Involution Codimensions of Finite Dimensional Algebras and Exponential Growth

Journal of Algebra 222, 471᎐484Ž. 1999 doi:10.1006rjabr.1999.8016, available online at http:rrwww.idealibrary.com on

A. Giambruno View metadata, citation and similar papers at core.ac.uk brought to you by CORE Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Palermo, 90123 Palermo, Italy provided by Elsevier - Publisher Connector E-mail: [email protected]

and

M. Zaicev

Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State Uni¨ersity, Moscow, 119899 Russia E-mail: [email protected]

Communicated by Susan Montgomery

Received March 26, 1999

Let F be a field of characteristic zero and let A be a finite dimensional algebra with involution ) over F. We study the asymptotic behavior of the sequence of n ) ) ) s ) -codimensions cAnnŽ., of A and we show that ExpŽ.A, lim ªϱ 'cAnŽ., exists and is an integer. We give an explicit way for computing ExpŽ.A, ) and as a consequence we obtain the following characterization of )-simple algebras: A is ) ) s ᮊ -simple if and only if ExpŽ.A, dim F A. 2000 Academic Press

1. INTRODUCTION

The purpose of this paper is to determine for any finite dimensional algebra with involution the exponential growth of its sequence of )-codimensions in characteristic zero. Let F be a field of characteristic zero and let A be an F-algebra with ) ) involution . Let VnŽ.be the space of multilinear polynomials in the UU ) variables x11, x ,..., xnn, x and let WA nŽ., be the subspace of multilin- ) ) ear -polynomial identities of A; then the codimension of WAnŽ., is ) cAnŽ., , the nth codimension of A.

471 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 472 GIAMBRUNO AND ZAICEV

A celebrated theorem of Amitsurwx 1 states that if an algebra with involution A satisfies a )-polynomial identity then A satisfies an ordinary Ž.non-involution polynomial identity. In light of this result inwx 7 it was noticed that, as in the ordinary case, if A satisfies a non-trivial )-poly- ) wx nomial identity then cAnŽ., is exponentially bounded. In 2 an explicit ) exponential bound for cAnŽ., was exhibited and this in turn gave a new proof of Amitsur’s theorem where now the degree of an ordinary identity for A is related to the degree of a given )-identity of A. We should also mention that inwx 10 we characterized finite dimensional ) algebras A such that cAnŽ., is polynomially bounded in terms of the representation theory of the hyperoctahedral group Hn. ) wx The asymptotic behavior of cAnŽ., was determined in 4 in case s = ) A MFkŽ.is the algebra of k k matrices over F and is either the ) transpose or the symplectic involution. It turns out that cMFnkŽŽ.., , t 2 n nªϱ Cn k for some explicit constants C and t. This result was achieved by proving that the )-trace codimensions and the )-codimensions are asymptotically equal and then combining a result of Loday and Procesiwx 12 ) on the -trace identities of MFkŽ.with some asymptotic computations of Regevwx 13 . Here we shall determine the exponential behavior of the sequence ) cAnŽ., for any finite-dimensional algebra A. To do so, we define nn ExpŽ.A, ) s lim sup cA Ž., ) , Exp Ž.A, ) s lim inf cA Ž., ) ''nnªϱ nªϱ n and, in case of equality,

ExpŽ.A, ) s Exp Ž.A, ) s Exp Ž.A, ) . ) Let cAnŽ.denote the usualŽ. without nth codimension of the algebra A and let ExpŽ.A denote the corresponding exponential growth. Inwx 8 and wx 9 it was proved that for any PI-algebra A, ExpŽ.A exists and is an integer. By exploiting the methods ofwx 8 we shall prove that if A is finite dimensional, ExpŽ.A, ) exists and can be explicitly computed: suppose U that F is algebraically closed and write A s B q J, where B s B is a maximal semisimple subalgebra and J is the Jacobson radical of A. Then ExpŽ.A, ) s max dim ŽC Ži.Žq иии qC i.., where C Ži.Ž,...,C i. are distinct iF1 tii1 t )-simple subalgebras of B and C Ži. JC Ži. J иии JC Ži. / 0. When F is an 12 ti arbitrary field of characteristic zero, among other consequences we shall prove that for a finite dimensional )-simple F-algebra A, ExpŽ.A, ) s q q q s dim Z A, where Z ZAŽ. is the symmetric center of A. It is worth ) s mentioning that in light of this result, the equality ExpŽŽ..MFk , s wx ExpŽŽ..MFkFdim MFk Ž.obtained in 4 is actually characterizing finite dimensional simple algebras with involution of the first kind i.e., Zqs Z. CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 473

Finally we remark that to compute the exponential behavior of arbitrary algebras with involution A seems to be not an easy task at present; in the ordinaryŽ. non-involution case by a basic reduction of Kemer Ž seewx 11. it is enough to study identities of the Grassmann envelope of a finite dimensional Z2-graded algebra. Unfortunately a similar reduction is not available for algebras with involution at present.

2. CODIMENSIONS AND COCHARACTERS

Throughout F is a field of characteristic zero and A is an F-algebra U U with involution ). Let Aqs Äa g Aa< s a 4 and Ays Äa g Aa< sya 4 be the sets of symmetric and skew elements of A, respectively. ) s UU We let FX²:², Fx11, x , x 22, x , . . .: be the free algebra with involu- ) UUg ) tion of countable rank. Recall that fxŽ 11, x ,..., xnn, x .²:FX, is a ) UUs g -polynomial identity for A if faŽ 11, a ,...,ann, a . 0 for all a1,...,an A. Let IdŽ.A, ) denote the ideal of all )-polynomial identities of A and ) ) U let VnŽ.denote the space of multilinear -polynomials in x11, x ,..., xn, U s q U s y U s xni. If we set s xixiiand k xixi, i 1, 2, . . . , then, since / ) char F 2, it is useful to write VnŽ.as ) s иии <␴ g s s s VnFŽ. Span Ä4w␴ Ž1. w␴ Žn. Sni, w s ior w ik i, i 1,...,n . s ; Let Hn be the hyperoctahedral group. Recall that Hn Z2 Sn is the s ) wreath product of Z2 Ä41, , the multiplicative group of order 2, and Sn. ␴ g ␴ g We write the elements of Hn as Ž.a1,...,ani; , where a Z2 , Sn. ) wx s ␴ g The group Hnnacts on V Ž.as follows Ž see 5.Ž : for h a1,...,an; . s s a␴ Ži. s " Hnidefine hs s␴ Ži., hki k␴ Ži. k␴ Ži. and then extend this action ) ) l ) diagonally to VnnŽ.. Since V Ž.Id ŽA, .is invariant under this action, ) r ) l ) ␹ ) we view VnnŽ.ŽV Ž.Id ŽA, ..as an Hn-module. Its character nŽ.A, is called the nth )-cocharacter of A and ) s ␹ ) s ) r ) l ) cAnnŽ., Ž.Ž.A, 1 dim FVnn Ž.Ž.Ž.Ž.V Id A, is the nth )-codimension of A. Recall that there is a one-to-one correspondence between irreducible ␭ ␮ ␭ & ␮ & y Hn-characters and pairs of partitions Ž., , where r, n r, for s ␹ all r 0, 1, . . . , n.If ␭, ␮ denotes the irreducible Hn-character corresponding to Ž.␭, ␮ , we write n ␹ ) s ␹ nŽ.A, ÝÝm␭, ␮␭, ␮ , rs0 ␭&r ␮&nyr G where m␭, ␮ 0 are the corresponding multiplicities. 474 GIAMBRUNO AND ZAICEV

For r s 0,...,n, we let

s иии <␴ g s s Vr , nyr SpanÄw␴ Ž1. w␴ Žn. Sni,w s ifor i 1,...,r and s s q wiik for i r 1,...,n4

be the space of multilinear polynomials in s1,...,srr, k q1,...,kn. j j To simplify the notation, in the sequel we shall write yii, y and z ii, z for symmetric and skew symmetric variables, respectively, that are indepen- s s s dent from each other. Therefore if we let s11y ,...,srry and k rq1 s z1,...,knnz yr , then

s Vr , nyr the space of multilinear polynomials in y1 ,..., yr , z1 ,..., znyr .

) l ) l It is clear that to study VnrŽ.Id ŽA, .it is enough to study V , nyr IdŽ.A, ) for all r.

If we let Sr act on the symmetric variables y1,..., yrnand S yr act = on the skew variables z1,..., znyrrn, we obtain an action of S S yr on r l ) = Vr, nyrrand V , nyrrŽŽ..V , nyrrnId A, is a left S S yr-module. Let ␺ ) ) s ␺ ) s r, nyrrŽ.A, be its character and let cA, nyrrŽ., , nyr Ž.Ž.A, 1 r l ) dim FrV , nyrrŽŽ..V , nyr Id A, . It is well known that there is a one-to-one correspondence between = ␭ ␮ irreducible SrnS yr-characters and pairs of partitions Ž., such that ␭ & r, ␮ & n y r. For any partition ␯ of an integer t let ␹␯ be the ␹ m ␹ corresponding irreducible St-character. Then ␭␮is the irreducible = ␭ ␮ SrnS yr-character associated to the pair Ž., . The following result holds

THEOREM 1wx 5, Theorem 1.3 . Let A be a PI-algebra with in¨olution; then, for all r F n,

n ␹ ) s ␹ nŽ.A, ÝÝm␭, ␮␭, ␮ and rs0 ␭&r ␮&nyr ␺ ) s ␹ m ␹ r , nyr Ž.A, Ý m␭, ␮␭Ž. ␮. ␭&r ␮&nyr

Moreo¨er

n n cAŽ., ) s cAy Ž., ) . nrÝ ž/r , n r rs0 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 475

3. GUESSING ExpŽ.A, )

Throughout A will be a finite dimensional algebra with involution over the field F. We write A s B q J, where B is a maximal semisimple subalgebra of A and J s JAŽ.is the Jacobson radical of A. Clearly U U J s J and, according towx 10, Theorem 4 we may assume that also B s B s [ иии [ is stable under the involution. Write B B1 Bm where B1,...,Bm ) s are -simple algebras. Recall that for each 1,...,m either Bi is a finite ( [ op dimensional simple algebra with induced involution or Bi C C , ) [ op where C is a simple homomorphic image of Bi and on C C is the U exchange involution Ž.a, b s Ž.Žb, a seewx 14, Proposition 2.13.24. . We now define an integer d s dAŽ.in the following way: we consider all possible nonzero products of the type

иии / CJCJ12 CJCky1 k 0, where C1,...,Ck are distinct subalgebras from the set Ä4B1,...,Bm and G s s F F k 1ifŽ.k 1 this means that C1 Bi for some 1 i n . We then s q иии q define d dAŽ.to be the maximal dimension of a subalgebra C1 Ck satisfying the above inequality. We shall prove that in case F is algebraically closed, d coincides with ExpŽ.A, ) . We note that if B ,...,B are not necessarily distinct among the B ’s ii1 t j and BJBJиии JB / 0 then dimŽ.ŽB q иии qB F d seewx 8, Lemma 3. . ii12 it i1 it

4. MULTIALTERNATING POLYNOMIALS G q ) LEMMA 1. Let t 0, a b d, and fŽ. y1,..., ya, z1,..., zb, x1,..., xt be a multilinear polynomial alternating onÄ4Ä4 y1,..., ya and on z1,..., zb . If g qyg g y1,..., ya B , z1,..., zb B , x1,..., xt A then fŽ y1,..., ya, z1,..., s zb, x1,..., xt . 0. s s q j y Proof. For each i 1,...,m, let EiiE E ibe a basis of Bi, q y where Eiiand E are sets of symmetric and skew elements, respectively. s D s qj y qs D q Then E Ei E E is a basis of B, where E Ei and ys D y E Ei . Since f is multilinear, it is enough to evaluate f on a basis of A. The proof is similar to the proof ofwx 8, Lemma 3 . If all the variables in f s / are basis elements of B, then, since BBij 0 for i j we will get a zero ) value unless all elements come from one -simple component, say Bt.In - q ) ) < q< ) this case, since dim Bttd and a b d, then either a E or b < y< Eti. Since f is alternating in the y ’s and the zi’s then the value of f will still be zero. Therefore, to get a non-zero value of f we should evaluate at least one variable on elements of J. This case also leads to a zero value of f by the maximality of d Žsee details inwx 8, Lemma 3. . 476 GIAMBRUNO AND ZAICEV

␭ For any partition of an integer t and a Young tableau T␭ we let RT␭

and CT␭ be the subgroups of row and column permutations of T␭, respectively; then e s Ý␴ g ␶ g Ž.sgn ␶␴␶is an essential idempotent of FS T␭ RTT, Ct␭

and FStT e ␭ is a minimal left ideal of FSt. ␭ & ␮ & y = Let r, n r, and W␭, ␮ be a left irreducible SrnS yr-module; if T␭ is a tableau of shape ␭ and T␮ is a tableau of shape ␮, then ( Ž.= W␭, ␮ FSrnSeeyrTT␭ ␮, where S rand S nyr act on disjoint sets of integers. ␭ ␭X s ␭XX␭ For a partition we denote with Ž 12, , . . .. the conjugate partition of ␭. ␭ & ␮ & y : REMARK 1. Let r, n r, and W␭, ␮ Vr, nyr be a left irre- = g ducible SrnS yr-module. Then there exists f W␭, ␮ such that

1 1 2 2 1 1 2 2 f s fy,..., y␭X , y ,..., y␭X ,..., z ,..., z␮X , z ,..., z␮X ,... Ž.1 121 1 11 2

¨ iiXXjjF F and f is alternating on each set of ariablesÄ y1,..., y␭ 4Ä, z1,..., z␮ 4,1 i ␭ F F ␮ ij 11,1 j . g Proof. Let f0 W␭, ␮ be a non-zero polynomial. Then there exist s / tableaux T␭, T␮ such that g eeTT␭ ␮ f0 0. Denote f s ÝÝŽ.Ž.sgn ␶ sgn ␳ g. ␶g ␳g CTT␭ C ␮

Then f is a polynomial with the prescribed property.

5. THE UPPER BOUND

␭ ␭ s ␭X Recall that for a partition , hŽ. 1 is the height of the corresponding diagram. ␭ & ␮ & y : LEMMA 2. Let r, n r, and W␭, ␮ Vr, nyr be a left irreducible = l ) ␭ F q ␮ SrnS yr-module. If W␭, ␮ Vr, nyr IdŽ.A, , then h Ž.dim A , hŽ. F y ␭X q ␮X F lq1 s ¨ F dim A , and lq1 lq1 d, where J 0. Moreo er dim W␭, ␮ a ␭␮X r X nyr G n Ž lq1. Ž lq1. for some a 1. Proof. Let f be the polynomial described in the previous remark. s = 11X Clearly W␭, ␮ FSŽ.rnSfyr . Since f is alternating on Ä y1,..., y␭ 4,it ␭X s ␭ F q ␮ F y 1 follows that 1 hŽ. dim A . Similarly, hŽ. dim A . Suppose that ␭X q ␮X ) ␭X q ␮X ) s lq1 lq1 d. Then iid for all i 1,...,l and, by the previous lemma, since f is not a )-identity for A, in each set of variables iiii Ä y ,..., y␭XX, z ,..., z␮ 4,1F i F l q 1, we must substitute at least one 1 ii1 element from J. But J lq1 s 0 leads to f being a )-identity for A. CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 477

Let dim Aqs p and dim Ays q. From the hook formula for the degrees of the irreducible representations of the symmetric group it follows ␹ F pl ␭X r ␹ F ql ␮X nyr s that ␭Ž.1 n Ž lq1. and ␮Ž.1 n Ž lq1. . Therefore dim W␭, ␮ ␹ ␹ F plqql␭␮XX r nyr ␭Ž.1 ␮ Ž.1 n Žlq1 .Žlq1 . .

THEOREM 2. If A is a finite dimensional algebra with in¨olution o¨er F ¨ ) F tn and d is the integer defined abo e, then cnŽ. A, an d , for some constants a, t. ␭ ␭X s ␭ Proof. For a partition let us write lq1 tŽ.. From the previous lemma we have

␺ ) s ␹ m ␹ r , nyr Ž.A, ÝÝm␭, ␮␭Ž. ␮. 0FtŽ.␭ qt Ž␮ .Fd ␭&r ␮&nyr

Hence, for some constant ␣,

) F ␣ ␭ rn␮ yr cAr , nyr Ž., ÝÝmnt␭, ␮ Ž.Ž.t . 0FtŽ.␭ qt Ž␮ .Fd ␭&r ␮&nyr

wx Since by 3 the multiplicities m␭, ␮ are polynomially bounded, from Theo- rem 1 we get

nn ) s nn) F brnyr cAnrŽ., ÝÝcA, nyr Ž., anÝ t12 t s ž/rrq F s ž/ r 0 t12t dr 0 G t12, t 0 s bbnq n F anÝ Ž. t12t an dd . q F t12t d G t12, t 0

6. CENTRAL POLYNOMIALS

The existence of multialternating central polynomials for k = k matrices proved inwx 6 easily leads to the following

LEMMA 3. Let C be a finite dimensional central )-simple algebra o¨er F, dim Cqs p and dim Cys q. For all m G 1 there exists a multilinear polynomial

s 1 1 2 m 2 m 1 1 2 m 2 m f fyŽ.1 ,..., yp ,..., y1 ,..., yp , z1 ,..., zq ,..., z1 ,..., zq 478 GIAMBRUNO AND ZAICEV

such that ¨ iiF F Ž.1 f is alternating on each set of ariablesÄ y1,..., yp4,1 i 2m, jjF F andÄ z1,..., zq4,1 j 2m; j g q j g y 12m 12m Ž.2 there exist yiiC , z C such that fŽ y1,..., yp , z1,..., zq . s 1.C Proof. Suppose first that C is simple. Bywx 6 for every m G 1 there 11 2m 2 m exists a multilinear polynomial fxŽ 1,..., x pqq ,..., x1 ,..., x pqq . alternat- iiF F ing on each set of variables Ä x1,..., x pqq4,1 i 2m, and f is a central polynomial for C. Since p q q s dim C it is clear that f can be viewed as alternating on 2m disjoint sets of symmetric or skew variables. s [ op In case C C11C with exchange involution and C1simple, then s s p q dim C1 and the polynomial

1 1 2 m 2 m 1 1 2 m 2 m fyŽ.1 ,..., yp ,..., y1 ,..., yp fzŽ.1 ,..., zp ,..., z1 ,..., zp is the required one. The polynomials found in the previous lemma will now be ‘‘glued together’’ to find multialternating polynomials of suitable degree non- vanishing in a finite dimensional algebra.

LEMMA 4. Let F be algebraically closed and let A be a finite dimensional ¨ иии / algebra o er F. Let C12 JC J CJCky1 k 0, where C1,...,Ck are distinct ) q иии q s [ иии [ s -simple subalgebras of A and C1 Ck C1 Ck . If p q иии q q s q иии q y G dimŽ.C1 Ck , q dimŽ.C1 Ck , then for all m 1 there exists a multilinear polynomial

s 1 1 2 m 2 m 1 1 2 m f fyŽ 1 ,..., yp ,..., y1 ,..., yp , z1 ,..., zq ,..., z1 ,...,

2 m zq , y12,..., y k , z12,..., z k .

such that ¨ iiF F Ž.1 f is alternating on each set of ariablesÄ y1,..., yp4,1 i 2m, jjF F andÄ z1,..., zq4,1 j 2m; j g q иии q q j g q иии q yqg Ž.2 there exist yi ŽC1 Cki.Ž, z C1 Cki., y A , g y 12m 12m / zi A such that fŽ. y1,..., yp , z1,..., zq , y12,..., y k , z12,..., z k 0. s s q s y Proof. For every i 1,...,k, let piiiidim C , q dim C , and let

f s fy1 ,..., y1 ,..., y 2 m ,..., y 2 m , z1 ,..., z1 ,..., z 2 m ,..., z 2 m iiiŽ.,1 i, piiiii,1 i, pi,1 i, qi,1 i, q CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 479 be the polynomial constructed in the previous lemma. We let

s иии XXиии Xиии X f˜ A12A m A12A m xfxxf11122 xfxxfky1 ky1 ky1 kk, where A means alternation on the p variables y i ,..., y i ,..., y i , i X 1, 1 1, pk1 ,1 ..., y i and A means alternation on the q variables z i ,..., z i ,..., k, pik 1, 1 1, q1 z i ,..., z i . k,1 k, qk We note that each polynomial fi corresponds to a pair of tableaux = = Ž.Pii, Q , where Piis a 2m pirectangle, Qiis a 2m qirectangle, and the variables in each column of PiiŽ.resp. Q are alternating. Now, f˜ corresponds to the pair Ž.P, Q obtained by gluing the rectangles Pi Žresp. Qi. one on top of the other and by alternating the variables in the columns; hence Aj is alternation on the variables in the jth column of P X and Aj is alternation on the variables in the jth column of Q. иии / g F F g Since CJCJ12 CJCky1 kii0, let c C Ž.1 i k , b1,...,bky1 J иии / be such that cbcb1122 bcky1 k 0. s t g q t g y For every i 1,...,k, let yi, jiiC , z , jiC be such that

fy12,..., y m , z12,..., z m s 1. iiž/,1 i, piii,1 i, qCi

s / Notice that since CCij 0 for i j, when evaluating the variables y and z on the Ci’s, alternation on the columns of the rectangle P Ž.resp. Q can be replaced with alternation on the columns of each subrectangle Pi Žresp. Qi.. Hence

12m 12m fy˜ ,..., y , z ,..., z , c ,...,c , b ,...,b y ž/1,1 k , pkk1,1 k , q 1 k 1 k 1 s иии иии 2 m иии Ž.p1 ! pk !q1 ! qk ! cfbcfb11122 2 cfbcfky1 ky1 ky1 kk s иии иии 2 m иии / Ž.p1 ! pk !q1 ! qk ! cbc112 bcky1 k 0.

g qj y We may clearly assume that c11, b ,...,bky1, ck A A ; suppose that q s y k12of them are symmetric and k of them are skew, k1k22k 1. Then

f s fy˜ 12,..., y m , z12,..., z m , y ,..., y , z ,..., z Ž.1,1 k , pkk1,1 k , q 1 k121 k

и y q иии yzq иии z k1212kk 12k does not vanish in A and is the desired polynomial. 480 GIAMBRUNO AND ZAICEV

7. THE LOWER BOUND

We are now in a position to find the lower bound for the exponential growth of the )-codimensions.

THEOREM 3. Let A be a finite dimensional algebra with in¨olution o¨er the algebraically closed field F and let d be the integer defined in Section 3. ) G bn Then cnŽ. A, an d , for some constants a, b. s q ) Proof. Let A B J and C1,...,Ck distinct -simple subalgebras of иии / s q иии q q s B such that CJCJ12 CJCky1 k 0. Let p dimŽ.C1 Ck , q q иии q y s q dimŽ.C1 Ck ; then d p q. Let n G 2 d q 4k and divide n y 4k by 2 d; then there exist m, t such that n y 4k s 2md q t,0F t - 2 d. Thus n s 2mpŽ.q q q 4k q t. Write s s 2mp q 2k q t,0F t - 2 d, n y s s 2mq q 2k.If f is the polynomial of the previous lemma of total degree 2mp q 2mq q 4k, then s иии g ) the polynomial g fy2 kq12y kqtsV , nys and g is not a -identity for A Žwe may make suitable substitutions for the extra variables: note that f иии has non-zero value which belongs to CJ1 JCk , hence we can take y q s иии s y q s 1.. 2 k 12k tCk s = We let the group G S2 mp S2 mq act on g by letting S2 mp act on the l l symmetric variables yi and S2 mqact on the skew variables z i. Let M be the G-submodule of Vs, nys generated by g. By complete reducibility M s Ž. contains an irreducible G-submodule of the form W␭, ␮ FGeTT␭ e␮ g , for some tableaux T␭ and T␮ , where ␭ & 2mp and ␮ & 2mq. Let lŽ.␭ be the length of the first row of ␭. Note that for every ␶ g ␶ S2 mp, Ž.g is still alternating on 2m disjoint sets of symmetric variables, and Ý␴ g ␴ acts by symmetrizing lŽ.␭ variables. It follows that if RT␭ Ž.␭ ) Ž.s Ž.␮ l 2m then egT␭ 0, a contradiction. Similarly for l . Therefore lŽ.␭ F 2m and l Ž␮ .F 2m. Suppose now that ␭ has height hŽ.␭ ) p. From the previous lemma we know that f Ž.hence g takes a non-zero value on A after replacing the j qs q иии q q variables yi with suitable elements from C Ž.C1 Ck . Since qs Ž. dim C p, the polynomial egT␭ vanishes on A since it is alternating on p q 1 variables. Hence hŽ.␭ F p. Similarly hŽ.␮ F q. We have proved that M contains an irreducible G-submodule of the p q form W␭ ␮ , where ␭ s ŽŽ2m . .ŽŽ.and ␮ s 2m . are two rectangles. , X b b 2 mp 2 mq Recall that as m ª ϱ,Ž.Ž.Ž. deg ␹␭ deg ␹␮ , a 2mp Ž.2mq p q for X some negative constants b, b and some a. It follows that

) G G c 2 mp 2 mq cAs, nys Ž., dim W␭, ␮ n p q . CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 481

Therefore we have

n nn cAŽ., ) s cAy Ž., ) G cAy Ž., ) nrÝ ž/rs, n rs ž/, n s rs0 n! G npqc 2 mp 2 mq. s!Ž.n y s !

Recalling that s s 2mp q 2k q t and n y s s 2mq q 2k, hence

n!2Ž.mp q 2mq ! G s!Ž.Ž.Ž.n y s !2mp !2mq ! and by Stirling formula we get

q Ž.2mp q 2mq 2 mp 2 mq ) G ␣ 2 mp 2 mq s ␣ q 2 mpq2 mq cAnŽ., npqnpŽ.q Ž.Ž.2mp 2 mp 2mq 2 mq n s an␣␣Ž. p q q s an d n , where a s Ž.p q q yŽ4 kqt. is a constant.

8. CONSEQUENCES

We record two immediate consequences of Theorem 2 and Theorem 3.

COROLLARY 1. If A is a finite dimensional algebra with in¨olution o¨er ) F the field F, then ExpŽ.A, exists and is an integer dim F A. ŽNote that the procedure of algebraic closure of the ground field does not affect the nth codimension..

COROLLARY 2. Let A be a finite dimensional algebra with in¨olution o¨er U the algebraically closed field F and let B s B be a maximal semisimple subalgebra of A. Then ExpŽ.A, ) s max dim ŽC Ži.Žq иии qC i.., where iF1 ti C Ži.,...,C Ži. are distinct )-simple subalgebras of B and C Ži. JC Ži. J иии JC Ži. 1 ti 12 ti / 0. Let F be the algebraic closure of the field F. For an algebra with involution A over F we let Z s ZAŽ.be the center of A and let Zqs ZAŽ.qqs Z l A be the symmetric center of A. Next two results give an exact estimate of ExpŽ.A, ) in the case of simple or semisimple algebras. 482 GIAMBRUNO AND ZAICEV

COROLLARY 3. Let A be a finite dimensional )-simple algebra with ¨ ¨ ) s q in olution o er F. Then ExpŽ.A, dim Z A. Proof. Suppose first that A is simple. If Zqs Z then

wxZ : F m ( m A FZF [ Ž.Z Fi, is1 ( m where FiZF and A Fiis a central simple algebra over F with induced involution. Moreover

wx s s m s wx m Z : F dim ZA dim FA dim FFŽ.A F Z : F dim FZi Ž.A F . s m s m From Corollary 2 it follows that dim ZFZA dim Ž.ŽA FiExp A F F, ).Ž.s Exp A, ) . In case Zq/ Z, i.e., ) is an involution of the second kind, then U Z s ZqŽ.␣ with ␣ sy␣ g Z is a quadratic extension of Zq and m q ( [ Z Z F F F with exchange involution. Hence

m qq( m m ( m [ ( m [ m A ZZZZF A Z F A Ž.F F Ž.Ž.A ZZF A F m q m UUs m q Um with involution given by Ž.a f12b f b f 1a f 2. As above it follows that

wZq : FZxwq : F x

m ( m qq( m A FZF A [[F iŽ.A ZiF ž/is1 is1 wZq : F x ( m [ m [ Ž.A ZiF A ZiF , is1 s wxq ( m [ m ) where, for i 1,..., Z : F , FiZF and A FiZA Fiis a -simple algebra with exchange involution. As in the previous case it follows that

q s m [ m s m ) s ) dim ZFZA dim Ž.A FiZA FiExpŽ.A FF, ExpŽ.A, .

Suppose now that A is not simple. Then A ( C [ C op, where C is simple and ) is the exchange involution. In this case since Zqs ZAŽ.q( ZCŽ., we get

wZCŽ.: F x m ( [ m [ op m A FZF ž/ž/C ŽC . FiZC ŽC . Fi , is1 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 483

( m [ op m ) where FiZF and Ž.Ž.C ŽC . FiZC ŽC . Fi is -simple over F with exchange involution. Thus as before we get

q s m [ m s m ) s ) dim ZFZA dim Ž.A FiZA FiExpŽ.A FF, ExpŽ.A, .

COROLLARY 4. Let A be a finite dimensional semisimple algebra with ¨ ¨ s [ ) in olution o er F. If A i Ai is the decomposition of A into -simple q q subalgebras, then ExpŽ.A, ) s max dim q A , where Z s ZAŽ.l Ais iZii i i i the symmetric center of Ai. m ([ m Proof. We have A FiF Ž.A FF and by the previous corollary m ( [ иии [ ( иии ( ) AiFF B i1 Bit, where B i1 Bit are -simple algebras s wxiiq ) s m central over F and tiiZ : F . It follows that ExpŽ.A, Exp ŽA F F, ). s max dim B ; since dim q A s dim B , the conclusion of the iFi1 Zii Fi1 corollary now follows.

COROLLARY 5. Let A be a finite dimensional algebra with in¨olution o¨er ) s ) s q F. Then ExpŽ.A, dim F A if and only if A is -simple and F Z . Proof. In light of Corollary 3 we only need to show that ExpŽ.A, ) s ) s q s m dim F A implies A -simple and F Z . Let A A F F with induced s s ) s ) involution. Then dim FFA dim A ExpŽ.A, Exp Ž.A, .If A is nilpotent, then ExpŽ.A, ) s 0, a contradiction. Hence A contains a maxi- s U mal semisimple subalgebra B B and by the previous corollary dim F A s ) s ) ExpŽ.A, dim FiB for a suitable -simple subalgebra Biof B. s ) ) Hence A Bi is -simple. This implies that A is -simple and by q s ) s q s Corollary 3 dim FZA ExpŽ.A, dim A implies F Z .

ACKNOWLEDGMENT

The first author was partially supported by CNR and MURST of Italy; the second author was partially supported by RBRF grants 96-01-00146 and 96-15-96050.

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