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Involution Codimensions of Finite Dimensional Algebras and Exponential Growth
Journal of Algebra 222, 471᎐484 Ž1999. doi:10.1006rjabr.1999.8016, available online at http:rrwww.idealibrary.com on
Involution Codimensions of Finite Dimensional Algebras and Exponential Growth A. Giambruno Dipartimento di Matematica e Applicazioni, Uni¨ ersita ` di Palermo, 90123 Palermo, Italy 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 )-codimensions c nŽ A, ). of A and we show that ExpŽ A, ). s lim n ª ⬁ c n Ž A, ) . 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 )-simple if and only if ExpŽ A, ). s dim F A. 䊚 2000 Academic Press
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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 variables x 1 , xU1 , . . . , x n , xUn and let WnŽ A, ). be the subspace of multilinear )-polynomial identities of A; then the codimension of WnŽ A, ). is c nŽ A, )., the nth codimension of A. 471 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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GIAMBRUNO AND ZAICEV
A celebrated theorem of Amitsur w1x states that if an algebra with involution A satisfies a )-polynomial identity then A satisfies an ordinary Žnon-involution. polynomial identity. In light of this result in w7x it was noticed that, as in the ordinary case, if A satisfies a non-trivial )-polynomial identity then c nŽ A, ). is exponentially bounded. In w2x an explicit exponential bound for c nŽ A, ). 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 in w10x we characterized finite dimensional algebras A such that c nŽ A, ). is polynomially bounded in terms of the representation theory of the hyperoctahedral group Hn . The asymptotic behavior of c nŽ A, ). was determined in w4x in case A s Mk Ž F . is the algebra of k = k matrices over F and ) is either the transpose or the symplectic involution. It turns out that c nŽ Mk Ž F ., ). ,nª⬁ Cn t k 2 n 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 Procesi w12x on the )-trace identities of Mk Ž F . with some asymptotic computations of Regev w13x. Here we shall determine the exponential behavior of the sequence c nŽ A, ). for any finite-dimensional algebra A. To do so, we define Exp Ž A, ) . s lim sup nª⬁
n
'c Ž A, ). , n
Exp Ž A, ) . s lim inf
nª⬁
n
'c Ž A, ). n
and, in case of equality, Exp Ž A, ) . s Exp Ž A, ) . s Exp Ž A, ) . . Let c nŽ A. denote the usual Žwithout ). nth codimension of the algebra A and let ExpŽ A. denote the corresponding exponential growth. In w8x and w9x it was proved that for any PI-algebra A, ExpŽ A. exists and is an integer. By exploiting the methods of w8x we shall prove that if A is finite dimensional, ExpŽ A, ). exists and can be explicitly computed: suppose that F is algebraically closed and write A s B q J, where B s BU is a maximal semisimple subalgebra and J is the Jacobson radical of A. Then ExpŽ A, ). s max i dim F Ž C1Ž i. q ⭈⭈⭈ qCtŽii. ., where C1Ž i., . . . , CtŽii. are distinct )-simple subalgebras of B and C1Ž i. JC 2Ž i. J ⭈⭈⭈ JCtŽii. / 0. When F is an arbitrary field of characteristic zero, among other consequences we shall prove that for a finite dimensional )-simple F-algebra A, ExpŽ A, ). s dim Z q A, where Zqs ZŽ A.q is the symmetric center of A. It is worth mentioning that in light of this result, the equality ExpŽ Mk Ž F ., ). s ExpŽ Mk Ž F .. s dim F Mk Ž F . obtained in w4x 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 ordinary Žnon-involution. case by a basic reduction of Kemer Žsee w11x. it is enough to study identities of the Grassmann envelope of a finite dimensional Z 2-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 with involution ). Let Aqs a g A < a s aU 4 and Ays a g A < a s yaU 4 be the sets of symmetric and skew elements of A, respectively. We let F² X, ): s F² x 1 , xU1 , x 2 , xU2 , . . . : be the free algebra with involution ) of countable rank. Recall that f Ž x 1 , xU1 , . . . , x n , xUn . g F² X, ): is a )-polynomial identity for A if f Ž a1 , aU1 , . . . , a n , aUn . s 0 for all a1 , . . . , a n g A. Let IdŽ A, ). denote the ideal of all )-polynomial identities of A and let VnŽ). denote the space of multilinear )-polynomials in x 1 , xU1 , . . . , x n , xUn . If we set si s x i q xUi and k i s x i y xUi , i s 1, 2, . . . , then, since char F / 2, it is useful to write VnŽ). as Vn Ž ) . s Span F w Ž1. ⭈⭈⭈ w Ž n. < g Sn , wi s si or wi s k i , i s 1, . . . , n4 . Let Hn be the hyperoctahedral group. Recall that Hn s Z 2 ; S n is the wreath product of Z 2 s 1, )4 , the multiplicative group of order 2, and Sn . We write the elements of Hn as Ž a1 , . . . , a n ; ., where a i g Z 2 , g Sn . The group Hn acts on VnŽ). as follows Žsee w5x.: for h s Ž a1 , . . . , a n ; . g a Hn define hsi s s Ž i. , hk i s k Ž i.Ž i. s "k Ž i. and then extend this action diagonally to VnŽ).. Since VnŽ). l IdŽ A, ). is invariant under this action, we view VnŽ).rŽ VnŽ). l IdŽ A, ).. as an Hn-module. Its character nŽ A, ). is called the nth )-cocharacter of A and c n Ž A, ) . s n Ž A, ) . Ž 1 . s dim F Vn Ž ) . r Ž Vn Ž ) . l Id Ž A, ) . . is the nth )-codimension of A. Recall that there is a one-to-one correspondence between irreducible Hn-characters and pairs of partitions Ž , ., where & r, & n y r, for all r s 0, 1, . . . , n. If , denotes the irreducible Hn-character corresponding to Ž , ., we write n
n Ž A, ) . s
Ý Ý rs0
&r &nyr
m , , ,
where m, G 0 are the corresponding multiplicities.
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GIAMBRUNO AND ZAICEV
For r s 0, . . . , n, we let Vr , nyr s Span w Ž1. ⭈⭈⭈ w Ž n. < g Sn ,wi s si for i s 1, . . . , r and wi s k i for i s r q 1, . . . , n4 be the space of multilinear polynomials in s1 , . . . , sr , k rq1 , . . . , k n . To simplify the notation, in the sequel we shall write yi , yij and z i , z ij for symmetric and skew symmetric variables, respectively, that are independent from each other. Therefore if we let s1 s y 1 , . . . , sr s yr and k rq1 s z1 , . . . , k n s z nyr , then Vr , nyr s the space of multilinear polynomials in y 1 , . . . , yr , z1 , . . . , z nyr . It is clear that to study VnŽ). l IdŽ A, ). it is enough to study Vr, nyr l IdŽ A, ). for all r. If we let S r act on the symmetric variables y 1 , . . . , yr and S nyr act on the skew variables z1 , . . . , z nyr , we obtain an action of Sr = Snyr on Vr, nyr and Vr, nyrrŽ Vr, nyr l IdŽ A, ).. is a left Sr = Snyr-module. Let r, nyr Ž A, ). be its character and let c r, nyr Ž A, ). s r, nyr Ž A, ).Ž1. s dim F Vr, nyrrŽ Vr, nyr l IdŽ A, )... It is well known that there is a one-to-one correspondence between irreducible Sr = Snyr-characters and pairs of partitions Ž , . such that & r, & n y r. For any partition of an integer t let be the corresponding irreducible St-character. Then m is the irreducible Sr = Snyr-character associated to the pair Ž , .. The following result holds THEOREM 1 w5, Theorem 1.3x. then, for all r F n,
Let A be a PI-algebra with in¨ olution;
n
n Ž A, ) . s
Ý Ý rs0
r , nyr Ž A, ) . s
&r &nyr
Ý
&r &nyr
m , ,
and
m , Ž m . .
Moreo¨ er n
c n Ž A, ) . s
Ý rs0
ž nr / c
r , nyr
Ž A, ) . .
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 J Ž A. is the Jacobson radical of A. Clearly J U s J and, according to w10, Theorem 4x we may assume that also BU s B is stable under the involution. Write B s B1 [ ⭈⭈⭈ [ Bm where B1 , . . . , Bm are )-simple algebras. Recall that for each s 1, . . . , m either Bi is a finite dimensional simple algebra with induced involution or Bi ( C [ C op , where C is a simple homomorphic image of Bi and ) on C [ C op is the exchange involution Ž a, b .U s Ž b, a. Žsee w14, Proposition 2.13.24x.. We now define an integer d s dŽ A. in the following way: we consider all possible nonzero products of the type C1 JC 2 J ⭈⭈⭈ Cky1 JCk / 0, where C1 , . . . , Ck are distinct subalgebras from the set B1 , . . . , Bm 4 and k G 1 Žif k s 1 this means that C1 s Bi for some 1 F i F n.. We then define d s dŽ A. to be the maximal dimension of a subalgebra C1 q ⭈⭈⭈ qCk satisfying the above inequality. We shall prove that in case F is algebraically closed, d coincides with ExpŽ A, ).. We note that if Bi1, . . . , Bi t are not necessarily distinct among the Bj ’s and Bi1 JBi 2 J ⭈⭈⭈ JBi t / 0 then dimŽ Bi1 q ⭈⭈⭈ qBi t . F d Žsee w8, Lemma 3x.. 4. MULTIALTERNATING POLYNOMIALS LEMMA 1. Let t G 0, a q b ) d, and f Ž y 1 , . . . , ya , z1 , . . . , z b , x 1 , . . . , x t . be a multilinear polynomial alternating on y 1 , . . . , ya4 and on z1 , . . . , z b 4 . If y 1 , . . . , ya g Bq, z1 , . . . , z b g By, x 1 , . . . , x t g A then f Ž y 1 , . . . , ya , z1 , . . . , z b , x 1 , . . . , x t . s 0. Proof. For each i s 1, . . . , m, let Ei s Eiq j Eiy be a basis of Bi , where Eiq and Eiy are sets of symmetric and skew elements, respectively. Then E s D Ei s Eqj Ey is a basis of B, where Eqs D Eiq and Eys D Eiy . Since f is multilinear, it is enough to evaluate f on a basis of A. The proof is similar to the proof of w8, Lemma 3x. If all the variables in f are basis elements of B, then, since Bi Bj s 0 for i / j we will get a zero value unless all elements come from one )-simple component, say Bt . In < or b ) this case, since dim Bt - d and a q b ) d, then either a ) < Eq t < Ey < . Since f is alternating in the y ’s and the z ’s then the value of f will t i i 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 in w8, Lemma 3x..
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GIAMBRUNO AND ZAICEV
For any partition of an integer t and a Young tableau T we let R T and CT be the subgroups of row and column permutations of T , respectively; then eT s Ý g R T , g C T Žsgn . is an essential idempotent of FS t and FS t eT is a minimal left ideal of FS t . Let & r, & n y r, and W, be a left irreducible S r = S nyr-module; if T is a tableau of shape and T is a tableau of shape , then W, ( F Ž Sr = Snyr . eT eT , where Sr and Snyr act on disjoint sets of integers. For a partition we denote with X s Ž X1 , X2 , . . . . the conjugate partition of . REMARK 1. Let & r, & n y r, and W, : Vr, nyr be a left irreducible Sr = Snyr-module. Then there exists f g W , such that f s f Ž y 11 , . . . , y1X1 , y 12 , . . . , y2X2 , . . . , z11 , . . . , z1X1 , z12 , . . . , z2X2 , . . . . and f is alternating on each set of ¨ ariables y 1i , . . . , yiXi 4 , z1j , . . . , zj Xj 4 , 1 F i F 1 , 1 F j F 1. Proof. Let f 0 g W , be a non-zero polynomial. Then there exist tableaux T, T such that g s eT eT f 0 / 0. Denote fs
Ý
Ý Ž sgn . Ž sgn . g .
gC T gC T
Then f is a polynomial with the prescribed property.
5. THE UPPER BOUND Recall that for a partition , hŽ . s X1 is the height of the corresponding diagram. LEMMA 2. Let & r, & n y r, and W, : Vr, nyr be a left irreducible Sr = Snyr-module. If W, Vr, nyr l IdŽ A, )., then hŽ . F dim Aq, hŽ . F dim Ay, and Xlq1 q Xlq1 F d, where J lq1 s 0. Moreo¨ er dim W, F n a Ž Xlq1 . r Ž Xlq1 . ny r for some a G 1. Proof. Let f be the polynomial described in the previous remark. Clearly W, s F Ž S r = S nyr . f. Since f is alternating on y 11 , . . . , y1X14 , it follows that X1 s hŽ . F dim Aq. Similarly, hŽ . F dim Ay. Suppose that Xlq1 q Xlq1 ) d. Then Xi q Xi ) d for all i s 1, . . . , l and, by the previous lemma, since f is not a )-identity for A, in each set of variables y 1i , . . . , yiX , z1i , . . . , zi X 4 , 1 F i F l q 1, we must substitute at least one i i element from J. But J lq1 s 0 leads to f being a )-identity for A.
477
CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
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 that Ž1. F n p l Ž Xlq1 . r and Ž1. F n ql Ž Xlq1 . ny r. Therefore dim W , s Ž1. Ž1. F n p lqql Ž Xlq1 . r Ž Xlq1 . nyr. THEOREM 2. If A is a finite dimensional algebra with in¨ olution o¨ er F and d is the integer defined abo¨ e, then c nŽ A, ). F ant d n, for some constants a, t. Proof. For a partition let us write Xlq1 s t Ž .. From the previous lemma we have
r , nyr Ž A, ) . s
Ý
0Ft Ž .qt Ž .Fd
Ý
&r &nyr
m , Ž m . .
Hence, for some constant ␣ , c r , nyr Ž A, ) . F
Ý
0Ft Ž .qt Ž .Fd
Ý
&r &nyr
r
m , n ␣ t Ž . t Ž .
nyr
.
Since by w3x the multiplicities m, are polynomially bounded, from Theorem 1 we get n
c n Ž A, ) . s
Ý rs0
s an b
ž nr / c
n r , nyr
Ž A, ) . F an b
Ý
Ý
t 1qt 2Fd rs0 t1 , t 2G0
ž nr / t t
r nyr 1 2
n Ý Ž t1 q t 2 . F anbdd n .
t1qt 2Fd t1 , t 2G0
6. CENTRAL POLYNOMIALS The existence of multialternating central polynomials for k = k matrices proved in w6x 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 f s f Ž y 11 , . . . , y p1 , . . . , y 12 m , . . . , y p2 m , z11 , . . . , z 1q , . . . , z12 m , . . . , z q2 m .
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GIAMBRUNO AND ZAICEV
such that Ž1. f is alternating on each set of ¨ ariables y 1i , . . . , y pi 4 , 1 F i F 2 m, and z1j , . . . , z qj 4 , 1 F j F 2 m; Ž2. there exist yij g Cq, z ij g Cy such that f Ž y 11, . . . , y p2 m , z11 , . . . , z q2 m . s 1C . Proof. Suppose first that C is simple. By w6x for every m G 1 there 2m . exists a multilinear polynomial f Ž x 11, . . . , x 1pqq , . . . , x 12 m , . . . , x pqq alternati i ing on each set of variables x 1 , . . . , x pqq 4 , 1 F i F 2 m, 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 2 m disjoint sets of symmetric or skew variables. In case C s C1 [ C1op with exchange involution and C1 simple, then p s q s dim C1 and the polynomial f Ž y 11 , . . . , y p1 , . . . , y 12 m , . . . , y p2 m . f Ž z11 , . . . , z 1p , . . . , z12 m , . . . , z p2 m . is the required one. The polynomials found in the previous lemma will now be ‘‘glued together’’ to find multialternating polynomials of suitable degree nonvanishing in a finite dimensional algebra. LEMMA 4. Let F be algebraically closed and let A be a finite dimensional algebra o¨ er F. Let C1 JC 2 J ⭈⭈⭈ Cky1 JCk / 0, where C1 , . . . , Ck are distinct )-simple subalgebras of A and C1 q ⭈⭈⭈ qCk s C1 [ ⭈⭈⭈ [ Ck . If p s dimŽ C1 q ⭈⭈⭈ qCk .q, q s dimŽ C1 q ⭈⭈⭈ qCk .y, then for all m G 1 there exists a multilinear polynomial f s f Ž y 11 , . . . , y p1 , . . . , y 12 m , . . . , y p2 m , z11 , . . . , z 1q , . . . , z12 m , . . . , z q2 m , y 1 , . . . , y 2 k , z1 , . . . , z 2 k . such that Ž1. f is alternating on each set of ¨ ariables y 1i , . . . , y pi 4 , 1 F i F 2 m, and z1j , . . . , z qj 4 , 1 F j F 2 m; Ž2. there exist yij g Ž C1 q ⭈⭈⭈ qCk .q, z ij g Ž C1 q ⭈⭈⭈ qCk .y, yi g Aq, z i g Ay such that f Ž y 11, . . . , y p2 m , z11 , . . . , z q2 m , y 1 , . . . , y 2 k , z1 , . . . , z 2 k . / 0. y Proof. For every i s 1, . . . , k, let pi s dim Cq i , qi s dim C i , and let 2m 1 1 2m 2m f i s f i Ž yi1, 1 , . . . , yi1, p i , . . . , yi2, m 1 , . . . , yi , p i , z i , 1 , . . . , z i , q i , . . . , z i , 1 , . . . , z i , q i .
CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
479
be the polynomial constructed in the previous lemma. We let f˜s A1 ⭈⭈⭈ A2 m A1X ⭈⭈⭈ A2X m x 1 f 1 xX1 x 2 f 2 ⭈⭈⭈ x ky1 f ky1 xXky1 x k f k , where Ai means alternation on the p variables y 1,i 1 , . . . , y 1,i p 1, . . . , y k,i 1 , . . . , y k,i p k and AiX means alternation on the q variables z1,i 1 , . . . , z1,i q1, . . . , z k,i 1 , . . . , z k,i q k. We note that each polynomial f i corresponds to a pair of tableaux Ž Pi , Q i ., where Pi is a 2 m = pi rectangle, Q i is a 2 m = qi rectangle, and the variables in each column of Pi Žresp. Q i . are alternating. Now, f˜ corresponds to the pair Ž P, Q . obtained by gluing the rectangles Pi Žresp. Q i . 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 and AjX is alternation on the variables in the jth column of Q. Since C1 JC 2 J ⭈⭈⭈ Cky1 JC k / 0, let c i g Ci Ž1 F i F k ., b1 , . . . , bky1 g J be such that c1 b1 c 2 b 2 ⭈⭈⭈ bky1 c k / 0. t y For every i s 1, . . . , k, let yi,t j g Cq i , z i, j g C i be such that f i yi1, 1 , . . . , yi2, mp i , z i1, 1 , . . . , z i2, mq i s 1 C i .
ž
/
Notice that since Ci C j s 0 for i / j, when evaluating the variables y and z on the Ci ’s, alternation on the columns of the rectangle P Žresp. Q . can be replaced with alternation on the columns of each subrectangle Pi Žresp. Q i .. Hence 1 2m 1 2m f˜ y 1, 1 , . . . , y k , p k , z 1, 1 , . . . , z k , q k , c1 , . . . , c k , b 1 , . . . , b ky1
ž
/
s Ž p1 ! ⭈⭈⭈ pk !q1 ! ⭈⭈⭈ qk ! .
2m
c1 f 1 b1 c 2 f 2 b 2 ⭈⭈⭈ c ky1 f ky1 bky1 c k f k
s Ž p1 ! ⭈⭈⭈ pk !q1 ! ⭈⭈⭈ qk ! .
2m
c1 b1 c2 ⭈⭈⭈ bky1 c k / 0.
We may clearly assume that c1 , b1 , . . . , bky1 , c k g Aqj Ay; suppose that k 1 of them are symmetric and k 2 of them are skew, k 1 q k 2 s 2 k y 1. Then 1 2m 1 2m f s f˜Ž y 1, 1 , . . . , y k , p k , z 1, 1 , . . . , z k , q k , y 1 , . . . , y k 1 , z 1 , . . . , z k 2 .
⭈ y k 1q1 ⭈⭈⭈ y 2 k z k 2q1 ⭈⭈⭈ z 2 k 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. Then c nŽ A, ). G an b d n, for some constants a, b. Proof. Let A s B q J and C1 , . . . , Ck distinct )-simple subalgebras of B such that C1 JC 2 J ⭈⭈⭈ Cky1 JCk / 0. Let p s dimŽ C1 q ⭈⭈⭈ qCk .q, q s dimŽ C1 q ⭈⭈⭈ qCk .y; then d s p q q. Let n G 2 d q 4 k and divide n y 4 k by 2 d; then there exist m, t such that n y 4 k s 2 md q t, 0 F t - 2 d. Thus n s 2 mŽ p q q . q 4 k q t. Write s s 2 mp q 2 k q t, 0 F t - 2 d, n y s s 2 mq q 2 k. If f is the polynomial of the previous lemma of total degree 2 mp q 2 mq q 4 k, then the polynomial g s fy 2 kq1 ⭈⭈⭈ y 2 kqt g Vs, nys and g is not a )-identity for A Žwe may make suitable substitutions for the extra variables: note that f has non-zero value which belongs to C1 J ⭈⭈⭈ JC k , hence we can take y 2 kq1 s ⭈⭈⭈ s y 2 kqt s 1 C k .. We let the group G s S2 m p = S2 m q act on g by letting S2 m p act on the symmetric variables yil and S2 m q act on the skew variables z il. Let M be the G-submodule of Vs, nys generated by g. By complete reducibility M contains an irreducible G-submodule of the form W, s FGeT eTŽ g ., for some tableaux T and T , where & 2 mp and & 2 mq. Let l Ž . be the length of the first row of . Note that for every g S2 m p , Ž g . is still alternating on 2 m disjoint sets of symmetric variables, and Ý g R T acts by symmetrizing l Ž . variables. It follows that if l Ž . ) 2 m then eTŽ g . s 0, a contradiction. Similarly for l Ž .. Therefore l Ž . F 2 m and l Ž . F 2 m. Suppose now that has height hŽ . ) p. From the previous lemma we know that f Žhence g . takes a non-zero value on A after replacing the variables yij with suitable elements from Cqs Ž C1 q ⭈⭈⭈ qCk .q. Since dim Cqs p, the polynomial eTŽ g . 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 form W, , where s ŽŽ2 m. p . and s ŽŽ2 m. q . are two rectangles. X Recall that as m ª ⬁, Ždeg .Ždeg . , aŽ2 mp. b Ž2 mq . b p 2 m p q 2 m q for X some negative constants b, b and some a. It follows that c s, nys Ž A, ) . G dim W , G n c p 2 m p q 2 m q .
481
CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
Therefore we have n
c n Ž A, ) . s
ž nr / c
Ý rs0
G nc
r , nyr
n! s! Ž n y s . !
Ž A, ). G ns c s, nys Ž A, ) .
ž /
p 2 m pq 2 m q .
Recalling that s s 2 mp q 2 k q t and n y s s 2 mq q 2 k, hence n! s! Ž n y s . !
G
Ž 2 mp q 2 mq . ! Ž 2 mp . ! Ž 2 mq . !
and by Stirling formula we get c n Ž A, ) . Gn
␣
Ž 2 mp q 2 mq . Ž 2 mp .
2m p
2 m pq2 m q
Ž 2 mq .
2mq
p 2 m pq 2 m q s n ␣ Ž p q q .
2 m pq2 m q
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 the field F, then ExpŽ A, ). exists and is an integer F 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 the algebraically closed field F and let B s BU be a maximal semisimple subalgebra of A. Then ExpŽ A, ). s max i dim F Ž C1Ž i. q ⭈⭈⭈ qCtŽii. ., where C1Ž i., . . . , CtŽii. are distinct )-simple subalgebras of B and C1Ž i. JC 2Ž i. J ⭈⭈⭈ JC tŽii. / 0. Let F be the algebraic closure of the field F. For an algebra with involution A over F we let Z s ZŽ A. be the center of A and let Zqs ZŽ A.qs Z l Aq be the symmetric center of A. Next two results give an exact estimate of ExpŽ A, ). in the case of simple or semisimple algebras.
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COROLLARY 3. Let A be a finite dimensional )-simple algebra with in¨ olution o¨ er F. Then ExpŽ A, ). s dim Z q A. Proof. Suppose first that A is simple. If Zqs Z then wZ : F x
A mF F (
[ Ž Z mZ Fi . , is1
where Fi ( F and A mZ Fi is a central simple algebra over F with induced involution. Moreover
w Z : F x dim Z A s dim F A s dim F Ž A mF F . s w Z : F x dim F Ž A mZ Fi . . From Corollary 2 it follows that dim Z A s dim F Ž A mZ Fi . s ExpŽ A mF F, ). s ExpŽ A, ).. In case Zq/ Z, i.e., ) is an involution of the second kind, then Z s ZqŽ ␣ . with ␣ U s y␣ g Z is a quadratic extension of Zq and Z mZ q F ( F [ F with exchange involution. Hence A mZ q F ( A mZ Z mZ q F ( A mZ Ž F [ F . ( Ž A mZ F . [ Ž A mZ F . with involution given by Ž a m f 1 q b m f 2 .U s bU m f 1 q aU m f 2 . As above it follows that w Zq : F x
A mF F ( A mZ q
ž[ /
w Zq : F x
Fi (
is1
[ Ž A mZ is1
q
Fi .
w Zq : F x
(
[ Ž A mZ
Fi [ A mZ Fi . ,
is1
where, for i s 1, . . . , w Zq : F x, Fi ( F and A mZ Fi [ A mZ Fi is a )-simple algebra with exchange involution. As in the previous case it follows that dim Z q A s dim F Ž A mZ Fi [ A mZ Fi . s Exp Ž A mF F , ) . s Exp Ž 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 ZŽ A.q( ZŽ C ., we get w ZŽC . : F x
A mF F (
[ is1
žC m
ZŽC .
Fi [ C op mZŽC . Fi ,
/ ž
/
CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
483
where Fi ( F and Ž C mZŽC . Fi . [ Ž C op mZŽC . Fi . is )-simple over F with exchange involution. Thus as before we get dim Z q A s dim F Ž A mZ Fi [ A mZ Fi . s Exp Ž A mF F , ) . s Exp Ž A, ) . .
COROLLARY 4. Let A be a finite dimensional semisimple algebra with in¨ olution o¨ er F. If A s [i A i is the decomposition of A into )-simple q Ž . subalgebras, then ExpŽ A, ). s max i dim Z qi A i , where Zq i s Z A i l A i is the symmetric center of A i . Proof. We have A mF F ( [Ž A i mF F . and by the previous corollary A i mF F ( Bi1 [ ⭈⭈⭈ [ Bit i , where Bi1 ( ⭈⭈⭈ ( Bit i are )-simple algebras x Ž . Ž central over F and t i s w Zq i : F . It follows that Exp A, ) s Exp A mF F, ). s max i dim F Bi1; since dim Z qi A i s dim F Bi1 , the conclusion of the corollary now follows. COROLLARY 5. Let A be a finite dimensional algebra with in¨ olution o¨ er F. Then ExpŽ A, ). s dim F A if and only if A is )-simple and F s Zq. Proof. In light of Corollary 3 we only need to show that ExpŽ A, ). s dim F A implies A )-simple and F s Zq. Let A s A mF F with induced involution. Then dim F A s dim F A s ExpŽ A, ). s ExpŽ A, ).. If A is nilpotent, then ExpŽ A, ). s 0, a contradiction. Hence A contains a maximal semisimple subalgebra B s BU and by the previous corollary dim F A s ExpŽ A, ). s dim F Bi for a suitable )-simple subalgebra Bi of B. Hence A s Bi is )-simple. This implies that A is )-simple and by Corollary 3 dim F A s ExpŽ A, ). s dim Z q A implies F s Zq.
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.
REFERENCES 1. S. A. Amitsur, Identities in rings with involution, Israel J. Math. 7 Ž1969., 63᎐68. 2. Y. Bahturin, A. Giambruno, and M. Zaicev, G-identities on associative algebras, Proc. Amer. Math. Soc. 127 Ž1999., 63᎐69. 3. A. Berele, Cocharacter sequences for algebras with Hopf algebra actions, J. Algebra 185 Ž1996., 869᎐885.
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4. A. Berele, A. Giambruno, and A. Regev, Involution codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 96 Ž1996., 49᎐62. 5. V. Drensky and A. Giambruno, Cocharacters, codimensions and Hilbert series of the polynomial identities for 2 = 2 matrices with involution, Canad. J. Math. 46 Ž1994., 718᎐733. 6. E. Formanek, A conjecture of Regev on the Capelli polynomials, J. Algebra 109 Ž1987., 93᎐114. 7. A. Giambruno and A. Regev, Wreath products and P.I. algebras, J. Pure Appl. Algebra 35 Ž1985., 133᎐149. 8. A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Ad¨ . Math. 140 Ž1998., 145᎐155. 9. A. Giambruno and M. Zaicev, Exponential codimension growth of PI-algebras: An exact estimate, Ad¨ . Math. 142 Ž1999., 221᎐243. 10. A. Giambruno and M. Zaicev, A characterization of algebras with polynomial growth of the codimensions, Proc. Amer. Math. Soc., to appear. 11. A. Kemer, ‘‘Ideals of Identities of Associative Algebras,’’ Translations of Mathematical Monograph, Vol. 87, American Mathematical Society, Providence, RI, 1988. 12. J. L. Loday and C. Procesi, Homology of symplectic and orthogonal algebras, Ad¨ . Math. 69 Ž1988., 93᎐108. 13. A. Regev, Asymptotic values for degrees associated with strips of Young diagrams, Ad¨ . Math. 41 Ž1981., 115᎐136. 14. L. H. Rowen, ‘‘Ring Theory,’’ Academic Press, New York, 1988.
Involution Codimensions of Finite Dimensional Algebras and Exponential Growth A. Giambruno Dipartimento di Matematica e Applicazioni, Uni¨ ersita ` di Palermo, 90123 Palermo, Italy 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 )-codimensions c nŽ A, ). of A and we show that ExpŽ A, ). s lim n ª ⬁ c n Ž A, ) . 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 )-simple if and only if ExpŽ A, ). s dim F A. 䊚 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 variables x 1 , xU1 , . . . , x n , xUn and let WnŽ A, ). be the subspace of multilinear )-polynomial identities of A; then the codimension of WnŽ A, ). is c nŽ A, )., the nth codimension of A. 471 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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A celebrated theorem of Amitsur w1x states that if an algebra with involution A satisfies a )-polynomial identity then A satisfies an ordinary Žnon-involution. polynomial identity. In light of this result in w7x it was noticed that, as in the ordinary case, if A satisfies a non-trivial )-polynomial identity then c nŽ A, ). is exponentially bounded. In w2x an explicit exponential bound for c nŽ A, ). 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 in w10x we characterized finite dimensional algebras A such that c nŽ A, ). is polynomially bounded in terms of the representation theory of the hyperoctahedral group Hn . The asymptotic behavior of c nŽ A, ). was determined in w4x in case A s Mk Ž F . is the algebra of k = k matrices over F and ) is either the transpose or the symplectic involution. It turns out that c nŽ Mk Ž F ., ). ,nª⬁ Cn t k 2 n 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 Procesi w12x on the )-trace identities of Mk Ž F . with some asymptotic computations of Regev w13x. Here we shall determine the exponential behavior of the sequence c nŽ A, ). for any finite-dimensional algebra A. To do so, we define Exp Ž A, ) . s lim sup nª⬁
n
'c Ž A, ). , n
Exp Ž A, ) . s lim inf
nª⬁
n
'c Ž A, ). n
and, in case of equality, Exp Ž A, ) . s Exp Ž A, ) . s Exp Ž A, ) . . Let c nŽ A. denote the usual Žwithout ). nth codimension of the algebra A and let ExpŽ A. denote the corresponding exponential growth. In w8x and w9x it was proved that for any PI-algebra A, ExpŽ A. exists and is an integer. By exploiting the methods of w8x we shall prove that if A is finite dimensional, ExpŽ A, ). exists and can be explicitly computed: suppose that F is algebraically closed and write A s B q J, where B s BU is a maximal semisimple subalgebra and J is the Jacobson radical of A. Then ExpŽ A, ). s max i dim F Ž C1Ž i. q ⭈⭈⭈ qCtŽii. ., where C1Ž i., . . . , CtŽii. are distinct )-simple subalgebras of B and C1Ž i. JC 2Ž i. J ⭈⭈⭈ JCtŽii. / 0. When F is an arbitrary field of characteristic zero, among other consequences we shall prove that for a finite dimensional )-simple F-algebra A, ExpŽ A, ). s dim Z q A, where Zqs ZŽ A.q is the symmetric center of A. It is worth mentioning that in light of this result, the equality ExpŽ Mk Ž F ., ). s ExpŽ Mk Ž F .. s dim F Mk Ž F . obtained in w4x 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 ordinary Žnon-involution. case by a basic reduction of Kemer Žsee w11x. it is enough to study identities of the Grassmann envelope of a finite dimensional Z 2-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 with involution ). Let Aqs a g A < a s aU 4 and Ays a g A < a s yaU 4 be the sets of symmetric and skew elements of A, respectively. We let F² X, ): s F² x 1 , xU1 , x 2 , xU2 , . . . : be the free algebra with involution ) of countable rank. Recall that f Ž x 1 , xU1 , . . . , x n , xUn . g F² X, ): is a )-polynomial identity for A if f Ž a1 , aU1 , . . . , a n , aUn . s 0 for all a1 , . . . , a n g A. Let IdŽ A, ). denote the ideal of all )-polynomial identities of A and let VnŽ). denote the space of multilinear )-polynomials in x 1 , xU1 , . . . , x n , xUn . If we set si s x i q xUi and k i s x i y xUi , i s 1, 2, . . . , then, since char F / 2, it is useful to write VnŽ). as Vn Ž ) . s Span F w Ž1. ⭈⭈⭈ w Ž n. < g Sn , wi s si or wi s k i , i s 1, . . . , n4 . Let Hn be the hyperoctahedral group. Recall that Hn s Z 2 ; S n is the wreath product of Z 2 s 1, )4 , the multiplicative group of order 2, and Sn . We write the elements of Hn as Ž a1 , . . . , a n ; ., where a i g Z 2 , g Sn . The group Hn acts on VnŽ). as follows Žsee w5x.: for h s Ž a1 , . . . , a n ; . g a Hn define hsi s s Ž i. , hk i s k Ž i.Ž i. s "k Ž i. and then extend this action diagonally to VnŽ).. Since VnŽ). l IdŽ A, ). is invariant under this action, we view VnŽ).rŽ VnŽ). l IdŽ A, ).. as an Hn-module. Its character nŽ A, ). is called the nth )-cocharacter of A and c n Ž A, ) . s n Ž A, ) . Ž 1 . s dim F Vn Ž ) . r Ž Vn Ž ) . l Id Ž A, ) . . is the nth )-codimension of A. Recall that there is a one-to-one correspondence between irreducible Hn-characters and pairs of partitions Ž , ., where & r, & n y r, for all r s 0, 1, . . . , n. If , denotes the irreducible Hn-character corresponding to Ž , ., we write n
n Ž A, ) . s
Ý Ý rs0
&r &nyr
m , , ,
where m, G 0 are the corresponding multiplicities.
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GIAMBRUNO AND ZAICEV
For r s 0, . . . , n, we let Vr , nyr s Span w Ž1. ⭈⭈⭈ w Ž n. < g Sn ,wi s si for i s 1, . . . , r and wi s k i for i s r q 1, . . . , n4 be the space of multilinear polynomials in s1 , . . . , sr , k rq1 , . . . , k n . To simplify the notation, in the sequel we shall write yi , yij and z i , z ij for symmetric and skew symmetric variables, respectively, that are independent from each other. Therefore if we let s1 s y 1 , . . . , sr s yr and k rq1 s z1 , . . . , k n s z nyr , then Vr , nyr s the space of multilinear polynomials in y 1 , . . . , yr , z1 , . . . , z nyr . It is clear that to study VnŽ). l IdŽ A, ). it is enough to study Vr, nyr l IdŽ A, ). for all r. If we let S r act on the symmetric variables y 1 , . . . , yr and S nyr act on the skew variables z1 , . . . , z nyr , we obtain an action of Sr = Snyr on Vr, nyr and Vr, nyrrŽ Vr, nyr l IdŽ A, ).. is a left Sr = Snyr-module. Let r, nyr Ž A, ). be its character and let c r, nyr Ž A, ). s r, nyr Ž A, ).Ž1. s dim F Vr, nyrrŽ Vr, nyr l IdŽ A, )... It is well known that there is a one-to-one correspondence between irreducible Sr = Snyr-characters and pairs of partitions Ž , . such that & r, & n y r. For any partition of an integer t let be the corresponding irreducible St-character. Then m is the irreducible Sr = Snyr-character associated to the pair Ž , .. The following result holds THEOREM 1 w5, Theorem 1.3x. then, for all r F n,
Let A be a PI-algebra with in¨ olution;
n
n Ž A, ) . s
Ý Ý rs0
r , nyr Ž A, ) . s
&r &nyr
Ý
&r &nyr
m , ,
and
m , Ž m . .
Moreo¨ er n
c n Ž A, ) . s
Ý rs0
ž nr / c
r , nyr
Ž A, ) . .
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 J Ž A. is the Jacobson radical of A. Clearly J U s J and, according to w10, Theorem 4x we may assume that also BU s B is stable under the involution. Write B s B1 [ ⭈⭈⭈ [ Bm where B1 , . . . , Bm are )-simple algebras. Recall that for each s 1, . . . , m either Bi is a finite dimensional simple algebra with induced involution or Bi ( C [ C op , where C is a simple homomorphic image of Bi and ) on C [ C op is the exchange involution Ž a, b .U s Ž b, a. Žsee w14, Proposition 2.13.24x.. We now define an integer d s dŽ A. in the following way: we consider all possible nonzero products of the type C1 JC 2 J ⭈⭈⭈ Cky1 JCk / 0, where C1 , . . . , Ck are distinct subalgebras from the set B1 , . . . , Bm 4 and k G 1 Žif k s 1 this means that C1 s Bi for some 1 F i F n.. We then define d s dŽ A. to be the maximal dimension of a subalgebra C1 q ⭈⭈⭈ qCk satisfying the above inequality. We shall prove that in case F is algebraically closed, d coincides with ExpŽ A, ).. We note that if Bi1, . . . , Bi t are not necessarily distinct among the Bj ’s and Bi1 JBi 2 J ⭈⭈⭈ JBi t / 0 then dimŽ Bi1 q ⭈⭈⭈ qBi t . F d Žsee w8, Lemma 3x.. 4. MULTIALTERNATING POLYNOMIALS LEMMA 1. Let t G 0, a q b ) d, and f Ž y 1 , . . . , ya , z1 , . . . , z b , x 1 , . . . , x t . be a multilinear polynomial alternating on y 1 , . . . , ya4 and on z1 , . . . , z b 4 . If y 1 , . . . , ya g Bq, z1 , . . . , z b g By, x 1 , . . . , x t g A then f Ž y 1 , . . . , ya , z1 , . . . , z b , x 1 , . . . , x t . s 0. Proof. For each i s 1, . . . , m, let Ei s Eiq j Eiy be a basis of Bi , where Eiq and Eiy are sets of symmetric and skew elements, respectively. Then E s D Ei s Eqj Ey is a basis of B, where Eqs D Eiq and Eys D Eiy . Since f is multilinear, it is enough to evaluate f on a basis of A. The proof is similar to the proof of w8, Lemma 3x. If all the variables in f are basis elements of B, then, since Bi Bj s 0 for i / j we will get a zero value unless all elements come from one )-simple component, say Bt . In < or b ) this case, since dim Bt - d and a q b ) d, then either a ) < Eq t < Ey < . Since f is alternating in the y ’s and the z ’s then the value of f will t i i 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 in w8, Lemma 3x..
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For any partition of an integer t and a Young tableau T we let R T and CT be the subgroups of row and column permutations of T , respectively; then eT s Ý g R T , g C T Žsgn . is an essential idempotent of FS t and FS t eT is a minimal left ideal of FS t . Let & r, & n y r, and W, be a left irreducible S r = S nyr-module; if T is a tableau of shape and T is a tableau of shape , then W, ( F Ž Sr = Snyr . eT eT , where Sr and Snyr act on disjoint sets of integers. For a partition we denote with X s Ž X1 , X2 , . . . . the conjugate partition of . REMARK 1. Let & r, & n y r, and W, : Vr, nyr be a left irreducible Sr = Snyr-module. Then there exists f g W , such that f s f Ž y 11 , . . . , y1X1 , y 12 , . . . , y2X2 , . . . , z11 , . . . , z1X1 , z12 , . . . , z2X2 , . . . . and f is alternating on each set of ¨ ariables y 1i , . . . , yiXi 4 , z1j , . . . , zj Xj 4 , 1 F i F 1 , 1 F j F 1. Proof. Let f 0 g W , be a non-zero polynomial. Then there exist tableaux T, T such that g s eT eT f 0 / 0. Denote fs
Ý
Ý Ž sgn . Ž sgn . g .
gC T gC T
Then f is a polynomial with the prescribed property.
5. THE UPPER BOUND Recall that for a partition , hŽ . s X1 is the height of the corresponding diagram. LEMMA 2. Let & r, & n y r, and W, : Vr, nyr be a left irreducible Sr = Snyr-module. If W, Vr, nyr l IdŽ A, )., then hŽ . F dim Aq, hŽ . F dim Ay, and Xlq1 q Xlq1 F d, where J lq1 s 0. Moreo¨ er dim W, F n a Ž Xlq1 . r Ž Xlq1 . ny r for some a G 1. Proof. Let f be the polynomial described in the previous remark. Clearly W, s F Ž S r = S nyr . f. Since f is alternating on y 11 , . . . , y1X14 , it follows that X1 s hŽ . F dim Aq. Similarly, hŽ . F dim Ay. Suppose that Xlq1 q Xlq1 ) d. Then Xi q Xi ) d for all i s 1, . . . , l and, by the previous lemma, since f is not a )-identity for A, in each set of variables y 1i , . . . , yiX , z1i , . . . , zi X 4 , 1 F i F l q 1, we must substitute at least one i i element from J. But J lq1 s 0 leads to f being a )-identity for A.
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CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
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 that Ž1. F n p l Ž Xlq1 . r and Ž1. F n ql Ž Xlq1 . ny r. Therefore dim W , s Ž1. Ž1. F n p lqql Ž Xlq1 . r Ž Xlq1 . nyr. THEOREM 2. If A is a finite dimensional algebra with in¨ olution o¨ er F and d is the integer defined abo¨ e, then c nŽ A, ). F ant d n, for some constants a, t. Proof. For a partition let us write Xlq1 s t Ž .. From the previous lemma we have
r , nyr Ž A, ) . s
Ý
0Ft Ž .qt Ž .Fd
Ý
&r &nyr
m , Ž m . .
Hence, for some constant ␣ , c r , nyr Ž A, ) . F
Ý
0Ft Ž .qt Ž .Fd
Ý
&r &nyr
r
m , n ␣ t Ž . t Ž .
nyr
.
Since by w3x the multiplicities m, are polynomially bounded, from Theorem 1 we get n
c n Ž A, ) . s
Ý rs0
s an b
ž nr / c
n r , nyr
Ž A, ) . F an b
Ý
Ý
t 1qt 2Fd rs0 t1 , t 2G0
ž nr / t t
r nyr 1 2
n Ý Ž t1 q t 2 . F anbdd n .
t1qt 2Fd t1 , t 2G0
6. CENTRAL POLYNOMIALS The existence of multialternating central polynomials for k = k matrices proved in w6x 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 f s f Ž y 11 , . . . , y p1 , . . . , y 12 m , . . . , y p2 m , z11 , . . . , z 1q , . . . , z12 m , . . . , z q2 m .
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GIAMBRUNO AND ZAICEV
such that Ž1. f is alternating on each set of ¨ ariables y 1i , . . . , y pi 4 , 1 F i F 2 m, and z1j , . . . , z qj 4 , 1 F j F 2 m; Ž2. there exist yij g Cq, z ij g Cy such that f Ž y 11, . . . , y p2 m , z11 , . . . , z q2 m . s 1C . Proof. Suppose first that C is simple. By w6x for every m G 1 there 2m . exists a multilinear polynomial f Ž x 11, . . . , x 1pqq , . . . , x 12 m , . . . , x pqq alternati i ing on each set of variables x 1 , . . . , x pqq 4 , 1 F i F 2 m, 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 2 m disjoint sets of symmetric or skew variables. In case C s C1 [ C1op with exchange involution and C1 simple, then p s q s dim C1 and the polynomial f Ž y 11 , . . . , y p1 , . . . , y 12 m , . . . , y p2 m . f Ž z11 , . . . , z 1p , . . . , z12 m , . . . , z p2 m . is the required one. The polynomials found in the previous lemma will now be ‘‘glued together’’ to find multialternating polynomials of suitable degree nonvanishing in a finite dimensional algebra. LEMMA 4. Let F be algebraically closed and let A be a finite dimensional algebra o¨ er F. Let C1 JC 2 J ⭈⭈⭈ Cky1 JCk / 0, where C1 , . . . , Ck are distinct )-simple subalgebras of A and C1 q ⭈⭈⭈ qCk s C1 [ ⭈⭈⭈ [ Ck . If p s dimŽ C1 q ⭈⭈⭈ qCk .q, q s dimŽ C1 q ⭈⭈⭈ qCk .y, then for all m G 1 there exists a multilinear polynomial f s f Ž y 11 , . . . , y p1 , . . . , y 12 m , . . . , y p2 m , z11 , . . . , z 1q , . . . , z12 m , . . . , z q2 m , y 1 , . . . , y 2 k , z1 , . . . , z 2 k . such that Ž1. f is alternating on each set of ¨ ariables y 1i , . . . , y pi 4 , 1 F i F 2 m, and z1j , . . . , z qj 4 , 1 F j F 2 m; Ž2. there exist yij g Ž C1 q ⭈⭈⭈ qCk .q, z ij g Ž C1 q ⭈⭈⭈ qCk .y, yi g Aq, z i g Ay such that f Ž y 11, . . . , y p2 m , z11 , . . . , z q2 m , y 1 , . . . , y 2 k , z1 , . . . , z 2 k . / 0. y Proof. For every i s 1, . . . , k, let pi s dim Cq i , qi s dim C i , and let 2m 1 1 2m 2m f i s f i Ž yi1, 1 , . . . , yi1, p i , . . . , yi2, m 1 , . . . , yi , p i , z i , 1 , . . . , z i , q i , . . . , z i , 1 , . . . , z i , q i .
CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
479
be the polynomial constructed in the previous lemma. We let f˜s A1 ⭈⭈⭈ A2 m A1X ⭈⭈⭈ A2X m x 1 f 1 xX1 x 2 f 2 ⭈⭈⭈ x ky1 f ky1 xXky1 x k f k , where Ai means alternation on the p variables y 1,i 1 , . . . , y 1,i p 1, . . . , y k,i 1 , . . . , y k,i p k and AiX means alternation on the q variables z1,i 1 , . . . , z1,i q1, . . . , z k,i 1 , . . . , z k,i q k. We note that each polynomial f i corresponds to a pair of tableaux Ž Pi , Q i ., where Pi is a 2 m = pi rectangle, Q i is a 2 m = qi rectangle, and the variables in each column of Pi Žresp. Q i . are alternating. Now, f˜ corresponds to the pair Ž P, Q . obtained by gluing the rectangles Pi Žresp. Q i . 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 and AjX is alternation on the variables in the jth column of Q. Since C1 JC 2 J ⭈⭈⭈ Cky1 JC k / 0, let c i g Ci Ž1 F i F k ., b1 , . . . , bky1 g J be such that c1 b1 c 2 b 2 ⭈⭈⭈ bky1 c k / 0. t y For every i s 1, . . . , k, let yi,t j g Cq i , z i, j g C i be such that f i yi1, 1 , . . . , yi2, mp i , z i1, 1 , . . . , z i2, mq i s 1 C i .
ž
/
Notice that since Ci C j s 0 for i / j, when evaluating the variables y and z on the Ci ’s, alternation on the columns of the rectangle P Žresp. Q . can be replaced with alternation on the columns of each subrectangle Pi Žresp. Q i .. Hence 1 2m 1 2m f˜ y 1, 1 , . . . , y k , p k , z 1, 1 , . . . , z k , q k , c1 , . . . , c k , b 1 , . . . , b ky1
ž
/
s Ž p1 ! ⭈⭈⭈ pk !q1 ! ⭈⭈⭈ qk ! .
2m
c1 f 1 b1 c 2 f 2 b 2 ⭈⭈⭈ c ky1 f ky1 bky1 c k f k
s Ž p1 ! ⭈⭈⭈ pk !q1 ! ⭈⭈⭈ qk ! .
2m
c1 b1 c2 ⭈⭈⭈ bky1 c k / 0.
We may clearly assume that c1 , b1 , . . . , bky1 , c k g Aqj Ay; suppose that k 1 of them are symmetric and k 2 of them are skew, k 1 q k 2 s 2 k y 1. Then 1 2m 1 2m f s f˜Ž y 1, 1 , . . . , y k , p k , z 1, 1 , . . . , z k , q k , y 1 , . . . , y k 1 , z 1 , . . . , z k 2 .
⭈ y k 1q1 ⭈⭈⭈ y 2 k z k 2q1 ⭈⭈⭈ z 2 k does not vanish in A and is the desired polynomial.
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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. Then c nŽ A, ). G an b d n, for some constants a, b. Proof. Let A s B q J and C1 , . . . , Ck distinct )-simple subalgebras of B such that C1 JC 2 J ⭈⭈⭈ Cky1 JCk / 0. Let p s dimŽ C1 q ⭈⭈⭈ qCk .q, q s dimŽ C1 q ⭈⭈⭈ qCk .y; then d s p q q. Let n G 2 d q 4 k and divide n y 4 k by 2 d; then there exist m, t such that n y 4 k s 2 md q t, 0 F t - 2 d. Thus n s 2 mŽ p q q . q 4 k q t. Write s s 2 mp q 2 k q t, 0 F t - 2 d, n y s s 2 mq q 2 k. If f is the polynomial of the previous lemma of total degree 2 mp q 2 mq q 4 k, then the polynomial g s fy 2 kq1 ⭈⭈⭈ y 2 kqt g Vs, nys and g is not a )-identity for A Žwe may make suitable substitutions for the extra variables: note that f has non-zero value which belongs to C1 J ⭈⭈⭈ JC k , hence we can take y 2 kq1 s ⭈⭈⭈ s y 2 kqt s 1 C k .. We let the group G s S2 m p = S2 m q act on g by letting S2 m p act on the symmetric variables yil and S2 m q act on the skew variables z il. Let M be the G-submodule of Vs, nys generated by g. By complete reducibility M contains an irreducible G-submodule of the form W, s FGeT eTŽ g ., for some tableaux T and T , where & 2 mp and & 2 mq. Let l Ž . be the length of the first row of . Note that for every g S2 m p , Ž g . is still alternating on 2 m disjoint sets of symmetric variables, and Ý g R T acts by symmetrizing l Ž . variables. It follows that if l Ž . ) 2 m then eTŽ g . s 0, a contradiction. Similarly for l Ž .. Therefore l Ž . F 2 m and l Ž . F 2 m. Suppose now that has height hŽ . ) p. From the previous lemma we know that f Žhence g . takes a non-zero value on A after replacing the variables yij with suitable elements from Cqs Ž C1 q ⭈⭈⭈ qCk .q. Since dim Cqs p, the polynomial eTŽ g . 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 form W, , where s ŽŽ2 m. p . and s ŽŽ2 m. q . are two rectangles. X Recall that as m ª ⬁, Ždeg .Ždeg . , aŽ2 mp. b Ž2 mq . b p 2 m p q 2 m q for X some negative constants b, b and some a. It follows that c s, nys Ž A, ) . G dim W , G n c p 2 m p q 2 m q .
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CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
Therefore we have n
c n Ž A, ) . s
ž nr / c
Ý rs0
G nc
r , nyr
n! s! Ž n y s . !
Ž A, ). G ns c s, nys Ž A, ) .
ž /
p 2 m pq 2 m q .
Recalling that s s 2 mp q 2 k q t and n y s s 2 mq q 2 k, hence n! s! Ž n y s . !
G
Ž 2 mp q 2 mq . ! Ž 2 mp . ! Ž 2 mq . !
and by Stirling formula we get c n Ž A, ) . Gn
␣
Ž 2 mp q 2 mq . Ž 2 mp .
2m p
2 m pq2 m q
Ž 2 mq .
2mq
p 2 m pq 2 m q s n ␣ Ž p q q .
2 m pq2 m q
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 the field F, then ExpŽ A, ). exists and is an integer F 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 the algebraically closed field F and let B s BU be a maximal semisimple subalgebra of A. Then ExpŽ A, ). s max i dim F Ž C1Ž i. q ⭈⭈⭈ qCtŽii. ., where C1Ž i., . . . , CtŽii. are distinct )-simple subalgebras of B and C1Ž i. JC 2Ž i. J ⭈⭈⭈ JC tŽii. / 0. Let F be the algebraic closure of the field F. For an algebra with involution A over F we let Z s ZŽ A. be the center of A and let Zqs ZŽ A.qs Z l Aq be the symmetric center of A. Next two results give an exact estimate of ExpŽ A, ). in the case of simple or semisimple algebras.
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COROLLARY 3. Let A be a finite dimensional )-simple algebra with in¨ olution o¨ er F. Then ExpŽ A, ). s dim Z q A. Proof. Suppose first that A is simple. If Zqs Z then wZ : F x
A mF F (
[ Ž Z mZ Fi . , is1
where Fi ( F and A mZ Fi is a central simple algebra over F with induced involution. Moreover
w Z : F x dim Z A s dim F A s dim F Ž A mF F . s w Z : F x dim F Ž A mZ Fi . . From Corollary 2 it follows that dim Z A s dim F Ž A mZ Fi . s ExpŽ A mF F, ). s ExpŽ A, ).. In case Zq/ Z, i.e., ) is an involution of the second kind, then Z s ZqŽ ␣ . with ␣ U s y␣ g Z is a quadratic extension of Zq and Z mZ q F ( F [ F with exchange involution. Hence A mZ q F ( A mZ Z mZ q F ( A mZ Ž F [ F . ( Ž A mZ F . [ Ž A mZ F . with involution given by Ž a m f 1 q b m f 2 .U s bU m f 1 q aU m f 2 . As above it follows that w Zq : F x
A mF F ( A mZ q
ž[ /
w Zq : F x
Fi (
is1
[ Ž A mZ is1
q
Fi .
w Zq : F x
(
[ Ž A mZ
Fi [ A mZ Fi . ,
is1
where, for i s 1, . . . , w Zq : F x, Fi ( F and A mZ Fi [ A mZ Fi is a )-simple algebra with exchange involution. As in the previous case it follows that dim Z q A s dim F Ž A mZ Fi [ A mZ Fi . s Exp Ž A mF F , ) . s Exp Ž 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 ZŽ A.q( ZŽ C ., we get w ZŽC . : F x
A mF F (
[ is1
žC m
ZŽC .
Fi [ C op mZŽC . Fi ,
/ ž
/
CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS
483
where Fi ( F and Ž C mZŽC . Fi . [ Ž C op mZŽC . Fi . is )-simple over F with exchange involution. Thus as before we get dim Z q A s dim F Ž A mZ Fi [ A mZ Fi . s Exp Ž A mF F , ) . s Exp Ž A, ) . .
COROLLARY 4. Let A be a finite dimensional semisimple algebra with in¨ olution o¨ er F. If A s [i A i is the decomposition of A into )-simple q Ž . subalgebras, then ExpŽ A, ). s max i dim Z qi A i , where Zq i s Z A i l A i is the symmetric center of A i . Proof. We have A mF F ( [Ž A i mF F . and by the previous corollary A i mF F ( Bi1 [ ⭈⭈⭈ [ Bit i , where Bi1 ( ⭈⭈⭈ ( Bit i are )-simple algebras x Ž . Ž central over F and t i s w Zq i : F . It follows that Exp A, ) s Exp A mF F, ). s max i dim F Bi1; since dim Z qi A i s dim F Bi1 , the conclusion of the corollary now follows. COROLLARY 5. Let A be a finite dimensional algebra with in¨ olution o¨ er F. Then ExpŽ A, ). s dim F A if and only if A is )-simple and F s Zq. Proof. In light of Corollary 3 we only need to show that ExpŽ A, ). s dim F A implies A )-simple and F s Zq. Let A s A mF F with induced involution. Then dim F A s dim F A s ExpŽ A, ). s ExpŽ A, ).. If A is nilpotent, then ExpŽ A, ). s 0, a contradiction. Hence A contains a maximal semisimple subalgebra B s BU and by the previous corollary dim F A s ExpŽ A, ). s dim F Bi for a suitable )-simple subalgebra Bi of B. Hence A s Bi is )-simple. This implies that A is )-simple and by Corollary 3 dim F A s ExpŽ A, ). s dim Z q A implies F s Zq.
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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4. A. Berele, A. Giambruno, and A. Regev, Involution codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 96 Ž1996., 49᎐62. 5. V. Drensky and A. Giambruno, Cocharacters, codimensions and Hilbert series of the polynomial identities for 2 = 2 matrices with involution, Canad. J. Math. 46 Ž1994., 718᎐733. 6. E. Formanek, A conjecture of Regev on the Capelli polynomials, J. Algebra 109 Ž1987., 93᎐114. 7. A. Giambruno and A. Regev, Wreath products and P.I. algebras, J. Pure Appl. Algebra 35 Ž1985., 133᎐149. 8. A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Ad¨ . Math. 140 Ž1998., 145᎐155. 9. A. Giambruno and M. Zaicev, Exponential codimension growth of PI-algebras: An exact estimate, Ad¨ . Math. 142 Ž1999., 221᎐243. 10. A. Giambruno and M. Zaicev, A characterization of algebras with polynomial growth of the codimensions, Proc. Amer. Math. Soc., to appear. 11. A. Kemer, ‘‘Ideals of Identities of Associative Algebras,’’ Translations of Mathematical Monograph, Vol. 87, American Mathematical Society, Providence, RI, 1988. 12. J. L. Loday and C. Procesi, Homology of symplectic and orthogonal algebras, Ad¨ . Math. 69 Ž1988., 93᎐108. 13. A. Regev, Asymptotic values for degrees associated with strips of Young diagrams, Ad¨ . Math. 41 Ž1981., 115᎐136. 14. L. H. Rowen, ‘‘Ring Theory,’’ Academic Press, New York, 1988.