SYMMETRY CLASSES OF TENSORS ASSOCIATED WITH PRINCIPAL INDECOMPOSABLE CHARACTERS AND OSIMA IDEMPOTENTS

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1 SYMMETRY CLASSES OF TENSORS ASSOCIATED WITH PRINCIPAL INDECOMPOSABLE CHARACTERS AND OSIMA IDEMPOTENTS RANDALL R. HOLMES AVANTHA KODITHUWAKKU Abstract. Two types of symmetry classes of tensors are studed. The frst s the symmetry class assocated wth a prncple ndecomposable character of the dhedral group. The second s the symmetry class assocated wth the Osma dempotent correspondng to a p-block of the dhedral group. In each case necessary and suffcent condtons are gven for the exstence of an orthogonal bass consstng of standard (decomposable) symmetrzed tensors. 0. Introducton The usual noton of symmetry class of tensors nvolves an ordnary rreducble character χ of a subgroup G of a symmetrc group S m. By defnton ths symmetry class V χ s the product s χ V m, where s χ s the central dempotent of the group algebra CG correspondng to the conjugate character χ and V s a fnte-dmensonal vector space over the feld C of complex numbers. One can replace s χ by any element s of CG to get a new noton of symmetry class of tensors. Modular representaton theory of the group G, whch requres the choce of a prme number p, provdes three natural choces for such an s: (1) s ϕ wth ϕ an rreducble Brauer character of G, (2) s Φ wth Φ a prncpal ndecomposable character of G, and (3) the Osma dempotent s B correspondng to a p-block B of G. Each of these choces results n a symmetry class of tensors that generalzes the usual noton n the sense that, f p does not dvde the order of G, then the symmetry class s the same as that correspondng to an ordnary rreducble character of G (see Remark 2.2). The frst of these three generalzatons s the focus of [HK13]. In that paper necessary and suffcent condtons are gven under whch the symmetry class of tensors assocated wth an rreducble Brauer character of the dhedral group D m (vewed as a subgroup of S m ) s guaranteed to have an orthogonal bass consstng of standard (decomposable) symmetrzed tensors a so-called o-bass Mathematcs Subject Classfcaton. 15A69, 20B05, 20C20, 20C15. Key words and phrases. Symmetry class of tensors, prncpal ndecomposable character, Osma dempotent. 1

2 2 RANDALL R. HOLMES AVANTHA KODITHUWAKKU In ths paper we take up the two remanng generalzatons, namely, the symmetry class of tensors assocated wth a prncple ndecomposable character of G and the symmetry class of tensors assocated wth the Osma dempotent correspondng to a p-block of G. Takng G to be the dhedral group agan we gve necessary and suffcent condtons for the exstence of an o-bass n each case. Results about o-bases of symmetry classes of tensors can be found n [BPR03, DP99, DdST05, Hol95, Hol04, HT92, SAJ01]. Symmetry classes of tensors assocated wth modular characters (.e., characters havng values n a feld of prme characterstc) were studed n [SS99]. 1. Character theory Let G be a fnte group and fx a prme number p. An element of G s p- regular f ts order s not dvsble by p. Denote by Ĝ the set of all p-regular elements of G. Let IBr(G) denote the set of rreducble Brauer characters of G. (A Brauer character s a certan functon from Ĝ to C assocated wth an F G-module where F s a sutably chosen feld of characterstc p. The Brauer character s rreducble f the assocated module s smple. For the theory of Brauer characters, see [Isa94, Ser77, CR62, Fe82].) Let Irr(G) denote the set of rreducble characters of G. (Unless preceded by the word Brauer, the word character always refers to an ordnary character.) If the order of G s not dvsble by p, then Ĝ = G and IBr(G) = Irr(G). For a character χ of G denote by ˆχ : Ĝ C the restrcton of χ to Ĝ. For each χ Irr(G) we have ˆχ = d ϕχ ϕ ϕ IBr for some nonnegatve ntegers d ϕχ, called the decomposton numbers of G wth respect to the prme number p. Correspondng to each ϕ IBr(G) s a prncpal ndecomposable character Φ ϕ : G C. (Ths s a character of G havng the property that ts restrcton ˆΦ ϕ to Ĝ s the Brauer character correspondng to the projectve cover of the smple F G-module correspondng to ϕ.) 1.1 Theorem. [Fe82, IV, 2.5, 3.1] For every ϕ IBr(G), () Φ ϕ (g) = 0 for all g G\Ĝ, () Φ ϕ = χ Irr(G) d φχχ, where the d φχ are the decomposton numbers of G wth respect to the prme number p. If p does not dvde, then the matrx [d ϕχ ] s the dentty matrx and Φ ϕ = ϕ for each ϕ IBr(G) (= Irr(G)). The Brauer graph of G (wth respect to the prme number p) s the graph wth vertex set Irr(G) and wth an edge jonng the vertces χ and ψ f and only f there exsts ϕ IBr(G) such that d ϕχ and d ϕψ are both nonzero.

3 SYMMETRY CLASSES OF TENSORS 3 A p-block of G s a subset B of Irr(G) IBr(G) wth the followng propertes [Isa94, 15.17, 15.27]: () B Irr(G) s a connected component of the Brauer graph of G, () B IBr(G) = {ϕ IBr(G) d ϕχ 0 for some χ B Irr(G)}. The set Bl(G) of p-blocks of G s a partton of the set Irr(G) IBr(G). If p does not dvde, then each p-block of G s a sngleton {χ} wth χ Irr(G). Let B Bl(G). The Osma dempotent correspondng to B s the element s B of the group algebra CG gven by or, equvalently, s B = 1 s B = 1 g G χ B Irr(G) g Ĝ ϕ B IBr(G) χ(e)χ(g)g Φ ϕ (e)ϕ(g)g, where e s the dentty element of G and bar denotes conjugaton [Isa94, 15.21, 15.26, 15.30]. It follows from the second expresson that the sum over g G n the frst expresson can be replaced by the sum over g Ĝ. The elements s B wth B Bl(G) are parwse orthogonal central dempotents of the algebra CG and they sum to 1. That s to say, for B, B Bl(G) we have s B s B = δ BB (Kronecker delta), B Bl(G) s B = 1, and each s B commutes wth every element of CG. 2. Symmetry classes of tensors Fx postve ntegers m and n and put Γ m,n = {γ Z m 1 γ n}. Fx a subgroup G of the symmetrc group S m of degree m. A rght acton of G on the set Γ m,n s gven by γσ = (γ σ(1),..., γ σ(m) ) (γ Γ m,n, σ G). For γ Γ m,n put G γ = {σ G γσ = γ}, the stablzer of γ. Let V be a complex nner product space of dmenson n and let {e 1 n} be an orthonormal bass for V. The nner product on V nduces an nner product on V m (the mth tensor power of V ) and, wth respect to ths nner product, the set {e γ γ Γ m,n } s an orthonormal bass for V m, where e γ = e γ1 e γm. The space V m s a left CG-module wth acton gven by σe γ = e γσ 1 (σ G, γ Γ m,n ), extended lnearly. The nner product on V m s G- nvarant, whch s to say (σv, σw) = (v, w) for all σ G and all v, w V m. Let s CG. We have s = σ G a σσ for some a σ C. For γ Γ m,n the standard (or decomposable) symmetrzed tensor correspondng to s and γ s gven by e s γ = se γ = σ G a σ e γσ 1.

4 4 RANDALL R. HOLMES AVANTHA KODITHUWAKKU The symmetry class of tensors assocated wth s s V s = sv m = e s γ γ Γ m,n. The orbtal subspace of V s correspondng to γ Γ m,n s V s γ = sv γ = e s γσ σ G, where V γ = e γσ σ G. An o-bass of a subspace W of V s s an orthogonal bass of W of the form {e s γ 1, e s γ 2,..., e s γ t } for some γ Γ m,n. By conventon, the empty set s regarded as an o-bass of the zero subspace of V s. Let = G be a set of representatves of the orbts of Γ m,n under the acton of G. 2.1 Theorem. [HK13, 1.1] For every s CG, V s = Vγ s γ (orthogonal drect sum). In partcular, V s has an o-bass f and only f Vγ s γ. has an o-bass for each Let S be a subset of G contanng the dentty permutaton 1 and let ϕ : S C be a functon. (For nstance, ϕ could be a character of G, n whch case S = G, or ϕ could be a Brauer character of G, n whch case S = Ĝ.) Put s ϕ = ϕ(1) S ϕ(σ)σ CG. σ S Let γ Γ m,n. We wrte e sϕ γ, V sϕ, and Vγ sϕ respectvely. Smlarly, for B Bl(G) we wrte e s B γ smply as e B γ, V B, and V B γ, respectvely. more smply as e ϕ γ, V ϕ, and V ϕ, V sb, and V s B γ more 2.2 Remark. Let ϕ IBr(G) and let B be the p-block of G contanng ϕ. Each of the symmetry classes of tensors V ϕ, V Φϕ, and V B can be regarded as a generalzaton of the classcal symmetry class V χ assocated wth an ordnary rreducble character χ of G. Indeed, f p does not dvde, then ϕ s an ordnary rreducble character and we have Φ ϕ = ϕ and B = {ϕ}, so these three symmetry classes reduce to the classcal one. (In ths case s B = s ϕ so V B = V ϕ. The conjugaton here could be avoded by replacng ϕ(σ) by ϕ(σ) n the defnton of the symmetrzer s ϕ. Ths would be more natural snce then s ϕ would be the central dempotent of CG correspondng to the character ϕ, but n the lterature the symmetrzer s ϕ s usually defned as above.) γ,

5 SYMMETRY CLASSES OF TENSORS Theorem. [Mer97, 6.6] We have V m = (orthogonal drect sum). χ Irr(G) V χ Ths theorem fals to hold f Irr(G) s replaced by IBr(G) [HK13, Example 2.5]. Fx a character ψ of G. We have ψ = t =1 m χ wth the m postve ntegers and wth χ 1,..., χ t dstnct rreducble characters of G, called the rreducble consttuents of ψ. 2.4 Theorem. Let ψ be as above. For every γ Γ m,n and σ G, Proof. Frst, Therefore, s ψ = ψ(1) = ψ(1) (e ψ γσ, e ψ γ ) = ψ(1)2 τ G t =1 t =1 ψ(τ)τ = ψ(1) m χ (1) χ (1) m 2 χ (1) τ G =1 τ G γ χ (τσ). t m χ (τ)τ χ (τ)τ = ψ(1) τ G (e ψ γσ, e ψ γ ) = (s ψ e γσ, s ψ e γ ) = ψ(1) 2 t = ψ(1) 2 t =1 t =1 j=1 m 2 χ (1) 2 (eχ γσ, e χ γ ) = ψ(1)2 t =1 m χ (1) s χ. m m j χ (1)χ j (1) (eχ γσ, e χ j γ ) t =1 m 2 χ (1) τ G γ χ (τσ), where the thrd equalty uses that V χ s orthogonal to V χj for j (Theorem 2.3) and the last equalty s from [Fre73, p. 339]. 2.5 Theorem. [HK13, 1.6] Let ψ be as above. For every γ Γ m,n, dm V ψ γ = t =1 dm V χ γ. The summands n the precedng theorem are gven by the followng wellknown formula due to Freese. 2.6 Theorem. [Fre73, p. 339] For every χ Irr(G) and γ Γ m,n, dm Vγ χ = χ(1) χ(σ) = χ(1)(χ, 1) Gγ. G γ σ G γ

6 6 RANDALL R. HOLMES AVANTHA KODITHUWAKKU r = 3. The dhedral group Assume that m 3 and defne r, s S m by ( ) m 1 m, s = m 1 ( m 1 m 1 m m Then G = D m = r, s s the dhedral group of degree m. The elements r and s satsfy the relatons r m = 1, s 2 = 1, and sr k s = r k for all k. It follows that G = {r k, sr k 0 k < m}. The order of G s 2m and the cyclc subgroup C of G generated by r has order m. The ordnary rreducble characters of G are [Ser77, pp ] ). r k sr k ψ ψ ψ 3 ( 1) k ( 1) k (m even) ψ 4 ( 1) k ( 1) k+1 (m even) χ j 2 cos 2πkj m 0 (1 j < m/2) Table 1. Let p be a fxed prme number. We have m = p q l for a nonnegatve nteger q and a postve nteger l not dvsble by p. Defne 4, l even, p 2, ε = 2, l odd, p 2, 1, p = 2. For each 1 ε and 1 j < l/2 put ϕ 1 = ˆψ, ϕ 2 j = ˆχ j. 3.1 Theorem [HK13, 2.1]. The complete lst of rreducble Brauer characters of G s ϕ 1 (1 ε), ϕ2 j (1 j < l/2). (The superscrpt n the notaton corresponds to the degree of the Brauer character.) Put J = {j Z 1 j < m/2} and note that {χ j j J} s the set of degree two rreducble characters of G. For Z put J = J {kl ± k Z}. 3.2 Theorem. The sets J, 0 l/2, form a partton of the set J.

7 SYMMETRY CLASSES OF TENSORS 7 Proof. Put U = l/2 =0 J. From the defnton of J we have U J. Let j J. By the dvson algorthm there exst ntegers k and t wth 0 t < l such that j = kl + t. If t l/2, then j J t U. If t > l/2, then puttng = l t we have 0 < < l/2 and j = kl + (l ) = (k + 1)l J U. Therefore, J = U. Next we check that the sets J are parwse dsjont. Let 0 l/2 and let j J J. Then kl ± = j = k l ± for some ntegers k and k. If both sgns are plus, we get (k k )l =, so 0 (k k )l l/2 mplyng 0 k k 1/2, whence k = k and =. If both sgns are mnus, a smlar argument agan yelds =. Now suppose kl + = k l. Then + = (k k)l, so 0 (k k)l l, mplyng + s 0 or l. The frst case yelds = 0 = ; the latter yelds = l/2 =. The remanng case kl = k l + s handled smlarly. The decomposton numbers d ϕψ of G can be read off from the equatons n the followng theorem. 3.3 Theorem. () For 1 η, ˆψ = { ϕ 1 1, f p = 2, ϕ 1, f p 2, where η s 2 or 4 accordng as m s odd or even. () For 1 j < m/2 we have j J for a unque 0 l/2 and { 2ϕ 1 1, f p = 2, ϕ ϕ1 2, f p 2, f = 0, ˆχ j = ϕ 2, ϕ ϕ1 4, f 0 < < l/2, f = l/2. Moreover, for each 0 l/2 and j J the character ˆχ j s as ndcated. Proof. () If p 2 the clam follows from the defnton of ϕ 1. Assume p = 2. For each 1 η the character ψ s lnear (.e., of degree one) so that ˆψ s lnear as well. Snce ϕ 1 1 s the only lnear Brauer character of G n ths case the clam follows. () The sets J, 0 l/2, form a partton of the set J = {j Z 1 j < m/2} (Theorem 3.2), so t suffces to prove the last sentence. Let 0 l/2 and let j J. It follows from Table 1 that n all cases both sdes of the equaton vansh on Ĝ\Ĉ = Ĝ\C (ths set beng empty when p = 2), so t suffces to show that both sdes agree on Ĉ = rpq. We have j = kl ± for some nteger k. Usng the cosne sum and dfference formulas

8 8 RANDALL R. HOLMES AVANTHA KODITHUWAKKU we get for each nteger t ( 2πp ˆχ j (r pqt q ) t(kl ± ) ) = 2 cos m ( 2πp q ) ( t 2πp q ) t = 2 cos(2πtk) cos sn(2πtk) sn m m ( 2πp q ) t = 2 cos m 2, f = 0, = ϕ 2 (rpqt ), f 0 < < l/2, 2( 1) pqt, f = l/2. (For the case = l/2 we have used the fact that l s even so that p s not 2 and s therefore odd.) Therefore, both sdes of the equaton agree on Ĉ and the proof s complete. We wrte Φ j for the prncpal ndecomposable character Φ ϕ correspondng to the rreducble Brauer character ϕ = ϕ j. 3.4 Theorem. () If p = 2, then Φ 1 1 = η ψ j + 2 χ j, j J 0 j=1 and f p 2, then { Φ 1 ψ + j J = 0 χ j, = 1, 2, ψ + j J l/2 χ j, = 3, 4 (m even), where η s 2 or 4 accordng as m s odd or even. () For each 1 < l/2, Φ 2 = j J χ j. Proof. Accordng to Theorem 1.1, for ϕ IBr(G) the multplcty of an rreducble character χ of G as a summand of Φ ϕ s the decomposton number d ϕχ, whch s the same as the multplcty of ϕ as a summand of ˆχ. Assume that p = 2. By Theorem 3.3, we see that ϕ 1 1 appears wth multplcty 1 n each of the characters ˆψ, 1 η, t appears wth multplcty 2 n each of the characters ˆχ j, j J 0, and t does not appear n any of the other restrcted rreducble characters. Therefore, by the precedng paragraph the decomposton of Φ 1 1 s as stated n (). The rest of the formulas are verfed n a smlar fashon. 3.5 Theorem.

9 SYMMETRY CLASSES OF TENSORS 9 () For 1 ε the space V Φ 1 has an o-bass f and only f at least one of the followng holds: (a) dm V = 1, (b) p = 2, (c) m s not dvsble by p. () For 1 < l/2 the space V Φ 2 has an o-bass f and only f ether dm V = 1 or l 0 mod 4, where l = l/ gcd(l, ). Proof. () Fx 1 ε. If dm V = 1, then V Φ 1 = e Φ1 γ, where γ = (1, 1,..., 1), so V Φ 1 has o-bass {e Φ1 γ } or accordng as dm V Φ 1 s 1 or 0. Assume p = 2 so that ε = 1. In ths case, Ĝ equals r pq, a subgroup of G. Let λ be the trval character of Ĝ. By Theorem 3.3 we have ˆψ = ϕ 1 1 for all 1 η, and ˆχ j = 2ϕ 1 1 for all j J 0. Snce ϕ 1 1 = ˆψ 1 = λ t follows from Theorem 3.4 that s Φ 1 1 s a nonzero scalar multple of s λ. In partcular, V Φ 1 1 = V λ. Now each orbtal subspace Vγ λ has dmenson at most one (Theorem 2.6) and therefore has an o-bass. By Theorem 2.1, V λ has an o-bass and therefore V Φ 1 1 does as well. Assume that m s not dvsble by p. If p = 2, then V Φ 1 has an o-bass as we have just seen, so assume that p 2. Then Ĝ = G and hence Φ1 = ψ. In partcular, V Φ 1 = V ψ. Snce ψ s of degree one, each orbtal subspace V ψ γ has dmenson at most one and hence V Φ 1 has an o-bass by an argument smlar to that n the precedng paragraph. Now assume that none of the three condtons stated n the theorem holds. Let γ = (1, 2,..., 2), whch s n Γ m,n snce n = dm V 2. Note that G γ = {1, s}. Clam: (e Φ1 γσ, e Φ1 γ ) 0 for every σ G. Let σ G. Snce G = C sc and s G γ we may assume that σ C. Put { J 0, f {1, 2}, T = J l/2, f {3, 4}. (If {3, 4}, then m s even and, snce p 2, t follows that l s even as well so that J l/2 s defned.) By Theorems 2.4 and 3.4 we get (3.5.1) c(e Φ1 γσ, e Φ1 γ ) = τ G γ ( 2ψ (τσ) + ) χ j (τσ), j T where c = 2/Φ 1 (1)2. And then usng Theorem 3.4 agan we get (3.5.2) c(e Φ1 ) = ( ) ψ (τσ) + Φ 1 (τσ). γσ, e Φ1 γ τ G γ

10 10 RANDALL R. HOLMES AVANTHA KODITHUWAKKU Assume that σ / Ĝ. From Equaton (3.5.2) and the fact that prncpal ndecomposable characters vansh on G \ Ĝ (Theorem 1.1), we get c(e Φ1 γσ, e Φ1 γ ) = (ψ + Φ 1 )(σ) + (ψ + Φ 1 )(sσ) = ψ (σ) + ( 2ψ + j T χ j ) (sσ) = ψ (σ) + 2ψ (sσ) 0, where we have used that σ C and Table 1. Now assume that σ Ĝ (wth the assumpton σ C stll n force). For j T, Theorem 3.3 gves { ψ 1 (σ) + ψ 2 (σ), {1, 2}, χ j (σ) = ψ 3 (σ) + ψ 4 (σ), {3, 4}, = 2ψ (σ), the second equalty from Table 1 and agan the fact that σ C. Also, from Table 1 we have ψ (sσ) = ( 1) +1 ψ (σ). So Equaton (3.5.1) gves c(e Φ1 γσ, e Φ1 γ ) = ( 2ψ + j T χ j ) (σ) + ( 2ψ + j T χ j ) (sσ) = 2ψ (σ) + j T 2ψ (σ) + 2( 1) +1 ψ (σ) ( = T + ( 1) +1) ψ (σ). Therefore, to show that the nner product s nonzero t s enough to show that T s nonempty. Frst, p 2, so l = m/p q < m/2, mplyng that l J 0 = T f {1, 2}. Now assume that {3, 4}. Then m s even and, snce p s odd, t follows that l = m/p q s even. In partcular, l/2 s an nteger. Snce l/2 = m/2p q < m/2, we have l/2 J l/2 = T. Ths establshes the clam. The precedng paragraph shows that the set T s nonempty, so there exsts j T. By Theorem 2.6 we have dm V χ j γ = 2. Now the rreducble character χ j s a consttuent of the character Φ 1 by Theorem 3.4, so Theorem 2.5 gves dm V Φ1 γ > 1. Therefore, by our clam above and because the nner product s G-nvarant, V Φ1 γ does not have an o-bass. Usng Theorem 2.1 we conclude that V Φ 1 does not have an o-bass and the proof of () s complete. () Let 1 < l/2. From Theorem 3.4 and then Theorem 3.3 we get ˆΦ 2 = j J ˆχ j = j J ϕ 2 = J ϕ 2. Snce Φ 2 vanshes on G\Ĝ (Theorem 1.1), t follows that s Φ 2 s a nonzero scalar multple of s ϕ 2. In partcular, V Φ 2 = V ϕ 2 and V Φ 2 has an o-bass f and only f V ϕ 2 has an o-bass. Snce V ϕ 2 has an o-bass f and only f ether

11 SYMMETRY CLASSES OF TENSORS 11 dm V = 1 or l 0 mod 4, where l = l/ gcd(l, ) [HK13, Theorem 2.3], the proof s complete. We now turn to a dscusson of the symmetry classes assocated wth Osma dempotents. Ths requres frst a determnaton of the p-blocks of G. Put { B1 1 {ψ, χ j, ϕ 1 1 = 1 η, j J 0}, f p = 2, {ψ, χ j, ϕ 1 1 2, j J 0}, f p 2, B 1 3 = {ψ, χ j, ϕ 1 3 4, j J l/2 } B 2 = {χ j, ϕ 2 j J }, 1 < l/2, where η s 2 or 4 accordng as m s odd or even. (m even), 3.6 Theorem. If p does not dvde, then the p-blocks of G are the sngleton sets {χ}, χ Irr(G). If p dvdes, then the p-blocks of G are B1 1, B1 3, B2 (1 < l/2). Note: The notaton B j s used to ndcate that ϕj s n the p-block and s the least ndex for whch ths s the case. Proof. It has already been noted (Secton 1) that f p does not dvde, then each p-block of G s a sngleton set as clamed. Assume that p dvdes. The followng arguments use Theorem 3.3 repeatedly wth no further menton. Let B be the p-block of G contanng ϕ 1 1. Assume that p = 2. Then ˆψ = ϕ 1 1 for each 1 η and ˆχ j = 2ϕ 1 1 for all j J 0, so that B contans B1 1. No ordnary rreducble character not n B1 1 has the property that ts restrcton to Ĝ has ϕ1 1 as a consttuent, so B = B1 1. Now assume that p 2. Snce p = 2m, we have p m, so 1 l < m/2. Therefore, χ l Irr(G) and, snce l J 0, we have ˆχ l = ϕ ϕ1 2. Ths mples that B contans χ l and hence ϕ 1 2 as well. Now, ˆψ = ϕ 1 ( = 1, 2) and ˆχ j = ϕ ϕ1 2 for all j J 0, so B contans B1 1. No ordnary rreducble character not n B1 1 has the property that ts restrcton to Ĝ has ether ϕ1 1 or ϕ 1 2 as a consttuent, so B = B1 1. Assume that m s even and let B be the p-block of G contanng ϕ 1 3. Then p 2 and l s even. Snce p = 2m, we have p m, so l < m and 1 l/2 < m/2. Therefore, χ l/2 Irr(G) and, snce l/2 J l/2, we have ˆχ l/2 = ϕ ϕ1 4. Ths mples that B contans χ l/2 and hence ϕ 1 4 as well. Argung as n the precedng case, we get B = B3 1. Fnally, let 1 < l/2 and let B be the p-block of G contanng ϕ 2. For each j J we have ˆχ j = ϕ 2, so B contans B2. No ordnary rreducble character not n B 2 has the property that ts restrcton to Ĝ has ϕ2 as a consttuent, so B = B 2. Ths completes the proof.

12 12 RANDALL R. HOLMES AVANTHA KODITHUWAKKU 3.7 Theorem. Let B be a p-block of G. If B contans a lnear character, then V B has an o-bass. Otherwse, B = B 2 for some 1 < l/2 and V B has an o-bass f and only f ether dm V = 1 or l 0 mod 4, where l = l/ gcd(l, ). Proof. Assume that B contans a lnear character. Assume frst that p does not dvde. Then B Irr(G) = {ψ} for some lnear ψ Irr(G), mplyng s B s a nonzero scalar multple of s ψ. In partcular, V B = V ψ. Each orbtal subspace Vγ ψ has dmenson ether 0 or 1 and therefore has an o-bass, so that V B has an o-bass by Theorem 2.1. Now assume that p dvdes. Then B s ether B1 1 or B1 3 by Theorem 3.6. Assume that B = B1 1. Then s B = 1 σ Ĝ where χ B Irr(G) θ = χ(1)χ(σ)σ = 1 σ Ĝ θ ψ (σ) + 2χ j (σ) σ, j J 0 =1 { 4, f p = 2 and m s even, 2, otherwse (see comment after the defnton of s B n Secton 1 about replacng G by Ĝ). It follows from Table 1 that θ =1 ψ (σ) = 0 and χ j (σ) = 0 for all σ G\C and all j, so n the sum above we can replace σ Ĝ by σ Ĉ. Clam: For each 1 θ we have ψ Ĉ = λ, where λ s the trval character of Ĉ. When {1, 2} ths follows mmedately from Table 1. Suppose θ = 4 and let {3, 4}. Then p = 2 and m s even. Snce m = p q l = 2 q l wth l odd, t follows that q > 0. Therefore ψ (r pq ) = ( 1) 2q = 1 and snce Ĉ = rpq and ψ s a homomorphsm, the clam follows. For j J 0, Theorem 3.3 gves f p = 2, and χ j Ĉ = (2ϕ 1 1) Ĉ = (2ψ 1 ) Ĉ = 2λ χ j Ĉ = (ϕ ϕ 1 2) Ĉ = (ψ 1 + ψ 2 ) Ĉ = 2λ f p 2. Ths, together wth the above clam, shows that s B s a nonzero scalar multple of s λ. In partcular, V B equals V λ. Snce λ s lnear, V λ has an o-bass (by an argument smlar to that n the frst paragraph), so V B does as well. Now assume that B = B3 1. Then s B = 1 σ Ĝ 4 ψ (σ) + 2χ j (σ) σ. =3 j J l/2 From Table 1 we see that 4 =3 ψ (σ) = 0 and χ j (σ) = 0 for all σ G\C and all j, so n the sum we may replace σ Ĝ by σ Ĉ as before. Now

13 SYMMETRY CLASSES OF TENSORS 13 for {3, 4} we have ψ Ĉ = µ, where µ s the character of Ĉ gven by µ(r k ) = ( 1) k. And for a J l/2, Theorem 3.3 gves χ j Ĉ = (ϕ ϕ 1 4) Ĉ = (ψ 3 + ψ 4 ) Ĉ = 2µ. Therefore, s B s a nonzero scalar multple of s µ. Snce µ s lnear, we conclude as n the precedng case that V B has an o-bass. Fnally, assume that B does not contan a lnear character. Then Theorem 3.6 gves B = B 2 for some 1 < l/2 (and ths holds even f p does not dvde, snce then B = {χ } = B 2 for some 1 < m/2 = l/2). Now B Irr(G) = {χ j j J }, and for all j J we have χ j = χ j (Table 1) and ˆχ j = ϕ 2 (Theorem 3.3), so s B = 1 2χ j (σ)σ = 1 j J σ Ĝ 2ϕ 2 (σ)σ, whch s a nonzero scalar multple of s ϕ 2. In partcular, V B equals V ϕ 2 and the clam follows from [HK13, Theorem 2.3]. References [BPR03] C. Bessenrodt, M. R. Pournak, and A. Refegerste, A note on the orthogonal bass of a certan full symmetry class of tensors, Lnear Algebra Appl. 370 (2003), MR (2004f:15052) [CR62] Charles W. Curts and Irvng Rener, Representaton theory of fnte groups and assocatve algebras, Pure and Appled Mathematcs, Vol. XI, Interscence Publshers, a dvson of John Wley & Sons, New York-London, MR (26 #2519) [DdST05] J. A. Das da Slva and Mara M. Torres, On the orthogonal dmenson of orbtal sets, Lnear Algebra Appl. 401 (2005), MR (2006a:05163) [DP99] M. R. Darafsheh and N. S. Poursalavat, Orthogonal bass of the symmetry classes of tensors assocated wth the drect product of permutaton groups, Pure Math. Appl. 10 (1999), no. 3, MR (2001f:15030) [Fe82] Walter Fet, The representaton theory of fnte groups, North-Holland Mathematcal Lbrary, vol. 25, North-Holland Publshng Co., Amsterdam-New York, MR (83g:20001) [Fre73] Ralph Freese, Inequaltes for generalzed matrx functons based on arbtrary characters, Lnear Algebra Appl. 7 (1973), MR (49 #5028) [HK13] Randall R. Holmes and Avantha Kodthuwakku, Orthogonal bases of Brauer symmetry classes of tensors for the dhedral group, Lnear Multlnear Algebra 61 (2013), no. 8, MR [Hol95] Randall R. Holmes, Orthogonal bases of symmetrzed tensor spaces, Lnear Multlnear Algebra 39 (1995), no. 3, MR (96k:15015) [Hol04], Orthogonalty of cosets relatve to rreducble characters of fnte groups, Lnear Multlnear Algebra 52 (2004), no. 2, MR (2004j:20011) [HT92] Randall R. Holmes and Tn Yau Tam, Symmetry classes of tensors assocated wth certan groups, Lnear Multlnear Algebra 32 (1992), no. 1, MR (93j:20019) [Isa94] I. Martn Isaacs, Character theory of fnte groups, Dover Publcatons Inc., New York, 1994, Corrected reprnt of the 1976 orgnal [Academc Press, New York; MR (57 #417)]. MR σ Ĝ

14 14 RANDALL R. HOLMES AVANTHA KODITHUWAKKU [Mer97] Russell Merrs, Multlnear algebra, Algebra, Logc and Applcatons, vol. 8, Gordon and Breach Scence Publshers, Amsterdam, MR (98:15002) [SAJ01] M. A. Shahab, K. Azz, and M. H. Jafar, On the orthogonal bass of symmetry classes of tensors, J. Algebra 237 (2001), no. 2, MR (2002e:20022) [Ser77] Jean-Perre Serre, Lnear representatons of fnte groups, Sprnger-Verlag, New York, 1977, Translated from the second French edton by Leonard L. Scott, Graduate Texts n Mathematcs, Vol. 42. MR (56 #8675) [SS99] M. Shahryar and M. A. Shahab, Modular symmetry classes of tensors, Turksh J. Math. 23 (1999), no. 3, MR (2002g:15063) Randall R. Holmes, Department of Mathematcs and Statstcs, Auburn Unversty, Auburn AL, 36849, USA holmerr@auburn.edu Avantha Kodthuwakku, Department of Mathematcs, Unversty of Colombo, Colombo, Sr Lanka kaa0006@auburn.edu