Multialternating graded polynomials and growth of polynomial identities

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. In particular we show it is an integer. The result was proviously known in case G is abelian.


Introduction
Let F be a field of characteristic zero and A an F -algebra graded by a finite group G. We shall assume throughout that A is a PI-algebra, i.e., it satisfies an ordinary polynomial identity. The graded polynomial identities satisfied by A and their growth have been extensively studied in the last years in an effort to develop a general theory that generalizes the theory of ordinary polynomial identities.
For instance in analogy with a basic result in Kemer's theory (see [15]), it was recently proved in [2] (and independently in [20] for G abelian) that if A is a finitely generated algebra, then A satisfies the same graded identities as a finite dimensional graded algebra. As a consequence one can reduce the study of an arbitrary G-graded PI-algebra to that of the Grassmann envelope of a finite dimensional Z 2 × G-graded algebra.
Also, in analogy with the results in [11] and [12] concerning the existence of the exponent of a PI-algebra, in [1] and [9] it was shown that when G is an abelian group exp G (A), the graded exponent of A, exists and is an integer.
Here we focus on the growth of the graded codimensions of A. Recall that if P G n is the space of multilinear graded polynomials of degree n and Id G (A) is the ideal of graded identities of A, then c G n (A), the nth G-codimension of A, measures the dimension of P G n mod. Id G (A). It was known that the sequence c G n (A), n = 1, 2, . . . , is exponentially bounded ( [10]), but only recently its exponential rate of growth was captured in case G is an abelian group. In fact in [1] for finitely generated algebras and in [9] in general, it was shown that if A is any G-graded PI-algebra and G is a finite abelian group, then exp G (A) = lim n→∞ n c G n (A) exists and is an integer called the G-exponent of A. Moreover the G-exponent can be explicitly computed and equals the dimension of a suitable finite dimensional semisimple graded algebra related to A.
In this note we shall extend the above result by proving that the G-exponent exists and is an integer in case of arbitrary (non necessarily abelian) groups G. We notice that the proof given in [9] works also when G is a non abelian group modulo two basic ingredients that we manage to prove here. The first result says that the multiplicities in the nth graded cocharacter of A are polynomially bounded, the second is of independent interest and consists on the construction of suitable multialternating graded polynomials non vanishing in a finite dimensional G-graded algebra which is simple as a graded algebra.
Throughout the paper F will be a field of characteristic zero and A an associative F -algebra satisfying a non-trivial polynomial identity. We assume that A is graded by a finite group G = {g 1 = 1, g 2 , . . . , g s } and we let A = ⊕ g∈G A g be its decomposition into a sum of homogeneous components.

Multialternating polynomials on G-simple algebras
We start by introducing some standard notation. We let F X, G be the free associative G-graded algebra of countable rank over F . The set X decomposes as X = s i=1 X gi , where the sets X gi = {x 1,gi , x 2,gi , . . .} are disjoint, and the elements of X gi have homogeneous degree g i . The algebra F X, G is endowed with a natural G-grading F X, G = ⊕ g∈G F g , where F g is the subspace spanned by the monomials x i1,gj 1 · · · x it,gj t of homogeneous degree g = g j1 · · · g jt .
Recall that a graded polynomial f ∈ F X, G is a graded (polynomial) identity of A if f vanishes under all graded substitutions x i,g = a g ∈ A g . Let also Id G (A) = {f ∈ F X, G | f ≡ 0 on A} be the ideal of graded identities of A.
We say that an algebra A is G-graded simple if A is a G-graded algebra which is simple as a graded algebra.
Let A be a finite dimensional G-graded simple algebra over an algebraically closed field of characteristic zero. The purpose of this section it to produce non identity G-graded polynomials with arbitrary many alternating sets of variables which correspond to the homogeneous components of A and with a bounded number of extra variables.
A key ingredient in the construction of these polynomials is a presentation of any G-graded simple algebra as a tensor product of two types of G-graded simple algebras, namely a twisted group algebra (with fine grading) and a matrix algebra with an elementary grading. Here is the precise statement. It is due to Bahturin, Sehgal and Zaicev. Theorem 1.1. [4] Let A be a finite dimensional G-graded simple algebra. Then there exists a subgroup H of G, a 2-cocycle α : H×H → F * where the action of H on F is trivial, an integer r and an r-tuple (g 1 , g 2 , . . . , g r ) ∈ G r such that A is G-graded In particular the idempotents 1 ⊗ e i,j as well as the identity element of A are homogeneous of degree e ∈ G.
Let t 1 , . . . , t s > 0 be integers and, for i = 1, . . . s, define X gi be another set of homogeneous variables disjoint from the previous sets.
This section is devoted to the proof of the following Theorem 1.2. Let F be an algebraically closed field of characteristic zero and A a finite dimensional G-graded simple algebra over F . For any t ≥ 1, there exist integers 2t ≤ t 1 , . . . , t s ≤ 2t|G| and a G-graded polynomial is not an identity of A; in particular it has an evaluation in A of the form 1 ⊗ e i,j .
(2) the cardinality of Y depends on the order of G and the dimension of A and not on the parameter t. In particular, the cardinality of Y is bounded. ; Y ) is alternating on each one of the sets X j g . In view of the theorem above we claim that it is sufficient to construct G-graded polynomials, which are non identities of A, and correspond to the cyclic subgroups of G.
In order to make the statement precise, let g be any element of G and let S = g be the subgroup it generates. We denote by d the order of S. Proposition 1.3. It is sufficient to construct, for any integer t ≥ 1, a G-graded polynomial (non identity of A) (3) f t,g (X S ; Y S ) is alternating on the set X i g j for every i = 1, . . . , 2t, and every 0 ≤ j ≤ d − 1.
(4) f t,g (X S ; Y S ) admits a non-zero G-graded evaluation on A of the form π g l ⊗ e 1,r .
Remark 1.4. Clearly, by adding an extra variable if necessary, we may assume that the value of the polynomial above is 1 ⊗ e 1,1 .
Proof. Indeed, having constructed the polynomials f t,g = f t,g (X S ; Y S ) above, the required polynomials are given by By [17, Theorem 1.6] (see also [16,Theorem 18.13]), it is semisimple and so it decomposes into the direct sum of S-graded simple algebras It follows from Bahturin, Sehgal and Zaicev' result that for every i = 1, . . . , l, there exists a subgroup C i ≤ S and a p i -tuple (w i,1 , . . . , w i,pi ) of elements in S (which determines the elementary grading on M pi (F )) such that Notice that since C i is cyclic, The structure of A S is given here up to an S-graded isomorphism. But we need more. In fact, in order to "bridge" the S-simple components B i by elements of A, we need to realize the algebra A S as a subalgebra of A in terms of its presentation (i.e. with the terminology of Theorem 1.1). Here is the precise statement.
First we make a definition: for a homogeneous subspace D of A we define Proposition 1.5. With the above notation: (1) For every i = 1, ..., l, (2) In terms of the presentation of A, after a possible permutation of the rtuple (g 1 , · · · , g r ) (the tuple which provides the elementary grading in the presentation of A), we have In particular p 1 + p 2 + · · · + p l = r.
Before proving the proposition let us show how to construct the polynomial f t,g (X S ; Y S ).
For each "page" B i,k we construct a Regev polynomial f k 2p 2 i (see [8]), with 2p 2 i variables whose homogeneous degrees are as the homogeneous degrees of B i,k and let i on a basis of B i,k gives an element of the form π b k ⊗ 1 pi×pi where π b k is a trivial unit of F C i . This is a slight abuse of notation since in fact the identity matrix 1 pi×pi is located in the block diagonal between rows p 1 + p 2 + · · · + p i−1 + 1 and p 1 + p 2 + · · · + p i . Remark 1.6. This is the place where we use the fact that C i is a cyclic group. Indeed, the evaluation of a Regev polynomial on the space B i,k has the same effect as the evaluation on p i × p i matrix algebra where all monomials are multiplied by the same trivial unit which is obtained as the product of commuting trivial units of F C i .
From the construction of f i we see that it has an evaluation of the form is a trivial unit of F C i . Note that since the homogeneous degrees of the spaces B i,k are disjoint for different k, the polynomial f i is alternating (in particular) on sets of g ν -variables, any g ν ∈ S, of cardinality which is equal to the dimension of the g ν -homogeneous subspace of B i .
In order to get arbitrary many alternating sets we let f t i be the product of tcopies (with disjoint sets of variables) of f i . Clearly, the evaluation of f t i on a basis of B i gives an element of the form π ai ⊗ 1 pi×pi where π ai is a trivial unit in F C i .
Thus we have constructed a polynomial f t i for any B i . We can now "bridge" these polynomials and get a polynomial . We observe that with suitable evaluations on the x's the polynomial φ S has an evaluation which is equal to 1 ⊗ e 1,1 . But we are not done yet. We still need to alternate variables with the same homogeneous degrees in the different polynomials f t i in a suitable way. More precisely, for every s = 1, . . . , t, we alternate all variables with the same homogeneous degrees which appear in the polynomials f s 1 , f s 2 , . . . , f s l . Clearly, because of the bridging variables x j , the resulting polynomial f t,g (X S ; Y S ) admits the value 1 ⊗ e 1,1 and has the required form.
Let us prove now the proposition above. To set up the notation again we recall that A has a presentation given by F α H ⊗ M r (F ) and the elementary grading is given by the r-th tuple (g 1 , g 2 , . . . , g r ). We let S be the cyclic group generated by g ∈ G and we denote by d its order. Let us introduce an equivalence relation on the index set {1, . . . , r} by setting i ∼ j if and only if g −1 i Hg j ∩ S = ∅. It is easy to see that this is indeed an equivalence relation and we let Ω 1 , . . . , Ω l be the equivalence classes. We may clearly assume (after reordering the tuple (g 1 , g 2 , ..., g r )) that equivalent indices are adjacent to each other. In other words we have integers p 1 , . . . , p l such that . . . , Ω l = {p 1 + · · · + p l−1 + 1, . . . , p 1 + · · · + p l = r}.
We shall replace (as we may) elements from the r-tuple which represent the same right H-coset by the same representative.
Consider indices i, j ∈ Ω k . We know (by the definition of the equivalence relation) that there exists an h ∈ H such that g −1 i hg j ∈ S. We claim that the number of elements h ∈ H with that property depends on the index k but not on i and j. In other words we claim the following.
Proof. Take z ∈ g −1 i Hg j ∩ S. For different elements q ∈ g −1 j Hg k0 ∩ S, we obtain different elements zq ∈ g −1 i Hg k0 ∩ S and hence |g −1 i Hg k0 ∩ S| ≥ |g −1 j Hg k0 ∩ S|. On the other hand, taking inverses we see that |g −1 j Hg k0 ∩ S| = |g −1 k0 Hg j ∩ S|. Being i, j and k 0 arbitrary the first part of the lemma follows. For the second part, note that g −1 i Hg i ∩ S and g −1 k0 Hg k0 ∩ S are subgroups of the cyclic group S. By the first part of the proof, they have the same order and hence they coincide. Following the first part of the proof we see that g −1 i Hg j ∩ S is a (right, and hence 2-sided (by commutativity)) g −1 i Hg i ∩ S-coset and the lemma is proved. Now, observe that for i = 1, . . . , l, the algebra is a direct summand of A S and so, the proof of the proposition will be completed if we show that U i , i = 1, . . . , l, is S-simple. To see this we exhibit an explicit presentation of U i as a S-simple algebra. Fix 1 ≤ k ≤ l and k 0 ∈ Ω k . Let w i−n (k−1) ∈ S be a g −1 k0 Hg k0 ∩ S-coset representative of g −1 k0 Hg i ∩ S, where p 1 + · · · + p k−1 + 1 ≤ i ≤ p 1 + · · · + p k , and n (k−1) = p 1 + · · · + p k−1 . Then the map determines an isomorphism of the S-graded algebra U k with F (g −1 k0 Hg k0 ∩ S) ⊗ M p k (F ). In the latter, the elementary grading is given by the p k -tuple (w 1 , . . . , w p k ). Details are omitted. Finally we note that Proposition 1.5 (b) follows from Lemma 1.7 (by ordering the coset's elements) and the proof of Proposition1.5 is completed. Remark 1.8. Note that since the group g −1 k0 Hg k0 ∩ S is cyclic, its cohomology vanishes and hence we may use group elements rather then representatives.

Graded exponent
Throughout F will be an algebraically closed field of characteristic zero and A a G-graded PI-algebra over F with G = {g 1 = 1, g 2 , . . . , g s } a finite group.
In this section we shall prove that the G-exponent of A exists and is an integer. This result was proved in case G is an abelian group in [1] for finitely generated algebras and in [9] in general. Here we shall follow closely that approach.
We start by recalling the general setting. The ideal of G-graded polynomial identities of A is denoted Id G (A). For every n ≥ 1, P G n = span F {x σ(1),gi σ(1) · · · x σ(n),gi σ(n) | σ ∈ S n , g i1 , . . . , g in ∈ G} is the space of multilinear G-graded polynomials in the homogeneous variables x 1,gi 1 , . . . , x n,gi n , g ij ∈ G. We construct the quotient space P G n (A) = P G n P G n ∩ Id G (A) and the non-negative integer c G n (A) = dim F P G n (A), n ≥ 1, is the nth G-graded codimension of A. Our aim is to prove that exp G (A) = lim n→∞ n c G n (A) exists and is an integer. Moreover we shall relate such an integer to the dimension of a finite dimensional semisimple algebra related to A.
For every n ≥ 1, write n = n 1 + · · · + n s a sum of non-negative integers and let P n1,...,ns ⊆ P G n be the space of multilinear graded polynomials in which the first n 1 variables have homogeneous degree g 1 , the next n 2 variables have homogeneous degree g 2 and so on. Then P G n is the direct sum of subspaces isomorphic to P n1,...,ns , for every choice of the integers n 1 , . . . , n s . Moreover such decomposition is inherited by P n1,...,ns ∩ Id G (A) and we define where n n 1 , . . . , n s = n! n 1 ! · · · n s ! denotes the multinomial coefficient. In order to compute an upper and a lower bound for c G n (A), it is enough to do so for c n1,...,ns (A) and then apply (1).
The space P n1,...,ns (A) is naturally endowed with a structure of S n1 × · · · × S nsmodule in the following way. The group S n1 × · · · × S ns acts on the left on P n1,...,ns by permuting the variables of the same homogeneous degree; hence S n1 permutes the variables of homogeneous degree g 1 , S n2 those of homogeneous degree g 2 , etc.. Since Id G (A) is invariant under this action, P n1,...,ns (A) inherits a structure of S n1 × · · · × S ns -module and we denote by χ n1,...,ns (A) its character.
Our first objective is to prove that the multiplicities in (2) are polynomially bounded.
Let E = e 1 , e 2 , . . . | e i e j = −e j e i be the infinite dimensional Grassmann algebra over F and let E = E 0 ⊕ E 1 be its standard Z 2 -grading. Now, if B = ⊕ (g,i)∈G×Z2 B (g,i) is a G × Z 2 -graded algebra, then B has an induced Z 2 -grading, ,0) and B 1 = ⊕ g∈G B (g,1) , and an induced Ggrading B = ⊕ g∈G B g where, for all g ∈ G, B g = B (g,0) ⊕B (g, 1) . Then one defines the Grassmann envelope of B as As in the ordinary case, the Grassmann envelope is an important object. In fact by a result of Aljadeff and Belov ( [2]), any variety of G-graded PI-algebras can be generated by the Grassmann envelope of a suitable finite dimensional G×Z 2 -graded algebra.
Let L be the relatively free G × Z 2 -graded algebra of V * on the (k + l)s graded generators [13,Theorem 4.8.2]).
Since L is a finitely generated G × Z 2 -graded PI-algebra, by [2, Theorem 1.1] there exists a finite dimensional G × Z 2 -graded algebra W such that var G×Z2 (L) = var G×Z2 (W ). Moreover L is a homomorphic image of a relatively free graded algebra T of such variety on ks even generators and ls odd generators.
The algebra T can be constructed by "generic" elements as follows: fix a basis {a 1 . . . , a m } of W of homogeneous elements. Let ξ ij , 1 ≤ t ≤ k, where the sum runs over all i j such that a ij is of homogeneous degree (g i , 0). Similarly define where the a ij are of homogeneous degree (g i , 1). Then T is generated by the "generic" elements Denote by L n the subspace of L spanned by all products w 1 · · · w i , 1 ≤ i ≤ n, where w 1 , . . . , w i run over the generators given in (3). Define T n accordingly on the relatively free generators given in (4). Since L is a homomorphic image of T , dim L n ≤ dim T n and we compute an upper bound of this last dimension.
Notice that every monomial of degree at most n in the generic elements in (4), i ] n is the subspace of polynomials of degree at most n in the commutative variables ξ i ] n = (k+l)sm+n n ≤ (n + (k + l)s) (k+l)sm , we get that (5) dim L n ≤ dim T n ≤ m(n + (k + l)s) (k+l)sm ≤ Cn t , for some constants C, t.
We are now ready to prove the following.
Lemma 2.1. There exist constants C, t such that for all n ≥ 1, m λ ≤ Cn t in (2).
Proof. Suppose that there exists n and λ ⊢ n such that m λ > Cn t ≥ dim L n , where C and t are defined in (5). Hence there exist m λ = r irreducible S n1 × · · · × S ns -modules M 1 , . . . , M r ∈ P n1,...,ns with character χ λ(1) ⊗ · · · ⊗ χ λ(s) and (M 1 ⊕ · · · ⊕ M r ) ∩ Id G (V) = 0. Now, as in [9, Lemma 4], we may take M i = F (S n1 × · · · × S ns )h i where, by eventually adding some empty sets of variables, we may assume that each h i is a polynomial in the homogeneous sets of variables Y j g1 . . . , Y j gs , Z p g1 , . . . , Z p gs , i ≤ j ≤ k, 1 ≤ p ≤ l, and h i is symmetric in each Y j gt and alternating in each Z p gt . Since (M 1 ⊕ · · · ⊕ M r ) ∩ Id G (V) = 0, for every choice of α 1 , . . . , α r ∈ F not all zero, we have that h = Let˜be the map defined in [9,Section 5]. Then, if we regard the variables of each set Y j gt as even variables and those of Z p gt as odd variables, we can construct the polynomialsh i , 1 ≤ i ≤ r. Thenh i is symmetric on each Y j gt and on each Z p gt . For every i and j, let Y j gi = {y j 1,gi , . . . , y j mi,gi } and Z j gi = {z j 1,gi , . . . , z j ri,gi }. Define S to be the relatively free G-graded algebra of the variety V on the graded generatorsȳ j p,gi ,z j q,gi , 1 ≤ i ≤ s, 1 ≤ p ≤ m i , 1 ≤ q ≤ r i , j = 1, 2, . . . . Then the algebra Q = (S ⊗ E 0 ) ⊕ (S ⊗ E 1 ) has a natural G × Z 2 -grading and we can take its Grassmann envelope Since E 0 ⊗ E 0 ⊗ E 1 ⊗ E 1 is commutative, E(Q) and S satisfy the same G-graded identities. Hence E(Q) ∈ V and, so, Q ∈ V * . Now in each polynomialh i , 1 ≤ i ≤ r, we identify the variables in each set Y j gi and in each set Z j gi and we leth • i be the corresponding polynomial. Since degh i = n, under the evaluation Since by hypothesis r > dim L n , there exist scalars α 1 , . . . , α r not all zero, such that ϕ( r i=1 α ih • i ) = 0 in L. Recalling that L is a relatively free algebra of V * , we obtain thath β j ′ t are disjoint monomials of E of the correct homogeneous degree (α j t of homogeneous degree 0 and β j ′ t of homogeneous degree 1), we get that ϕ(h • ) = 0. By computing explicitly we obtain 0 = ϕ(h • ) = ϕ ′ (h) ⊗ γ where 0 = γ ∈ E and ϕ ′ is an evaluation such that ϕ ′ (y j p,gi ) =ȳ j p,gi , ϕ ′ (z j ′ q,gi ) =z j ′ q,gi , Now, recall that the elementsȳ j p,gi ,z j q,gi , j ≥ 1, generate the relatively free Ggraded algebra of V of countable rank. Hence ϕ ′ (h) = 0 says that h = r i=1 α i h i ∈ Id G (V), and this contradiction completes the proof.
Next we shall prove the existence of the G-exponent of A. Let B be a finite dimensional G × Z 2 -graded algebra such that var G (A) = var G (E(B)). By the Wedderburn-Malcev theorem [7] and the result in [21], we can write B = B 1 ⊕ · · · ⊕ B k + J, where B 1 , . . . , B k are G × Z 2 -graded simple algebras and J is the Jacobson radical of B. Recall that according to [9, Section 3], a semisimple subalgebra D = D 1 ⊕ · · · ⊕ D h , where D 1 , . . . , D h ∈ {B 1 , . . . , B k } are distinct, is admissible if D 1 JD 2 J · · · JD h = 0. Then one defines where D runs over all admissible subalgebras of B.
We shall prove that p coincides with the G-exponent of A. In fact we have.
Theorem 2.2. Let B be a finite dimensional G × Z 2 -graded algebra over an algebraically closed field of characteristic zero. Then there exist constants C 1 > 0, C 2 , k 1 , k 2 such that C 1 n k1 p n ≤ c G n (E(B)) ≤ C 2 n k2 p n , where p = p(B) is the integer defined in (6).
Proof. This theorem is proved in [9] for G abelian but the proof carries over to the non abelian case by making the following changes.
In the computation of the upper bound c G n (E(B)) ≤ C 2 n k2 p n one should use Lemma 2.1 above instead of [9, Remark 1].
For the computation of the lower bound C 1 n k1 p n ≤ c G n (E(B)) one should use Theorem 1.2 instead of [1, Lemma 18], concerning the construction of multialternating polynomials of arbitrary large degree for finite dimensional G-graded simple algebras. We should point out that while the polynomial constructed in [1, Lemma 18] depends on a parameter t, the one constructed in Theorem 1.2 depends on s parameters t 1 , . . . , t s (each corresponding to a homogeneous component of the algebra) and these parameters are squeezed between 2t and 2|G|t. Then one notices that in [9, Section 6] the proofs are carried over for each homogeneous component separately. This completes the proof of the theorem.
Since graded codimensions do not change by extension of the base field, we get the following. Theorem 2.3. Let G be a finite group and A a G-graded PI-algebra over any field F of characteristic zero. Then exp G (A) = lim n→∞ n c G n (A) exists and is an integer.