Journal of Algebra 239, 356᎐364 Ž2001. doi:10.1006rjabr.2000.8656, available online at http:rrwww.idealibrary.com on
Exponent Reduction for Projective Schur Algebras1 Eli Aljadeff and Jack Sonn Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel E-mail: [email protected]
, [email protected]
Communicated by Alexander Lubotzky Received July 10, 2000
In this paper it is proved that the ‘‘exponent reduction property’’ holds for all projective Schur algebras. This was proved in an earlier paper of the authors for a special class, the ‘‘radical abelian algebras.’’ The precise statement is as follows: let A be a projective Schur algebra over a field k and let k Ž . denote the maximal cyclotomic extension of k. If m is the exponent of A mk k Ž ., then k contains a primitive mth root of unity. One corollary of this result is a negative answer to the question of whether or not the projective Schur group PSŽ k . is always equal to Br Ž Lrk ., where L is the composite of the maximal cyclotomic extension of k and the maximal Kummer extension of k. A second consequence is a proof of the ‘‘Brauer᎐Witt analogue’’ in characteristic p: if charŽ k . s p / 0, then every projective Schur algebra over k is Brauer equivalent to a radical abelian algebra. 䊚 2001 Academic Press
1. INTRODUCTION The projective Schur group of a field k, denoted by PSŽ k ., is the subgroup of the Brauer group Br Ž k . generated by Žin fact, consisting of. all classes that may be represented by a projective Schur algebra A. By definition, a finite dimensional k-central simple algebra A is projective Schur if AU , the group of units of A, contains a subgroup ⌫ which spans A as a k-vector space and is finite modulo the center, that is, kU ⌫rkU is a finite group. The notions of projective Schur algebras and projective Schur group are the projective analogues of Schur algebras and the Schur group of k Ždenoted by SŽ k ... These notions Žof projective Schur algebra and of projective Schur group. were introduced in 1978 by Lorenz and Opolka w6x. One of the main points for making this generalization is that PSŽ k . 1
The research was supported by the Fund for the Promotion of Research at the Technion. 356
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contains Žby abuse of language. all symbol algebras. More precisely, any symbol algebra is a projective Schur algebra in an obvious way. ŽIndeed, let A s Ž a, b . n be the symbol algebra generated by x and y subject to the relations x n s a g kU , y n s b g kU , yx s n xy, where n g kU is a primitive nth root of unity. It is easy to see that ⌫ s ² x, y : spans A as a k-vector space and kU ⌫rkU ( ⺪rn⺪ = ⺪rn⺪. In particular, kU ⌫rkU is a finite group.. Invoking the Merkurev᎐Suslin Theorem, one deduces that if k contains all roots of unity Žresp. contains the nth roots of unity., then PSŽ k . s Br Ž k . Žresp. PSŽ k . = Br Ž k . n . where the subscript n denotes n-torsion. The subgroup PSŽ k . may be large even if roots of unity are not present in k. Indeed, if k is a number field, then PSŽ k . s Br Ž k . as shown in w6x. Here one uses the fact that for k a number field, any element in Br Ž k . is split by a cyclic extension which is contained in a cyclotomic extension of k. In general, the projective Schur group PSŽ k . is properly contained in Br Ž k ., e.g., when k is a rational function field k 0 Ž x . over any number field k 0 . For power series fields k s k 0 ŽŽ x .. Žover a number field k 0 . the situation depends on the field k 0 . For instance if k 0 is a number field which is not totally real, then PSŽ k . / Br Ž k .. On the other hand, the Kronecker᎐Weber Theorem implies that PSŽ⺡ŽŽ x ... s Br Ž⺡ŽŽ x .... As mentioned above, symbol algebras are ‘‘natural’’ members of PSŽ k . and clearly, any symbol algebra is split by a Kummer extension of k. In general, Kummer extensions are not sufficient to split all elements in PSŽ k .. For instance if k is a number field, Kummer extensions do not split any element whose order is prime to the number of roots of unity in k Žrestriction-corestriction argument.. As mentioned earlier, when k is a number field, every element in PSŽ k . s Br Ž k . is split by a cyclotomic extension of k. In w2x the following is proved: THEOREM 1.1. E¨ ery element in PSŽ k . has a splitting field which is the composite of a cyclotomic extension of k and a Kummer extension of k. This was a key result used to show that in general PSŽ k . is properly contained in Br Ž k ., since one shows that Žin general. Br Ž Lrk . is properly contained in Br Ž k ., where L denotes the composite of all cyclotomic and Kummer extensions of k. It is now natural to ask w4, p. 528x: Is PSŽ k . s Br Ž Lrk . in general? In this paper we show that this is false: THEOREM 1.2. There are fields k for which PSŽ k . is properly contained in Br Ž Lrk . where L is the composite of the maximal cyclotomic extension of k and the maximal Kummer extension of k. This will follow Žas shown below. from the following theorem which is the main result of this paper.
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THEOREM 1.3. Let k Ž . be the maximal cyclotomic extension of k and let res : PSŽ k . ª PSŽ k Ž .. be the restriction map. Then for all ␣ g PSŽ k ., the exponent expŽresŽ ␣ .. of resŽ ␣ . ‘‘di¨ ides the number of roots of unity in k.’’ More precisely, if m s expŽresŽ ␣ .., then k contains a primiti¨ e mth root of unity. In terms of algebras the theorem says that if A is a projective Schur algebra over k, then expŽ A mk k Ž .. divides the number of roots of unity in k. We say that projective Schur algebras satisfy the exponent reduction property. To show how Theorem 1.2 follows from Theorem 1.3, take k s ⺡Ž x, t ., the rational function field in two indeterminates over the rationals. Let Kr⺡Ž t . be a regular extension Žover ⺡. which is cyclic of degree four Žsee, for example, w7, p. 224x and let D s Ž K Ž x .rQŽ x, t ., , x . be the corresponding cyclic algebra.. It is clear that the order of D is not reduced by any cyclotomic extension, that is, expŽ D mk k Ž .. s 4, whereas k does not contain the fourth roots of unity. By Theorem 1.3, w D x f PSŽ k .. On the other hand, D is split by a Kummer ⺪r2⺪ = ⺪r2⺪-extension, so w D x g Br Ž Lrk .. In w4x it is shown that certain projective Schur algebras, so-called radical abelian algebras, satisfy the exponent reduction property. A projective Schur algebra is radical if it is isomorphic to a crossed product algebra A s Ž Krk, G, ␤ . where Krk is a Ž G-Galois. radical extension Ž K s nr . k Ž x 11r n1 , x 21r n 2 , . . . , x 1r , x i g kU . and the class ␤ is represented by a r 2-cocycle f whose values in K U are finite modulo kU . If G is abelian we call A radical abelian. A radical algebra should be viewed as the ‘‘natural’’ way to construct projective Schur algebras. This is analogous to the construction of cyclotomic algebras Žsee w10x.. It is conjectured in w1x that every element in PSŽ k . may be represented by a radical algebra, even Žas conjectured in w4x. by a radical abelian algebra. In this paper we show the stronger conjecture holds if the field k has positive characteristic. In fact we show that every element in PSŽ k . may be represented by the tensor product of algebras which are ‘‘natural’’ examples of radical abelian algebras. Let A s Ž Krk, Gal Ž Krk ., ␣ . be a crossed product algebra over the field k. It is well known that if Krk is cyclic, one may represent the class ␣ g H 2 Ž Gal Ž Krk ., K U . by a 2-cocycle whose values are in kU Žrather than K U .. If in addition the field K can be embedded in a radical extension L of k, then we may represent the algebra A by the crossed product algebra Ž Lrk, Gal Ž Lrk ., infŽ ␣ .. where infŽ ␣ . denotes the image of ␣ under the inflation map. The point of this is that also infŽ ␣ . may be represented by a 2-cocycle with values in kU Žin fact, the same set of values., so we see that Ž Lrk, Gal Ž Lrk ., infŽ ␣ .. is a radical algebra. Note that if the extension
Lrk is abelian, then Ž Lrk, Gal Ž Lrk ., infŽ ␣ .. is a radical abelian algebra. We see that if an element w A x in Br Ž k . is split by a cyclic extension of k which is contained in a radical Žresp. radical abelian . extension of k then A is equivalent to a radical Žresp. radical abelian . algebra. Here we consider two types of such algebras. Type I. Symbol algebras. These are split by a cyclic Kummer extension of k. Type II. Algebras that are split by a cyclic extension of k which is contained in a cyclotomic extension of k. Note that algebras of type I and type II are radical abelian. THEOREM 1.4. Let k be a field of positi¨ e characteristic. Then e¨ ery element in PSŽ k . may be represented by an algebra which is the tensor product of algebras of types I and II. Using the fact that a tensor product of radical Žradical abelian. algebras is equivalent to a radical Žradical abelian . algebra Žsee w3, Lemma 2.4x, where it is proved for radical algebras, but the same proof applies for radical abelian algebras., we obtain: COROLLARY 1.5. If char Ž k . / 0, e¨ ery element in PSŽ k . is represented by a radical abelian algebra.
2. PROOFS OF THE RESULTS We are given a projective Schur algebra A over k. By definition, AU Žthe group of units of A. contains a subgroup ⌫ which spans A as a k-vector space and which is finite modulo kU . We write A s k Ž ⌫ .. For any subgroup H of ⌫ we may consider the subalgebra spanned by H and denote it by k Ž H .. Recall that ⌫ is center by finite, so by a theorem due to Schur, ⌫X , the commutator subgroup of ⌫, is finite. The subalgebras k Ž H . may not be simple, but if we restrict ourselves to subgroups H, with ⌫X : H : ⌫ we have the following reduction w1x: THEOREM 2.1. Let k Ž ⌫ . be a projecti¨ e Schur algebra o¨ er k. Then it is Brauer equi¨ alent to a projecti¨ e Schur algebra k ŽU ., UrkU finite, and such that for e¨ ery subgroup H of U with U X : H : U, k Ž H . is a simple algebra. We say the algebra k ŽU . is reduced. Thus, for the proof of Theorem 1.3, we shall assume that k Ž ⌫ . is a reduced algebra. Consider the subalgebra k Ž ⌫X . and let K = k be its center. Since the group ⌫X is finite, it follows that k Ž ⌫X . is a Schur algebra over K. Furthermore it is a simple component of the group algebra k⌫X and therefore K : k Ž . where k Ž . is a
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cyclotomic extension of k Ž a root of unity. w2x. In particular, the family ⍀ [ k Ž H . : ⌫X : H : ⌫, k Ž H . is a Schur algebra over its center L H , and L H is contained in a cyclotomic extension of k 4 is not empty. Let k Ž H . be a maximal element in ⍀. Observe that such an element exists because k Ž ⌫ . is finite dimensional over k. Let k Ž . be a cyclotomic extension of k which contains L H s ZŽ k Ž H ... By Brauer’s Splitting Theorem w5, pp. 385, 418x, k Ž H . is split by a cyclotomic extension of L H , so we may assume that k Ž . splits k Ž H .; that is, k Ž . mL H k Ž H . ( Mr Ž k Ž .. for some integer r. We are to show that expŽ k Ž ⌫ . mk k Ž .. divides the number of roots of unity in k. To start with, note that the group ⌫ acts on k Ž H . by conjugation, hence on its center L H . LEMMA 2.2. The fixed field L⌫H of L H under ⌫ is k. Proof. This is clear, since L⌫H ; ZŽ k Ž ⌫ .. s k. Let Hˆ s C ⌫ Ž L H ., the centralizer of L H in ⌫. By the definition of Hˆ it follows that ⌫X : H : Hˆ and L H : ZŽ k Ž Hˆ .. s L Hˆ . We claim that L H s Hˆ L Hˆ . Indeed, the group ⌫rHˆ acts faithfully on L H , and since L⌫r s k we H see that < ⌫rHˆ < s w L H : k x. But ⌫rHˆ acts faithfully on L Hˆ as well, so by a Hˆ similar argument, we have L⌫r s k. This implies w L Hˆ : k x s < ⌫rHˆ < s Hˆ w L H : k x, proving the claim. In order to simplify notation let L s L H s L Hˆ . PROPOSITION 2.3.
k Ž ⌫ . mk L is Brauer equi¨ alent to k Ž Hˆ ..
Proof. Consider k Ž ⌫ . as a module over k Ž Hˆ .. Let us show it is free of rank < ⌫rHˆ <. Let g 1 , . . . , g n4 be representatives of the left cosets of Hˆ in ⌫. Clearly, they span k Ž ⌫ . as a left k Ž Hˆ . module. To show they are independent we let w s z1 g i1 q z 2 g i 2 q ⭈⭈⭈ qz s g i s s 0 be a nontrivial relation of shortest length. Clearly s ) 1 since the g i are invertible. Take y1 Ž x g L with g i1 xgy1 such an element exists since ⌫rHˆ acts i 1 / g i 2 xg i 2 faithfully on L., and consider wx y g i1 Ž x . w s wx y g i1 xgy1 i1 w s z1 g i1Ž x . g i1 q z 2 g i 2Ž x . g i 2 q ⭈⭈⭈ qz s g i sŽ x . g i s y z1 g i1Ž x . g i1 q z 2 g i1Ž x . g i 2 q ⭈⭈⭈ qz s g i1Ž x . g i s s z 2 g i 2Ž x . y g i1Ž x . g i 2 q z 3 g i 3Ž x . y g i1Ž x . g i 3 q ⭈⭈⭈ qz s g i sŽ x . y g i1Ž x . g i s s 0, yielding a shorter nontrivial relation, since 0 / g i 2Ž x . y g i1Ž x . g L, a contradiction.
To prove the proposition we show L mk k Ž ⌫ . ( End kŽ Hˆ . Ž k Ž ⌫ . . s Hom kŽ Hˆ . Ž k Ž ⌫ . , k Ž ⌫ . . ( Mn Ž k Ž Hˆ . . Ž k Ž ⌫ . with the left k Ž Hˆ . structure .. Indeed, the algebra k Ž ⌫ . acts on k Ž ⌫ . from the right and L acts on k Ž ⌫ . from the left. Clearly the actions commute and both commute with the k Ž Hˆ . left action. ŽOf course, here we are using that L s ZŽ k Ž Hˆ ... This gives a nontrivial homomorphism from L mk k Ž ⌫ . into End kŽ Hˆ .Ž k Ž ⌫ ... The map is 1᎐1 Žsince L mk k Ž ⌫ . is simple. and therefore surjective Žthe dimensions over k are equal.. This completes the proof of the proposition. Observe that the algebras k Ž H . and k Ž Hˆ . have the same center L, and so Žby the double centralizer theorem. we have k Ž Hˆ . , k Ž H . mL A Ž A central simple over L.. Theorem 1.3 will follow if we show that expŽ A. Žin Br Ž L.. divides the number of roots of unity in k. We start with the following key lemma. LEMMA 2.4. Let z g Hˆ and let n be its order modulo k Ž H .U . Then n di¨ ides the number of roots of unity in k. Proof. We assume the lemma is false and get a contradiction. Taking a power of z we may, for some prime p, assume the order of z modulo k Ž H .U is p rq1 and k contains the p r th roots of unity but not the p rq1 th root of unity, r G 0. Consider the action of z on k Ž H . induced by conjugation. Since this action centralizes L, by the Skolem᎐Noether Theorem there is an element a g k Ž H .U such that w s zay1 centralizes k Ž H .. In particular, it centralizes a, so z and a commute in k Ž Hˆ .. It follows that the element w is in the center of the subalgebra B s k Ž² H, z :. s k Ž w .Ž H . s LŽ w .Ž H . Žin k Ž Hˆ ... Clearly LŽ w . is contained in the center of B. In fact LŽ w . is precisely the center of B since there is an obvious map of algebras from LŽ w . mL k Ž H . Žsimple with center LŽ w .. onto B. rq 1
Claim. The element w satisfies an equation of the form X p y e s 0 where e g LU . Furthermore p rq1 is the smallest possible. To see this, recall that w centralizes a so that a, z, w commute. We therefore have rq 1 rq 1 rq 1 rq 1 rq 1 w p s Ž zay1 . p s z p ayp g k Ž H .U . But w p centralizes k Ž H ., rq 1 hence w p g L s ZŽ k Ž H ... Finally, it is easy to see that the order of w modulo LU equals the order of z modulo k Ž H .U Žs p rq1 .. This proves the claim. Observe that the algebra B is a Schur algebra over its center LŽ w . since k Ž H . is, and moreover it is of the form k Ž H˜ ., where H˜ s ² H, i i z :. Set B0 s B and for i s 1, . . . , r let Bi s k Ž² H, z p :. s k Ž w p .Ž H . s i i LŽ w p .Ž H .. As for B, one shows easily that LŽ w p . s ZŽ Bi .. Also, note
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that Br strictly contains k Ž H ., so we will reach a contradiction to the r maximality of k Ž H . if we show that LŽ w p . is contained in a cyclotomic extension of k. Let us show that this is indeed the case. First recall that the field LŽ w . is the center of the algebra B s k Ž² H, z :.. By w2, Corollary 2.3x, LŽ w . is contained in the composite of the maximal cyclotomic extension k Ž . of k and a Žfinite. Kummer extension k ŽU . of k. Since k contains no primitive p rq1 root of unity, the p-primary component of the Galois group Gal Ž k ŽU .rk . has exponent F p r. It follows that the p-primary component of Gal Ž k Ž , U .rk Ž .. has exponent at most p r. On the other hand the field k Ž , w . is a Kummer r extension of k Ž . and is contained in k Ž , U .. Hence if w p f k Ž ., rq1 Gal Ž k Ž , w .rk Ž .. is cyclic of order p . This is impossible of course. It r r follows that w p g k Ž . and hence LŽ w p . : k Ž .. This completes the proof of the lemma. Our next step will be to decompose the algebra k Ž Hˆ . as a tensor product k Ž H . mL A, with expŽ A. dividing the number of roots of unity in k. To do this recall that PAbŽ F . is the subgroup of PSŽ F . consisting of classes which may be represented by a projective Schur algebra F Ž ⌫ . and ⌫rF U s G abelian. In this case we say that F Ž ⌫ . is of abelian type. Obviously, the natural examples are the symbol algebras. We recall from w3x the following. PROPOSITION 2.5. If F Ž ⌫ . is a projecti¨ e Schur algebra of abelian type, then it is Brauer equi¨ alent to the tensor product of symbol algebras. Furthermore, if expŽ G . s n, then the exponent of F Ž ⌫ . in Br Ž F . di¨ ides n. LEMMA 2.6. Let k Ž H . and k Ž Hˆ . be as abo¨ e. Then k Ž Hˆ . ( k Ž H . mL LŽ ⌳ . where LŽ ⌳ . is a projecti¨ e Schur algebra of abelian type and expŽ ⌳rLU . di¨ ides the number of roots of unity in k. Furthermore, LŽ ⌳ . is isomorphic to a product of symbol algebras. Let us postpone the proof of the lemma and complete the proof of Theorem 1.3. By Lemma 2.6, expŽ LŽ ⌳ .. Žas an element in Br Ž L.. divides expŽ ⌳rLU ., hence expŽ LŽ ⌳ .. divides the number of roots of unity in k. But k Ž ⌫ . mk k Ž . ; k Ž Hˆ . mL k Ž . Ž L : k Ž . and Proposition 2.3; ; denotes Brauer equivalence. ( k Ž H . mL LŽ ⌳ . mL k Ž . ; LŽ ⌳ . mL k Ž . Ž k Ž H . is Schur over L, hence is split by k Ž ... This implies that expŽ k Ž ⌫ . mk k Ž .. s expŽ LŽ ⌳ . mL k Ž .. which divides expŽ LŽ ⌳ ... It follows that k Ž ⌫ . mk k Ž . divides the number of roots of unity in k and the theorem is proved. Proof of Lemma 2.6 Ž See w3, Lemma 2.3x... Recall that Hˆ normalizes ˆ Ž H .U of the units of k Ž H .U ; hence we may consider the subgroup Hk
ˆ Ž H .U . Since ⌳ l k Ž H .U s k Ž Hˆ .. Let ⌳ be the centralizer of k Ž H . in Hk ˆ Ž H .U rk Ž H .U which is a quotient of HrL ˆ U LU , it follows that ⌳rLU ; Hk U ˆ l H. This implies ⌳rL is finite. Furthermore, the commutator subgroup HˆX of Hˆ is contained in k Ž H .U Žsince HˆX ; ⌫X ; H ., so ⌳rLU is abelian. Finally, we recall from Lemma 2.4 that the order of any element in Hˆ modulo k Ž H .U , hence expŽ ⌳rLU ., divides the number of roots of unity in k. By Proposition 2.5 it follows that the order of LŽ ⌳ . in Br Ž L. divides expŽ ⌳rLU ., hence it also divides the number of roots of unity in k. In the proof of Proposition 2.5, it is shown that LŽ ⌳ . is isomorphic to a product of symbol algebras. To complete the proof of Lemma 2.6, we need to show that k Ž Hˆ . s k Ž H . mL LŽ ⌳ .. By the double centralizer theorem, we have k Ž Hˆ . = k Ž H . mL LŽ ⌳ . s k Ž H . LŽ ⌳ .. In order to prove the reverse incluˆ z normalizes k Ž H . and centralizes its center sion, take an element z g H. L. Therefore, there is an element eŽ z . g k Ž H .U such that zy1 eŽ z . centralizes k Ž H .. This puts zy1 eŽ z . in ⌳, and the lemma is proved. COROLLARY 2.7. If the field k contains all roots of unity then e¨ ery element in PSŽ k . is equi¨ alent to a product of symbol algebras. ŽThis is of course a direct consequence of the Merkurev᎐Suslin Theorem.. Proof. Let k Ž ⌫ . be a projective Schur algebra. We may assume it is reduced. If k contains all roots of unity then the field L above must be k, so the algebra k Ž Hˆ . coincides with the entire algebra k Ž ⌫ .. From the proof above, it follows that k Ž ⌫ . is the tensor product of a Schur algebra k Ž H . over k and symbol algebras. But k Ž H . must be split ŽBrauer’s theorem., hence k Ž ⌫ . ; product of symbol algebras. We turn now to the proof of Theorem 1.4. Let k be a field of characteristic p ) 0 and let A s k Ž ⌫ . be a projective Schur algebra over k. We are to show that A is represented by the tensor product of symbol algebras and an algebra which is split by a cyclotomic extension of k Žnote that in positive characteristic every finite cyclotomic extension is cyclic.. Since the tensor product of such algebras Žtensor product of symbol algebras and an algebra which is split by a cyclotomic extension of k . is again such an algebra, we may assume expŽ A. s q t , where q is a prime number and t ) 0. If k contains no nontrivial qth roots of unity Žin particular if q s p ., A is split by a cyclotomic extension by Theorem 1.1 Ž k has no Kummer q-extension.. On the other hand if k contains all q-power roots of unity, then A is similar to a tensor product of symbol algebras by w8x or Corollary 2.7. We therefore assume k contains a primitive q r th root of unity Ž r ) 0. but not a primitive q rq1 th root of unity. By exponent reduction ŽTheorem 1.3. there is a finite cyclotomic extension k Ž . of k such that expŽ A mk k Ž .. s q s, 0 F s F r. It follows
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that Amq is split by k Ž . and is therefore equivalent to a cyclic algebra Ž k Ž .rk, , a. where Žby abuse of notation. the 2-cocycle is determined by the relation n s a, n s w k Ž . : k x. The key observation here is that the s algebra Amq has a q s th root in Br Ž k . which is split by a cyclotomic extension of k. In other words, there is a cyclic algebra B s Ž k Ž 1 .rk, , b . s s with Bmq ; Amq . Of course B is a radical abelian algebra. Let us complete the proof of the theorem assuming such a B exists. Consider the s algebra C s By1 mk A. Clearly w C x q s 1 in Br Ž k .. Furthermore k cons tains a primitive q th root of unity and so by the Merkurev᎐Suslin Theorem, C may be represented by a product of symbol algebras. The theorem is now proved since A ; B mk C. It remains to show the existence of B. Keeping in mind that charŽ k . s p, we see that the cyclic cyclotomic extension k Ž .rk can be embedded into a cyclic cyclotomic extension k Ž 1 .rk of degree q s n. By w9, p. 262x, the algebra B s Ž k Ž 1 .rk, , a. Žsame a as in A. has the desired property. s
Remark. From the proof we see that if charŽ k . / 0, then a central simple k-algebra has the exponent reduction property if and only if it is a projective Schur algebra over k.
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