have for all ae L and d e D a (g> φ ( d) = φ ( a <g) d) = $((α <g) d)^) = £>(α <g) d*) = a i.e. Let /i e H OΠ IF(Z, F ) be any non- zero functional. HomF(D, F). Since X Φ We set X = µ φ e 0, th e skew- field Z) is a symmetric F~algebra with v respect to X. As X(d ) = X(d) for all de D, th e assertion of th e Proposition holds in th e first case. Case 2. Suppose now t h at η is n ot th e identity on Z. Letx e HomF(JD, F) be any symmetric functional on D. instead ψ e H om F (D , F) <p(dv) = (ψ d). If φ and we are done. If X is n ot ^- invariant, we consider defined by φ (d) = %(d) + χ(d'), d e D. We may th us assume t h at X(dη ) — —X{d) for all and also t h at char (F) Φ deD, 2. We consider th e F - linear map η exists 0 Φ Clearly, 0, then (Z>, 37, ^) is a symmetric in volutary F - algebra, Φ z η x e Z such t h at x = —x. e Ή .om Since η (Z, Z). F z Φ id z , there By Lemma 2.4, Xx e H o m F φ , F) de- fined by Xx(d) = X(xd), d e D, is a symmetric functional on Z) as well. N ow χ^ Φ ) = X(xΦ ) = X(xd) = Xx(d) for all d e D , and = X(—χiφ )=—X((xdy) the proof is complete. • In order to extend Proposition 2.5 to simple algebras S = M at w(D ), we use the fact th at any involutary antiisomorphism on S can be written as an involutary antiisomorphism on D followed by transposition and conjugation of matrices. 2.6 THEOREM. L et S be a simple finite- dimensional F- algebra which admits an F- linear involutary antiisomorphism τ. a skew- field D, Z = Z(S) = Z(D)Ί (a) [Al; X, Thm. 11] Then τ Set S = Mat w(Z)) with τ S and K = {ze Z\ z = z). induces an involutary antiisomorphism η 7] on D such that K — {ze Z\ z — z). (b) (dυ y [Al X, Thm. 10] = (cuy\ d]d{cu) 2.7 THEOREM. There exists a non- singular (ckl) e S such that for all (di3) e S = M at w(D ). L et S be a finite- dimensional simple F- algebra which admits an F- linear involutary antiisomorphism τ . H om F (S, F) such that (S, r, )ψ Then there exists ψ e is a symmetric involutary F- algebra. Proof. Set S = Matm(Z)) for a finite- dimensional skew- field D. By Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYMMETRIC FORMS, IDEMPOTENTS 43 Theorem 2.6, there exist an invertiblezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE (cu) e S and an F - linear involutary antiisomorphism η on D such t h at (d 4 i )' - (ckl)- \ dh)(cu) for all (dtj) e S. By Proposition 2.5, there exists X e H om F (D , F) such t h at (D , 37, %) is a symmetric involutary F - algebra. We set φ = X- tr eΐlom F(S, F). Then φ is a symmetric functional on S, and for (d*j) e S we have ?((cU r ) = ?>((<*?,)) = Σ « = Σ *(<***) = φ iidij)) . This establishes the claim. • 2.8 COROLLARY. Let (S, τ ) be an involutary F- algebra. If S is semisimple and char (F) Φ 2, then the following assertions are equivalent for an irreducible S- module M. (1) M carries a symmetric τ - form. (2) There exists an idempotent e e S such that M = Se and eτ = e. Proof. Let 1 = βj + + ε t be th e decomposition of 1 e S in to Wedderburn idempotents ε^. Then τ permutes th e εέ. Observe t h at under each of the conditions (1) and (2), the idempotent ε t corresponding to M is fixed. Thus th e assertion follows from Theorems 1.8 and 2.7. • F or more examples of involutary algebras we refer to [Al; chap. X]. § 3.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Absolutely irreducible (7- modules 3.1 EXAM PLES, (a) Let (A, τ ) be an involutary F - algebra and M an absolutely irreducible A- module. If M carries a symplectic r- form < , ), then it is very easy to see t h at there does not exist an idempotent e e A such t h at eT = e and M = Ae: Suppose there is such an idempotent e. Since < , ) is non- degenerate, we find ae A such t h at (ae, e) Φ 0. Since M is absolutely irreducible, eae = µ efor some µ e F. Consequently, 0Φ (ae, e> = <eταβ, β> = (eae, e) = µ (e,e> = 0, a contradiction. (b) Let (A, τ ) be an involutary F - algebra, c h a r ( F ) = 2 and M an absolutely irreducible A- module with symmetric r- form ( , ) . If dim F M > 2, counting yields a non- zero isotropic vector in M. Since the isotropic Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 44 PETER LAN D ROCK AN D OLAF M AN Z vectors in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA M form a submodule of M, th e form < , > is symplectic. By (a), there does not exist ee A such t h at eτ — e and M — Ae. Thus th e assertion of Theorem 1.3 is false for c h a r ( F ) = 2. (c) N evertheless, if F is not a splitting field, there might exist such an idempotent. As a trivial example, consider A — F2[CZ] can let M be its 2- dimensional irreducible module. Clearly, there exists exactly one primitive idempotent ee A such t h at M^ Ae. H ence eτ — e, and M carries a symmetric A- form, by Lemma 1.5. F or char (F) = 2, one might have to consider quadratic forms instead of bilineai ones. F or more results in this direction, we refer to th e (unpublished) dissertation of W. Willems [Wi]. In the following, we restrict ourselves to group algebras F[G] with symmetric functional λ x e H om F (F [G ], F) and involutary antiisomorphism τ = A. Since the A- forms are just the G - invariant ones, we shall speak henceforth of G- forms. We next slightly sharpen the assertion of 3.1 (a) in case of group algebras. To do so, we need the following result (see [H B; VII, 8.12]). 3.2 TH EOREM. Let M be an absolutely irreducible self- dual F[G]module. Then to within an F- scalar multiple, there exists only one G- form on M. If char (F) Φ 2, this form is either symmetric or symplectic, 3.3 COROLLARY. Let M be an absolutely irreducible F[G]- module. If char (F) Φ 2, then the following asesertίons are equivalent (1) M carries a symplectic G- form. (2) ff= 0 for all idempotents feF[G] which satisfy M ^ F[G]f Proof. (1) iφ (2): Suppose there exists an / such t h at ffφ 0. Then Lemma 1.2 (b) implies t h at there also exists an idempotent ee A such t h at M = F[G]e and e = e. By Lemma 1.5 and Proposition 1.7, the module M carries a symmetric G- form. Since M is absolutely irreducible, M cannot carry a symplectic G- form, by Theorem 3.2. This contradicts (1). (2) => (1): Suppose t h at M carries a symmetric G- form. As char (F) Φ 2, Theorem 1.3 yields an idempotent eeF[G] such t h at M ^ F[G]e and e = e. I n particular, ee = e Φ 0, contradicting (2). By Theorem 3.2, M carries a symplectic G- form. • The next lemma, which we only state under th e conditions needed later on, is probably well- known. Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYMMETRIC FORMS, IDEMPOTENTS 45 3.4 LEMMA.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Let e be an idempotent in F[G], where c h a r ( F ) ||G |. Then λ 1(e) = (l/ |G |)dim F (F [G ]e). Proof. Let L ΞΞ> F be an algebraically Then dimF(F[G]e) = άim L (L®FF[G]e) closed extension field of F. = dimL (L[G]e). Let e = /Ί + + fs be a decomposition of e into primitive idempotents ft in L[G ], and / = fx. We denote the character of L[G]f by X, and the corresponding Wedderburn idempotent by ε e L[G]. Thus 2 and ^(ε) = %(1) / |G|. Since al] primitive idempotents corresponding to ε ι are conjugate to / , and since λ x(u~ fu) = λ {(f) (for units u in L[G\ ), we conclude λ x(f) = (l/ χ(l))λ,(e) = X(1)I\ G\ = (1/ |G |) dim L(L[G ]/ ). Consequently, Ue) = iUfd = (l/ |G |)gdim L (L[G ]/ , ) = (1/ |G |) dim L(L[G]β) = (l/ |G D dim F (F [G ]e). D As a disadvantage of Theorem 1.3 we recall th at its proof does not yield an explicit formula for a r- invariant idempotent in terms of the given τ- form. U nder certain circumstances, we can do better. Let M be an F[G ]- module with G- form < , ). F or an element xeM, we define geσ Then cx has the following properties. (1) λ , (c xa) = (ax, Xs) for all a e F[G]. (N amely just observe th at λ x(cxh) = (hx, x) for all h e G.) (2) If fe F[G] satisfies fx = x, then also fcx = c^. (To see this, note th at χ x( Cχ a) (3) (ax, x) = (afx, x) = ^ifeα/ ) = ί^/ c^α) for all aeF[G], and therefore fcx — cx — 0.) If < , > is symmetric, then cx = iΣ geo (gx, x)g and cx = cx. 3.5 Remark. Before we proceed, we recall what Lemma 1.2 says in our present context. Let M be an irreducible F[G]- module which carries a G- form and which contains a non- isotropic vector x. Then there exist Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 46 PETER LAN DROCK AN D OLAF M AN Z primitive idempotents / , zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED e e F[G] with t h e following properties: ( i ) M^ F[G]f^ F[G]e. ( ii) fx = x. (iii) e — e. (iv) e is t h e 1- element of fF[G]f, hence fF[G]f = eF[G]e. 3.6 PROPOSITION . L et M be an irreducible F[G]- module which carries a symmetric G- form <( , ) . Suppose that M contains a non- isotropic vector x> and let e = e be chosen according to Remark 3.5. / / {veeF[G]e\ ϋ then e = γ c x = v} = Fe, for some γ e F. Proof. We choose th e idempotent / as in Remark 3.5. By (ii), fx = x implies fcx = cx. Since < , ) is symmetric, (iv) yields cx = cx = cj = cj = fcJefF[G]f = eF[G\ e, and cx = βe for some βe F. I t remains to show th at cx Φ because λ x(cx) = <x, x} Φ 0. 0. This follows, Π 3.7 TH EOREM. L et F[G] be semi- simple, and suppose that the absolutely irreducible F[G]- module M carries a symmetric G- form < , >. Suppose that M contains a non- isotropic vector x (, which holds provided that chΆγ( F)φ 2). Then e = (dimFM)l(\ G\ .(x, x}) cx = (dim F M )/ (|G |.< *, x» Σ <gx, *>g geo is an idempotent such that M = F[G]e and e = e. Proof. Let e = e be chosen as in Remark 3.5. Since F[G] is semisimple and M is absolutely irreducible, we have M = F[G]e and F = E n d P [ β ] (M ) ^ &ntleF[G]e. I n particular, {ueβF [G ]e|ί} = v} = F e, an d Proposition 3.6 implies t h at e = γ c x for some ^ 6 F. I t remains to determine the scalar γ . By Lemma 3.4, and t h e assertion follows. • 3.8 Remarks, (a) I t should be clear t h at Theorem 3.7 still holds if we drop t h e hypothesis "semi- simple" and assume instead t h at M h as defect 0 (i.e. t h e block ideal of F[G] corresponding to M is simple). Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYM M ETRIC F ORM S, I D EM P OTEN TS 47 (b) IfzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA M h as positive defect however, th en cx definitely is no candidate for an idempotent. To see this note t h at λ x(cxj) = <jx, x) = 0 for all j e J(F[G]). Consequently, cx e ann (J(F[G])) = soc (F[G\ ). Thus cx = fcj eeF[G]e is in th e socle and hence in th e radical of th e block ideal corresponding to M. Therefore, c2x — 0. (c) We do not know how to generally proceed if F is not a splitting field for M. The case F — R however will be treated in t h e next section. § 4.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Real orthogonal representations Let M be an R[G ]- module and fix a symmetric, positive definite bilinear form ( , ) on M. Then [ , ] defined by [v, w] = Σ (gυ , gw), υ, w e M, obviously is a symmetric, positive definite G- form. Theorem 1.3. I t th us follows from 4.1 COROLLARY. L et M be an irreducible ~R[G]- module. Then there exists an idempotent e e R[G ] such that M ^ R[G ]β and e = e. 4.2 LEMMA. L et e be a primitive idempotent in R[G ] with e = e. Then / : = {vee~R[G]e\ ϋ = v} = R e . Proof. By Lemma 1.5, <u, w) — λ x{vw), υ , w e R[G ]β, is a symmetric G- form on R[G ]β. Moreover, < , ) is positive definite, and it holds t h at <y, wά) = λ i(vάw) = <uα, w;> for all a e eR[G ]e. Suppose th at dim R / > 2 and recall t h at eR[G ]β = R, C or H , where H denotes th e quaternion skew- field. If eR[G ]β ^ C, we choose i e eH[G]e corresponding to th e complex unit in C. I t then follows for all 0 Φ υ e R[G ]β t h at 0 < (υ i, vίy = (υ ii, u> = (vi2, u) = —(y, u> < 0, a contradiction. We may thus assume th at βR[G]β = H . If dim R 7 = 2, then / = span R<β, x) for some xeeR.[G]e and th e elements of I pairwise commute. Therefore, I is closed under multiplication and I = C. Consequently, if 2 dim R 7 = 2 or 4, then J contains an element ί such t h at i = — e and we proceed as in th e last paragraph. We still have to consider th e case Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 48 PETER LAN D ROCK AN D OLAF M AN Z dim R 7 = 3, sayzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I = span R <e, x,y}. Let {e, i,j, k} be the canonical R- basis of eR[G ]e = H . After a suitable basis transformation we may assume Sin ce x2 = — e(l + µ2), we t h a t x = i + µ k an d y — j + vk for µ ,> ι e R. obtain for 0 Φ υ e R[G ]e 2 2 0 < (υ x, vx) = (vx , v} = - ( 1 + ^ )<u, u> < 0, again a contradiction. This completes the proof. • Let V be an F[G]- module (for an arbitrary field F) and < , ) a G- form on V. Then the mapping (x • - * < > >« > where <y, ι^>α = <y, αw;> , yields an isomorphism between End F [ ί ?] (V) and the F- space BG(V) of all G - invariant bilinear forms on V (possibly degenerate). Assume in addition th at V = F[G]e for an idempotent e. The isomorphism eF[G]e ^ E n d ^ / V) is given by a H- > α α , where # α(ι>) = uα. H ence « •- > < » >α , where <ϋ, w;>α = <ϋ, wa), induces a vector space isomorphism between eF[G]e and BG(V). following serves as a substitute for Theorem 3.2. The 4.3 PROPOSITION . Let M be an irreducible TR[G]- module. Then M has exactly one symmetric G- form (up to R- scαZαr factors). Proof. By Corollary 4.1, we may assume t h at M = R[G ]e for an idempotent e = e. Consider the symmetric G- form < , ) on M induced by λ ι (see Lemma 1.5). Then every other G - invariant bilinear form on M is given by < , > α for a unique a e eR[G\ e. N ow < , > α is symmetric if and only if <u, wa) = (v, υι) a = (w, v}a = (w, va} = (wά, v) = (v, wά} for all v, w e M. This happens if and only if d = a, and Lemma 4.2 implies t h at a = γe for some γ e R. Consequently, < , ) α = γ( , ), which was to be proved. D Let M be an irreducible R[G ]- module. U sing th e form [ , ] and Proposition 4.3, any given symmetric G- form < , ) on M may be assumed to be positive definite. The group G is th en said to be represented orthogonally on M. It makes sense now to consider the unit sphere Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYMMETRIC FORMS, IDEMPOTENTS 49 {xe M\ (x, x} = 1} on M. Also a distance function d( , ) can be introduced in the usual way by d(x, yf = (x - y, x - y) , x, y 6 Λf. 4.4 TH EOREM. Let G be represented irreducίbly and orthogonally on the R- space M with respect to the form < , >. a) Given xe M with (x, x} = 1, is an idempotent satisfying both M ^ R[G ]e an d e = e. Here, elements of M in the same G- orbit lead to G- conjugate d ί empotents. b) Conversely, given e = Σ gς G<xgg an idempotent with M = R[G ]β and e = e, ίΛere, exists xe M with (x, x} = 1 an d <^x, Λ:> = IGI a^/ dimRAf, geG. Proof, a) Consider first xe M with <#, x> = 1, and choose the idempotent e = β eR [G ] with M ^ R[G ]β according to Remark 3.5. By Lemma 4.2, {v e eR[G]e\ ΰ = v} = Re, and Proposition 3.6 yields e = γc x = γ Σ (gx, x}g for some γ e R. gea The scalar γ again is determined by Lemma 3.4, namely dim R M / |G | = λ 1(e) = γ λ x{cx) = r(x,x) = r. F inally observe th at replacing x by hx (he G) replaces e by heh~\ b) Assume conversely th at e — e is given. Then Lemma 1.5 asserts t h at υ , weR[G]e, defines a symmetric G- form < , ) ' on R[G ]β = M. In particular, (geG), and Lemma 3.4 yields (e, ey — 1. By Proposition 4.3, there is a non- zero γ e R such th at <ι>, α;) 7 = γ( v, w} for all v, w 6 R[G ]β. Then 1 = <e, β) x = γ( β 9 e), and ^ > 0, since < , > is positive definite. H ence we may take x to be V 7 e. • F or data transmission via a G aussian channel, it turned out to be successful to consider the codewords as G - orbits on the un it sphere of Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 50 PETER LAN D ROCK AN D OLAF M AN Z some Euclidian space R \ The question about reasonable lower bounds for the minimal Euclidian distance—actually our motivation for this paper—has only got partial answers. The following result was first proved by D. Splpian in 1968. (See [BM; chap. 6] for this result and some background in Coding Theory.) 4.5 COROLLARY (Slepian).zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Let G be represented irreducibly and orthogonally, but non- trίva ί lly, on the -ίΐ space M with respect to the form < , >. Let xeM with (x, x> = 1. Then a) * Σ eβd(gx,xY = 2\ G\ . b) / / 9ΐ denotes any conjugacy class in G and ke% then gΣ em d(gx, x) 2 = 2 |9 ΐ |( l - X(k)lX(ί) \ where X is the character of M. Proof, By Theorem 4.4, e = dim R M/ |G ] g Σ ec (gx, x)g is an idempotent affording M. a) Since M is not th e trivial module, we have o = ( Σ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA < gχ, * > £ ) ( Σ h) = Σ < gχ, * > Σ gQG hβG gGG hβG 2 and therefore g Σ eod(gx, x) = g Σ ea^O- - (gx, x» = 2\ G\ . b) Since d(gx, x) 2 = 2(1 — (gx, x}) 9 it amounts to show t h at Σ <gχ, χ> = mW)IO) • a n d P ut S = εΣ evg observe t h at e = X(ΐ) l\ G\ SΣ GG (gx, x}g~ ι . Thus ^(βSR) = %(1)/ |G| g Σ em (gx, x) and it is therefore sufficient to show t h at λ ^ effi) = ι (\ m\ l\ G\ )X(k). Let d = dim R En d R [ σ] (M )). Then ε = Z(l)/ (|G |- d) Σ geG t{g)g~ is the Wedderburn idempotent corresponding to M. We decompose ε = e1 + - - - + es into primitive idempotents et e R[G ], where e = ex and s — X(ϊ) / d. Then ei — u^eUi for units ut e R[G ]. Since we obtain (χ(i)/ d)Λ(e3l) = Λ(εS) = χ(i)/ (|G i d) Σ *(g) = (x(i)ld)(\ m\ \ι G\ )x(k)), gem which establishes the claim. • ACKN OWLED G EMEN T. This paper was written when the first author was visiting Mainz and the second autor was visiting Aarhus and H aifa. We th an k th e Deutsche Forschungsgemeinschaft (DFG), th e Danish Nat- Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883 SYMMETRIC FORMS, IDEMPOTENTS 51 ural Science Research Council and the Technίon (Haifa) for th eir financial support. We are also indebted to P. F leischmann, R. Kn orr and A. Juh asz for helpful discussions. REF EREN CES [Al] A. Albert, Structure of algebras, AMS Colloquium Publications, 1939. [BM] I. F . Blake, R. C. Mullin, The mathematical theory of coding1, Academic Press, New York, 1975. [CR] C. Curtis, I. Reiner, Methods of representation theory, John Wiley, New York 1981. [HΓB] B. H uppert, N. Blackburn, F inite groups I I , Springer- Verlag, Berlin, 1982. [La] P. Landrock, F inite group algebras and their modules, Cambridge U niversity Press, Cambridge, 1983. [OT] T. Okuyama, Y. Tsushima, On a conjecture of P. Landrock, J. Algebra 104 (1986), 203- 208. [Wi] W. Willems, Metrische Moduln ϋber G ruppenringen, D issertation, Mainz, 1976. Peter Landrock Department of Mathematics University of Aarhns DK- 8000 Aarhus Denmark Olaf Manz Department of Mathematics (IWR) University of Heidelberg D- 6900 Heidelberg Germany Current Address: UCI Utility Consultants International D- 6000 Frankfurt 71 Downloaded from https://www.cambridge.org/core. IP address: 181.215.217.119, on 12 Jun 2019 at 12:11:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000003883