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Volume 2012 (2012), Article ID 170697, 26 pages
Growth for Algebras Satisfying Polynomial Identities
Mathematics Department, The Weizmann Institute, 76100 Rehovot, Israel
Received 6 September 2012; Accepted 25 September 2012
Academic Editors: E. Aljadeff and F. Marko
Copyright © 2012 Amitai Regev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The th codimension of a PI algebra measures how many identities of degree the algebra satisfies. Growth for PI algebras is the rate of growth of as goes to infinity. Since in most cases there is no hope in finding nice closed formula for , we study its asymptotics. We review here such results about , when is an associative PI algebra. We start with the exponential bound on then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of . The representation theory of the symmetric group is an important tool in this theory.
We study algebras satisfying polynomial identities (PI algebra). A natural question arises: give a quantitative description of how many identities such algebra satisfies? We assume that is associative, though the general approach below can be applied to nonassociative PI algebras as well.
Denote by the ideal of identities of in the free algebra . If then its dimension is always infinite, hence dimension by itself is essentially of no use here. In order to overcome this difficulty we now introduce the sequence of codimensions.
1.1. Growth for PI Algebras
Given , we let denote the multilinear polynomials of degree in , so in the associative case . Identities can always be multilinearized, hence the subset and its dimension give a good indication as to how many identities of degree satisfies. In fact, in characteristic zero the ideal is completely determined by the sequence , but we make no use of that remark in the sequel.
To study , we introduce the quotient space and its dimension The integer is the th codimension of . Clearly determines since is known.
The study of growth for PI algebra is mostly the study of the rate of growth of the sequence of its codimensions, as goes to infinity. We have the following basic property.
Theorem 1.1 (see ). In the associative case, is always exponentially bounded.
This theorem implies several key properties for PI algebras. And it fails in various nonassociative cases.
Various recent results indicate that in general there is no hope to find a closed formula for . Instead, one therefore tries to determine the asymptotic behavior of that sequence, as goes to infinity. We mention here three such results.
Recall that for two sequences of numbers, if .
(1) The asymptotics for the matrices , see Section 6.
Theorem 1.2 (see ). When goes to infinity,
(2) The integrality theorem of Giambruno-Zaicev, see Section 10.
Theorem 1.3 (see ). Let be an associative PI -algebra with , then the limit exists and is an integer. We denote , so .
(3) The “1/2” theorem of Berele, see Section 11.
1.2. Structure of the Paper
The paper reviews some of the main results about the asymptotics of codimensions. It does not contain full proofs but rather, it indicates some of the key ideas in the proofs of the main results.
We start by introducing Kemer's classification of the verbally prime -ideals. After introducing the codimensions, two proofs of their exponential bound are reviewed. The character of that space, is denoted as follows: and is called the th cocharacter of . Since , cocharacters are refinement of codimensions, and are important tool in their study. By a theorem of Amitsur-Regev and of Kemer, is supported on some hook. Shirshov's Height Theorem then implies that the multiplicities in the cocharacters are polynomially bounded.
We then review the proof of the Giambruno-Zaicev Theorem in the finite dimensional case, and the proof of Berele's “1/2” Theorem in the case of a Capelli identity.
The question of the algebraicity of the generating function is examined in Section 12.
In the last section, we review a construction of nonassociative algebras where the integrality property of the exponent fails.
2. PI Algebras and Ideals
Let be a field. In most cases we assume that . We begin by studying associative -algebras. Analogue theories exist for nonassociative algebras. Let denote the algebra of associative and noncommutative polynomials in the countable sequence of variables . The polynomial is an identity of the -algebra if for every . The algebra satisfies polynomial identities, or in short is PI, if there exist a nonzero polynomial which is an identity of . For example, any commutative algebra is PI since it satisfies . Applying alternating polynomials imply that every finite dimensional algebra, associative or nonassociative is PI; see Section 3.1. The class of the PI algebras is both large and interesting! We remark that the algebra of the matrices over plays a basic role in PI theory.
Definition 2.1. (1) Let be the subset of the identities of the algebra .
(2) T ideals. The set is a two sided ideal in , with the additional property that it is closed under substitutions. Such ideals in are called T ideals. Thus, is a ideal. It is easy to show that a ideal is always the ideal of identities of some PI algebra .
(3) Varieties of PI algebras. Given a -ideal , the class of the PI algebras satisfying is the variety corresponding to . We use here the language of ideals, rather than that of varieties.
2.2. Kemer's Theory for -Ideals (See )
The Specht Problem
One of the main problems in PI theory is the Specht problem: Are ideals always finitely generated as ideals?
Kemer  developed a powerful structure theory for ideals which allowed him to prove that if then ideals are indeed finitely generated. We review some of the ingredients of that theory.
Amitsur  proved that the ideal is prime in the following sense. If then either or ; moreover, the only prime ideals in are .
Kemer introduced the notion of verbally prime ideals as follows.
Definition 2.2. The ideal is verbally prime if it satisfies the following condition: Let and be polynomials in disjoint sets of variables. If Kemer then classified the verbally prime ideals in , see Theorem 2.3 below.
2.2.1. The Algebras , , and
Let be an infinite dimensional vector space, and let be the corresponding infinite dimensional Grassmann (Exterior) algebra. Then and , where Now are the matrices over , while are the matrices over .
We have . The elements of are block matrices where , , is and is , both with entries from . For example:
Theorem 2.3 (see ). The following are the three families of the verbally prime ideals: , , and .
The importance of the verbally prime ideals is demonstrated in the following theorem.
Theorem 2.4 (see ). Let be a ideal. Then there exist verbally prime ideals such that
3. Multilinear Polynomials
As usual, denotes the th symmetric group.
Definition 3.1. The polynomial is multilinear (in ) if for some coefficients . Let denote the vector space of multilinear polynomials in , so . Extending the map by linearity yields the vector space isomorphism , where is the group algebra of . We will identify By the process of multilinearization [8, 9] one proves the following theorem.
Theorem 3.2. Let the PI algebra satisfy an identity of degree , then satisfies a multilinear identity of degree . Moreover, if then the ideal of identities is determined by its multilinear elements.
It follows that if satisfies an identity of degree then satisfies an identity of the form This fact is applied in Section 4.1 in proving the exponential bound for the codimensions.
3.1. Example: Standard and Capelli Identities
The polynomial is alternating in if for every permutation , . For example, the polynomial is alternating. It is called the th Standard polynomial.
Similarly the polynomial which is called the (th) Capelli polynomial, is multilinear of degree and is alternating in .
It is rather easy to show that if then satisfies both and , and hence every finite dimensional algebra is PI. And the same argument applies in the nonassociative case. In particular the algebra of the matrices satisfies . The celebrated Amitsur-Levitzki Theorem [10–12] states that satisfies the standard identity . Of course, for large the degree is much smaller than .
Many infinite dimensional algebras are PI. For example, any infinite dimensional commutative algebra is PI. We remark that obviously, the free algebra itself is not PI.
4. The Codimensions
Question. How many identities are satisfied by a given PI algebra, namely, how large are ideals?
Computing dimensions might seem useless at first sight, since if the -ideal is nonzero then . To answer the above question we introduce below the notion of codimensions. Given a PI algebra , we would like to study its multilinear identities of degree , namely, the space as in (3.2). Note that if are any variables then . A first step is the study of the dimensions . Now , and since , clearly Thus, knowing is equivalent to knowing . This leads us to introduce the codimensions as follows.
Definition 4.1. Let be an -algebra, then is called the th codimension of , and is the sequence of codimensions of .
For example, satisfies (i.e., is nilpotent) if and only if . And is commutative if and only if for all .
Remark 4.2. Note that if is not PI then , hence for all . In fact, the algebra is PI (i.e., ) if and only if there exist such that . This follows directly from the definition.
4.1. Exponential Bound for the Codimensions
A basic property of the codimensions sequence in the associative case is, that it is bounded by exponential growth. Applications of this fact are given in the sequel.
Theorem 4.3 (see ). Assume the (associative) algebra satisfies an identity of degree , then for all .
Proof. We sketch two proofs (both different from the original proof). Both proofs apply the notion of a -good permutation, which we now review.
Call -bad if there are indices with . Otherwise is -good, and we denote By a Shirshov-Latyshev argument [13, 14], if satisfies an identity of degree , then satisfies an identity (3.3) which, by a certain inductive argument implies that .
By an argument based on Dilworth Theorem in Combinatorics , Latyshev then showed that , thus completing the (first) proof.
A second proof of the bound goes as follows . First, by the RSK correspondence , where is the number of standard Young taableaux of shape , and is the number of parts of . Then by the Schur-Weyl theory, where and . The (second) proof now follows since .
4.1.1. Application: The Theorem
Codimensions where introduced, in , in order to prove the following theorem.
Theorem 4.4 (see ). If and are (associative) PI algebras then is PI.
Proof. It is not too difficult to show that . Assume now that satisfies an identity of degree , and satisfies an identity of degree . Together with Theorem 4.3, this implies that hence for a large enough , . Finally by Remark 4.2, is PI if and only if there exist such that , and the proof follows.
Remark 4.5. For explicit identities for see Remark 8.2 below, which also implies the following. Again let satisfy an identity of degree , and an identity of degree . Then satisfies an identity of degree about , where .
The above results motivate the study of the following problem.
Question 1. Given a PI algebra , find a formula for the sequence . In most cases this seems to be too difficult, and one tries to, at least, get a “nice” asymptotic approximation of .
In Section 6 we determine such asymptotics for the algebras . We also give such partial results for the other verbally prime algebras and .
Recall the identification (see (3.2)) and let be a PI algebra. The regular left action of on is as follows. Let and , then This makes into a left module, with a submodule of .
Definition 5.1. The -character of the quotient module is called the th cocharacter of .
Cocharacters where introduced in  in the study of standard identities.
Since , from the ordinary representation theory of [19, 20] it is well known that the irreducible characters are parametrized by the partitions on : Also, decomposes as follows: where the are the minimal two sided ideals of , and there are natural bijections Here and where, again, is the number of standard Young taableaux of shape .
Given a tableau of shape , it determines the semi-idempotent , and is a minimal left ideal.
The -character of the regular representation is . It follows that the th cocharacter of can be written as for some multiplicities , and .
Clearly, the degree of is the corresponding codimension , and since , hence For example  let be the infinite dimensional Grassmann (Exterior) algebra, then
Remark 5.2. Properties of the identification imply the following characterization of Capelli identities (Section 3.1) by cocharacters . Let be a PI algebra with its cocharacter. Then satisfies if and only if whenever .
As an application of cocharacters one can prove the following.
Proposition 5.3 (see ). If satisfies and satisfies then satisfies .
We remark that a proof of this result without applying cocharacters is yet unknown.
Question 2. Given a PI algebra , find a formula for the multiplicities . In most cases this is too difficult, and one tries to at least get some approximate description of .
Remark 5.4. The approach of codimensions and of cocharacters in the study of PI algebras applies also in the nonassociative case (though with different phenomena).
5.1. The Cocharacters of Matrix Algebras
In the case of the matrices there is the following nice formula for the multiplicities of .
Example 5.5 (see , [25, Theorem 12.6.5], ). Denote . First, if then . So let . If then . If with then . If (so ) then . And in all other cases. Recent results of Berele  and of Drensky and Genov  indicate that when , there is no hope in getting nice formulas for the cocharacter-multiplicities . A somewhat similar phenomena is discussed in Section 12.
5.2. Trace Identities, Codimensions, and Cocharacters
In the case of matrices, the following is an extension of the previous theory of codimensions and cocharacters.
Instead of ordinary polynomials we can consider trace polynomials, namely, polynomials involving variables and traces, for example is a (mixed) trace polynomial. These trace polynomials can be evaluated on the algebra (or on any algebra with a trace) hence yielding trace identities. For example, it can be proved that the trace polynomial is a trace identity of . This is an example of a mixed trace polynomial, while the polynomial is a pure trace polynomial, which is also an identity of . We then have trace identities of , “pure” and “mixed,” hence trace codimensions and , and trace cocharacters and .
Theorem 5.6. We have
This implies that the trace codimensions are given by the following formula: This formula is the starting point for computing the asymptotic formula of given in Section 6.
6. Asymptotics of the Codimensions
For the matrices, Procesi  proved the following formula for .
Theorem 6.1 (see ). We have
It was already mentioned that when , it most likely is impossible to find an exact formula for the multiplicities , and the same is probably true about . Instead of giving up, one looks for the asymptotic of .
Here we have the folowing theorem.
Theorem 6.2 (see ). When goes to infinity,
We review the major steps toward the proof of Theorem 6.2. First, it follows from deep results of Formanek [2, 26, 35] that the th codimensions and the -st pure trace codimensions are asymptotically equal: We saw that by Theorem 5.6, Thus, The last major step here is the computation of the asymptotic behavior of the following sum: The asymptotics, as , of the more general sums is given in . That asymptotic is of the form , where , and , all given explicitly in . We remark that the constant term is evaluated by applying the Selberg integral (6.9).
Theorem 6.3 (see ). The Selberg integral
Together, the above steps yield the asymptotic value of Theorem 6.2.
6.1. The Other Verbally Prime Algebras
For the other verbally prime algebras (see Section 2.2.1), we quote the following partial asymptotic results.
7. Shirshov's Height Theorem (See )
7.1. The Theorem
A powerful tool in the study of PI algebras is Shirshov's Height Theorem, which we now quote. We consider the alphabet of letters; is the set of all words (i.e., monomials) in ; the subset of the words of length ≤. We consider finitely generated PI algebra .
Theorem 7.1 (Shirshov's Height Theorem). Consider a PI algebra satisfying the identity (3.3):
There exists large enough such that any finitely generated algebra that satisfies the identity (3.3), satisfies the following condition.
Modulo , is spanned by the elements
7.2. Application: Bounds on the Cocharacters
7.2.1. The Hook Theorem
Denote by the partitions of in the hook: Let be characters, . We say that is supported on , and denote , if for all , Similar terminology applies when is replaced by another family of subsets of partitions. We have the following theorem.
Theorem 7.2 (see [6, 38]). Let be any (associative) PI algebra, then there exist such that its cocharacters are supported on the hook .
Explicitly, let satisfy an identity of degree , and let , where is the base of the natural logarithms. Then , namely,
7.2.2. An Application of Shirshov's Theorem
8. Explicit Identities
Amitsur  proved that any PI algebra satisfies a power of a standard identity, namely, an identity of the form , where is the th standard polynomial. Amitsur's proof, which applies Structure Theory of Rings , yields a bound on the index but not on . A recent proof of Amitsur's theorem  applies the identification , together with the exponential bound on and yields a combinatorial proof of that theorem, a proof which gives bounds on both and . Moreover, the same arguments yield explicit identities in various other cases, for example in the case.
Theorem 8.1 (see [38, 43]). Let be a PI algebra satisfying , and let and be integers satisfying , then . In particular if satisfies an identity of degree , and , then (of degree which is about .)
In fact, with these and satisfies the power of the Capelli identity: .
Remark 8.2. Let satisfy an identity of degree and an identity of degree . Denote then . Thus, if then satisfies the identities and . Note that here the degree of , for example, is about .
Also, satisfies some identity of degree , where is about . Indeed, let , then the classical inequality implies that . Thus, if then, for that , and by Remark 4.2 satisfies an identity of degree where is about .
9. Nonidentities for Matrices: The Polynomial
Usually, lower bounds for codimensions are harder to obtain than upper bounds. Given a PI algebra , a lower bound for can be obtained by the following technique. Find a polynomial which is a nonidentity of , namely, . In addition, with the identification (see (5.4)), verify that for some . Then . This follows since , so for some minimal left ideal with , hence . We now construct such a polynomial via the polynomial , when [44, Definition 2.3].
Corresponding to the sum , construct the monomial of degree : For example, . Now alternate the 's and alternate the 's to obtain : Let be the conjugate tableau of , and Then and in that sense corresponds to the tableau where . It is not difficult to show that takes central values on .
Theorem 9.1 (see ). The plolynomial , which corresponds to the rectangle , is a nonidentity of . Hence is a central polynomial.
By Young's rule it follows that where and , see (5.4). Since satisfies the Capelli identity , it follows that for all with , . And since , hence , where is of the rectangular shape . Thus, also is a central polynomial for . The fact that is a rectangle plays an important role in proving lower bounds for codimensions, since the following is applied: two or more rectangles of same height can be glued together horizontally, while two or more rectangles of same width can be glued together vertically.
For , Theorem 9.1 and further results of Formanek [2, 26, 35] imply that the ordinary cocharacters and the trace cocharacters are nearly equal, hence they have same asymptotic. Since the trace codimensions are much easier to handle than the ordinary codimensions (see Theorem 5.6), this fact allows the computation—in the next section—of the exact asymptotic of . Also, the fact that is a nonidentity of has applications in proving lower bounds for various other types of codimensions.
10. The Giambruno-Zaicev Theorem:
For most PI algebras it seems hopeless to find a precise, or even asymptotic, formula for the codimensions . We therefore ask a much more restricted question.
Question 3. Given the associative PI algebra , what can be said about the asymptotic behavior of the codimensions ?
Theorem 10.1 (see ). Let be an associative PI algebra (with ), then the limit
exists and is an integer.
We denote , so .
10.1. Review of the Proof When
When is finite dimensional, the number can be calculated as follows. We may assume that is algebraically closed. First, by a classical theorem of Wedderburn and Malcev [45, Theorem 3.4.3], , where is the Jacobson radical of , and is semisimple. Thus where are simple, namely, . Consider now all possible nonzero products of the type where the are distinct, and for such nonzero products let Then , and is the limit in Theorem 10.1: Namely, .
For example, consider the algebra of upper block triangular matrices where for , and are rectangular matrices of the corresponding sizes. Then the Wedderburn-Malcev decomposition of is where is the semisimple part and is the Jacobson radical. Notice that the matrix units in satisfy so and also, this is the maximal such nonzero product. It follows that here
10.1.1. The Upper Bound
In the general case the exponent is given by (10.3). To prove this, one first proves the following upper bound.
Lemma 10.2. There exist constants , such that for all , .
Let denote the partitions of . Given , define the subsets by so is nearly a strip of height (with at most cells below the row). The proof of Lemma 10.2 follows by showing that the cocharacters are supported on such nearly a strip , and by the polynomial bound on the multiplicities in the cocharacters, see Theorem 7.3. Thus , where the sum is supported on such nearly a strip . Similar to the estimates in , such sum is bounded by , and the proof of Lemma 10.2 follows.
10.1.2. The Lower Bound
Here we prove the folowing lemma.
Lemma 10.3. There exist constants and such that for all , .
A key ingredient in proving the lower bound is the polynomial which is a nonidentity for , see Section 9. We already noted that has a component which is central (nonidentity) for , where is the rectangle , see Theorem 9.1.
We start with a single matrix algebra , with the corresponding central polynomials and . Rectangles of the same height (width) can be glued horizontally (vertically). Gluing horizontally to itself times yields the rectangle , with a corresponding -power , which is central and nonidentity for , and . Thus . For that , , the asymptotic of then yields the lower bound for some constants and .
In the general case we are given as in (10.2). To each corresponds the nonidentity polynomial , with the corresponding rectangular tableaux , all with the same width . These tableaux can be glued vertically, thus yielding the rectangular tableau , where . To that tableau there corresponds a polynomial which is essentially the product of the polynomials , with the corresponding component which is the product of the corresponding polynomials , hence that polynomial is central nonidentity of , and similarly for powers of these polynomials. Similar to the above case of a single matrix algebra , the asymptotic of then yields the lower bound for some constants and . This proves the lower bound.
Corollary 10.4. Putting together the lower and the upper bounds, Theorem 10.1 then follows.
11. Berele's “1/2” Theorem (See )
Applying Theorem 10.1, Berele proved the following remarkable theorem.
Based on various examples, that theorem was conjectured for some time. It was first proved in  under the hypothesis that satisfies a Capelli identity, and later in general in . Here is an outline of the proof in the Capelli case.
Proof . Recall that , where is the multiplicity of . If satisfies a Capelli identity then there exists an integer such that only for partitions with at most parts (see Remark 5.2). The proof is now based on investigating the asymptotics of the and , and we proceed to do so.
Let be the universal PI algebra for in generators. This algebra has an grading by degree and a corresponding Poincaré series . It also has a finer grading by multidegree with corresponding Poincaré series . Belov  proved that is (the Taylor series of) a rational function with coefficients in (see also , Theorem 9.44). It is not difficult to adapt that proof to show that is also a rational function with integer coefficients. The proof also implies that the denominator of this rational function can be taken to be a product of terms of the form . We call such a rational function “nice.”
The Poincaré series is related to the cocharacters via where is the Schur function of .
Quite a lot is known about Taylor series of nice rational functions, see for example . If is any nice rational function with Taylor series , then the coefficients can be described using a finite set of polynomials. Namely, can be partitioned into regions with corresponding polynomials such that for each , Using properties of Schur functions it can be shown that an analogue formula holds for the . Hence we may write The regions , in general, are defined by linear inequalities and modular linear equations. Here is a simple example to make this more clear. Let . Then is given by Now back to the general case. There exists a number called the essential height of the cocharacter defined to be the largest number such that there exist nonzero multiplicities in which and can be taken arbitrarily large. Giambruno and Zaicev proved that . This number, denoted , is also important for the proof of Theorem 11.1. Let be the vector in whose coordinates are ones followed by zeros. Then in (11.4) certain summands will be on regions of the form where is an integer lattice. Let be the regions of this form. Then in the computation of these terms dominate and we can refine (11.4) to The rest of the proof closely immitates the computation of . The main theorem is that if is as above and is a polynomial then summed over partitions of in will be asymptotic to a constant—times to an integer power—times . Hence, the cocharacter will be a sum of such terms.
There remains the problem that the powers of might not be equal, and the way around this difficulty is to use the fact that if then the cocharacter sequence is Young derived, see . Namely, the Poincaré series can be written as , where has all the nice properties of .
We conclude this section with the following general remark.
12. Algebraicity of Some Generating Functions
Definition 12.1 (see [60, 61]). (1) Given the sequence , then is its corresponding ordinary generating function. In particular is the generating function of the codimensions of the algebra .
(2) The function is algebraic if there exist polynomials such that Algebraicity or nonalgebraicity of the generating function is an indication of the complexity of the sequence .
Note that when , and , hence which is algebraic.
12.1. Nonalgebraicity of Some Generating Functions
Theorem 12.3 (see ). Let , , and assume that as goes to infinity, where and are complex constants and is a real number. For to be algebraic it is necessary that be rational; and if then must also be non-integral.
Applying this theorem to the codimensions of matrices we deduce the following theorem.
Theorem 12.4. Let be the generating functions of the (ordinary) codimensions of . If and is odd then is not algebraic.
Conjecture 12.5. If then the generating function is not algebraic.
For the other verbally prime algebras (see Section 2.2.1), recall from Theorem 6.4 the following partial asymptotic results: where the constant is yet unknown. Also, where the constants and are yet unknown, and .
Corollary 12.6. If then the generating function of the codimensions in not algebraic.
Example 12.7 (see ). We apply here a theorem due to Kemer  which says that the algebras and have the same identities, hence the same codimensions. Now see , and hence which is clearly algebraic. Note that compare with (6.3) of Section 6. See also .
Conjecture 12.8. If and then is not algebraic.
13. Nonassociative with a Non Integer
As remarked before, the notions of codimensions and cocharacters also apply to nonassociative PI algebras. Here, in the most general case, the free associative algebra is replaced by the free algebra . Now is a nonassociative PI algebra and again, are the identities of . Since we are dealing now with nonassociative polynomials, hence different parenthesizes in a monomial yield different monomials. It follows that in the nonassociative case, the space of the multilinear polynomials of degree is now of dimension , where is the th Catalan number (which counts the number of different parenthesizes of a monomial of degree ). The definition of the codimensions is, formally, the same as that in the associative case.
compare with Definition 4.1.
Similarly for the cocharacters, the action of on is again given by (5.1), and one introduces cocharacters precisely as in the associative case, see Definition 5.1. However, some phenomena here are rather different from those in the associative case. For example, a counter-example to Theorem 4.4, hence also to Theorem 4.3, was given in .
A counter-example to Theorem 10.1 in the nonassociative case was first constructed in . Recently, Giambruno et al.  constructed a family of nonassociative PI algebras such that for every there is such an algebra for which the limit exists and is equal to . We briefly describe that construction.
13.1. The Algebra
Let be an infinite word on the alphabet . Given an integer , define the sequence by Construct the algebra as follows. Its basis is where and multiplication is given as follows: and all other products are zero.
Let be an infinite word. The notion of the complexity of is classical. For each , is the number of distinct subwords of of length . The algebra depends on the integer and on the complexity of the word , and the following theorem is proved.
Theorem 13.2 (see ). Given , we can choose and a word such that
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