Growth for Algebras Satisfying Polynomial Identities

Abstract

The 𝑛th codimension 𝑐_𝑛(𝐴) of a PI algebra 𝐴 measures how many identities of degree 𝑛 the algebra 𝐴 satisfies. Growth for PI algebras is the rate of growth of 𝑐_𝑛(𝐴) as 𝑛 goes to infinity. Since in most cases there is no hope in finding nice closed formula for 𝑐_𝑛(𝐴), we study its asymptotics. We review here such results about 𝑐_𝑛(𝐴), when 𝐴 is an associative PI algebra. We start with the exponential bound on 𝑐_𝑛(𝐴) then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of 𝑐_𝑛(𝐴). The representation theory of the symmetric group 𝑆_𝑛 is an important tool in this theory.

1. Introduction

We study algebras 𝐴 satisfying polynomial identities (PI algebra). A natural question arises: give a quantitative description of how many identities such algebra 𝐴 satisfies? We assume that 𝐴 is associative, though the general approach below can be applied to nonassociative PI algebras as well.

Denote by Id(𝐴) the ideal of identities of 𝐴 in the free algebra 𝐹{𝑥}. If Id(𝐴)≠0 then its dimension dimId(𝐴) 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 𝑛, we let 𝑉_𝑛 denote the multilinear polynomials of degree 𝑛 in 𝑥₁,…,𝑥_𝑛, so in the associative case dim𝑉_𝑛=𝑛!. Identities can always be multilinearized, hence the subset Id(𝐴)∩𝑉_𝑛 and its dimension give a good indication as to how many identities of degree 𝑛𝐴 satisfies. In fact, in characteristic zero the ideal Id(𝐴) is completely determined by the sequence {Id(𝐴)∩𝑉_𝑛}_𝑛≥1, but we make no use of that remark in the sequel.

To study dim(Id(𝐴)∩𝑉_𝑛), we introduce the quotient space 𝑉_𝑛/(Id(𝐴)∩𝑉_𝑛) and its dimension 𝑐_𝑛(𝐴)𝑉=dim^𝑛Id(𝐴)∩𝑉_𝑛. (1.1) The integer 𝑐_𝑛(𝐴) is the 𝑛th codimension of 𝐴. Clearly 𝑐_𝑛(𝐴) determines dim(Id(𝐴)∩𝑉_𝑛) since dim𝑉_𝑛 is known.

The study of growth for PI algebra 𝐴 is mostly the study of the rate of growth of the sequence 𝑐_𝑛(𝐴) of its codimensions, as 𝑛 goes to infinity. We have the following basic property.

Theorem 1.1 (see [1]). In the associative case, 𝑐_𝑛(𝐴) 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 𝑐_𝑛(𝐴). Instead, one therefore tries to determine the asymptotic behavior of that sequence, as 𝑛 goes to infinity. We mention here three such results.

Recall that for two sequences of numbers, 𝑎_𝑛∼𝑏_𝑛 if lim_𝑛→∞𝑎_𝑛/𝑏_𝑛=1.

(1) The asymptotics for the 𝑘×𝑘 matrices 𝑀_𝑘(𝐹), see Section 6.

Theorem 1.2 (see [2]). When 𝑛 goes to infinity, 𝑐_𝑛𝑀_𝑘(𝐹∼1)√2𝜋^𝑘−112^{(𝑘2−1)/2}⋅1!2!⋯(𝑘−1)!⋅𝑘^(𝑘2/2)⋅1𝑛^{(𝑘2−1)/2}⋅𝑘^2(𝑛+1). (1.2)

(2) The integrality theorem of Giambruno-Zaicev, see Section 10.

Theorem 1.3 (see [3]). Let 𝐴 be an associative PI 𝐹-algebra with char(𝐹)=0, then the limit lim_𝑛→∞𝑐_𝑛(𝐴)^1/𝑛 (1.3) exists and is an integer. We denote exp(𝐴)=lim_𝑛→∞(𝑐_𝑛(𝐴))^1/𝑛, so exp(𝐴)∈ℕ.

(3) The “1/2” theorem of Berele, see Section 11.

Theorem 1.4 (see [4, 5]). Let 𝐴 be a PI algebra with 1∈𝐴. Then as 𝑛 goes to infinity, 𝑐_𝑛(𝐴)∼𝑎⋅𝑛^𝑡⋅ℎ^𝑛, moreover, ℎ∈ℕ (ℎ is given by the previous theorem) and 𝑡∈(1/2)ℤ, namely, 𝑡 is an integer or a half integer.

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 𝑇-ideals. After introducing the codimensions, two proofs of their exponential bound are reviewed. The 𝑆_𝑛 character of that space, 𝜒_𝑆𝑛(𝑉_𝑛/(Id(𝐴)∩𝑉_𝑛)) is denoted as follows: 𝜒_𝑛(𝐴)=𝜒_𝑆𝑛𝑉_𝑛Id(𝐴)∩𝑉_𝑛 (1.4) and is called the 𝑛th cocharacter of 𝐴. Since 𝑐_𝑛(𝐴)=deg𝜒_𝑛(𝐴), cocharacters are refinement of codimensions, and are important tool in their study. By a theorem of Amitsur-Regev and of Kemer, 𝜒_𝑛(𝐴) is supported on some (𝑘,ℓ) 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 ∑_𝑛𝑐_𝑛(𝐴)⋅𝑥^𝑛 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 𝑇 Ideals

2.1. 𝑇-ideals

Let 𝐹 be a field. In most cases we assume that char(𝐹)=0. We begin by studying associative 𝐹-algebras. Analogue theories exist for nonassociative algebras. Let 𝐹{𝑥}=𝐹{𝑥₁,𝑥₂,…} denote the algebra of associative and noncommutative polynomials in the countable sequence of variables 𝑥₁,𝑥₂,…. The polynomial 𝑓(𝑥₁,…,𝑥_𝑛)∈𝐹{𝑥} is an identity of the 𝐹-algebra 𝐴 if 𝑓(𝑎₁,…,𝑎_𝑛)=0 for every 𝑎₁,…,𝑎_𝑛∈𝐴. The algebra 𝐴 satisfies polynomial identities, or in short is PI, if there exist a nonzero polynomial 0≠𝑓(𝑥)=𝑓(𝑥₁,…,𝑥_𝑛)∈𝐹{𝑥} which is an identity of 𝐴. For example, any commutative algebra is PI since it satisfies 𝑥₁𝑥₂−𝑥₂𝑥₁. 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 𝑀_𝑘(𝐹) of the 𝑘×𝑘 matrices over 𝐹 plays a basic role in PI theory.

Definition 2.1. (1) Let Id(𝐴)⊆𝐹{𝑥} be the subset of the identities of the algebra 𝐴. Id(𝐴)=𝑓∈𝐹{𝑥}∣𝑓(𝑥)=0isanidentityof𝐴. (2.1)
 (2) T ideals. The set Id(𝐴) is a two sided ideal in 𝐹{𝑥}, with the additional property that it is closed under substitutions. Such ideals in 𝐹{𝑥} are called T ideals. Thus, Id(𝐴) is a 𝑇 ideal. It is easy to show that a 𝑇 ideal is always the ideal of identities of some PI algebra 𝐴.
 (3) Varieties of PI algebras. Given a 𝑇-ideal 𝐼⊆𝐹{𝑥}, the class of the PI algebras 𝐴 satisfying 𝐼⊆Id(𝐴) is the variety corresponding to 𝐼. We use here the language of 𝑇 ideals, rather than that of varieties.

2.2. Kemer's Theory for 𝑇-Ideals (See [6])

The Specht Problem
One of the main problems in PI theory is the Specht problem: Are 𝑇 ideals always finitely generated as 𝑇 ideals?

Kemer [6] developed a powerful structure theory for 𝑇 ideals which allowed him to prove that if char(𝐹)=0 then 𝑇 ideals are indeed finitely generated. We review some of the ingredients of that theory.

Amitsur [7] proved that the 𝑇 ideal 𝐼=Id(𝑀_𝑘(𝐹)) is prime in the following sense. If 𝑓𝑔∈𝐼 then either 𝑓∈𝐼 or 𝑔∈𝐼; moreover, the only prime 𝑇 ideals in 𝐹{𝑥} are 𝐼=Id(𝑀_𝑘(𝐹)).

Kemer introduced the notion of verbally prime ideals as follows.

Definition 2.2. The 𝑇 ideal 𝐼 is verbally prime if it satisfies the following condition: Let 𝑓(𝑥₁,…,𝑥_𝑚) and 𝑔(𝑥_𝑚+1,…,𝑥_𝑚+𝑛) be polynomials in disjoint sets of variables. If 𝑓𝑥₁,…,𝑥_𝑚⋅𝑔𝑥_𝑚+1,…,𝑥_𝑚+𝑛∈𝐼then𝑓∈𝐼or𝑔∈𝐼. (2.2) Kemer then classified the verbally prime ideals in 𝐹{𝑥}, see Theorem 2.3 below.

2.2.1. The Algebras 𝑀_𝑘(𝐹), 𝑀_𝑘(𝐺), and 𝑀_𝑘,ℓ

Let 𝑈=span_𝐹{𝑢₁,𝑢₂,…} be an infinite dimensional vector space, and let 𝐺=𝐺(𝑈) be the corresponding infinite dimensional Grassmann (Exterior) algebra. Then 𝐺=span𝑢_𝑖1⋯𝑢_𝑖𝑟∣𝑟=1,2,…and1≤𝑖₁<⋯<𝑖_𝑟, (2.3) and 𝐺=𝐺₀⊕𝐺₁, where 𝐺₀=span𝑢_𝑖1⋯𝑢_𝑖𝑟,𝐺∣𝑟even₁=span𝑢_𝑖1⋯𝑢_𝑖𝑟∣𝑟odd. (2.4) Now 𝑀_𝑘(𝐹) are the 𝑘×𝑘 matrices over 𝐹, while 𝑀_𝑘(𝐺) are the 𝑘×𝑘 matrices over 𝐺.

The Algebra 𝑀_𝑘,ℓ
We have 𝑀_𝑘,ℓ⊆𝑀_𝑘+ℓ(𝐺). The elements of 𝑀_𝑘,ℓ are block matrices 𝐴𝐵𝐶𝐷, (2.5) where 𝐴∈𝑀_𝑘(𝐺₀), 𝐷∈𝑀_ℓ(𝐺₀), 𝐵 is 𝑘×ℓ and 𝐶 is ℓ×𝑘, both with entries from 𝐺₁. For example: 𝑀_1,1=𝐺₀𝐺₁𝐺₁𝐺₀. (2.6)

Theorem 2.3 (see [6]). The following are the three families of the verbally prime 𝑇 ideals: Id(𝑀_𝑘(𝐹)), Id(𝑀_𝑘(𝐺)), and Id(𝑀_𝑘,ℓ).

The importance of the verbally prime ideals is demonstrated in the following theorem.

Theorem 2.4 (see [6]). Let 𝐼⊂𝐹{𝑥} be a 𝑇 ideal. Then there exist verbally prime 𝑇 ideals 𝐽₁,…,𝐽_𝑟 such that 𝐽₁⋯𝐽_𝑟⊆𝐼⊆𝐽₁∩⋯∩𝐽_𝑟. (2.7)

3. Multilinear Polynomials

As usual, 𝑆_𝑛 denotes the 𝑛th symmetric group.

Definition 3.1. The polynomial 𝑓(𝑥₁,…,𝑥_𝑛)∈𝐹{𝑥} is multilinear (in 𝑥₁,…,𝑥_𝑛) if 𝑓𝑥₁,…,𝑥_𝑛=_{𝜎∈𝑆𝑛}𝛼_𝜎𝑥_𝜎(1)⋯𝑥_𝜎(𝑛) (3.1) for some coefficients 𝛼_𝜎∈𝐹. Let 𝑉_𝑛=𝑉_𝑛(𝑥₁,…,𝑥_𝑛) denote the vector space of multilinear polynomials in 𝑥₁,…,𝑥_𝑛, so dim𝑉_𝑛=𝑛!. Extending the map 𝜎→𝑥_𝜎(1)⋯𝑥_𝜎(𝑛) by linearity yields the vector space isomorphism 𝑉_𝑛≅𝐹𝑆_𝑛, where 𝐹𝑆_𝑛 is the group algebra of 𝑆_𝑛. We will identify 𝐹𝑆_𝑛≡𝑉_𝑛. (3.2) By the process of multilinearization [8, 9] one proves the following theorem.

Theorem 3.2. Let the PI algebra 𝐴 satisfy an identity of degree 𝑑, then 𝐴 satisfies a multilinear identity of degree 𝑑. Moreover, if char(𝐹)=0 then the ideal of identities Id(𝐴) is determined by its multilinear elements.

It follows that if 𝐴 satisfies an identity of degree 𝑑 then 𝐴 satisfies an identity of the form 𝑦₁⋯𝑦_𝑑−_{1≠𝜋∈𝑆𝑑}𝛼_𝜋𝑦_𝜋(1)⋯𝑦_𝜋(𝑑),𝛼_𝜋∈𝐹. (3.3) 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 𝑓(𝑥₁,…,𝑥_𝑛) is alternating in 𝑥₁,…,𝑥_𝑛 if for every permutation 𝜋∈𝑆_𝑛, 𝑓(𝑥_𝜋(1),…,𝑥_𝜋(𝑛))=sgn(𝜋)𝑓(𝑥₁,…,𝑥_𝑛). For example, the polynomial St_𝑛[𝑥]=St_𝑛𝑥₁,…,𝑥_𝑛=_{𝜎∈𝑆𝑛}sgn(𝜎)⋅𝑥_𝜎(1)⋅𝑥_𝜎(2)⋯𝑥_𝜎(𝑛) (3.4) is alternating. It is called the 𝑛th Standard polynomial.

Similarly the polynomial Cap_𝑛𝑥;𝑦=_{𝜎∈𝑆𝑛}sgn(𝜎)⋅𝑥_𝜎(1)⋅𝑦₁⋅𝑥_𝜎(2)⋅𝑦₂⋯𝑦_𝑛−1⋅𝑥_𝜎(𝑛), (3.5) which is called the (𝑛th) Capelli polynomial, is multilinear of degree 2𝑛−1 and is alternating in 𝑥₁,…,𝑥_𝑛.

It is rather easy to show that if dim𝐴=𝑑<∞ then 𝐴 satisfies both Cap_𝑑+1[𝑥] and St_𝑑+1[𝑥], and hence every finite dimensional algebra is PI. And the same argument applies in the nonassociative case. In particular the algebra 𝑀_𝑘(𝐹) of the 𝑘×𝑘 matrices satisfies St_𝑘²₊₁[𝑥₁,𝑥₂,…,𝑥_𝑘²₊₁]. The celebrated Amitsur-Levitzki Theorem [10–12] states that 𝑀_𝑘(𝐹) satisfies the standard identity St_2𝑘[𝑥₁,…,𝑥_2𝑘]=0. Of course, for large 𝑘 the degree 2𝑘 is much smaller than 𝑘²+1.

Many infinite dimensional algebras are PI. For example, any infinite dimensional commutative algebra is PI. We remark that obviously, the free algebra 𝐹{𝑥} itself is not PI.

4. The Codimensions

Question. How many identities are satisfied by a given PI algebra, namely, how large are 𝑇 ideals?

Computing dimensions might seem useless at first sight, since if the 𝑇-ideal 𝐼 is nonzero then dim𝐼=∞. To answer the above question we introduce below the notion of codimensions. Given a PI algebra 𝐴, we would like to study its multilinear identities of degree 𝑛, namely, the space Id(𝐴)∩𝑉_𝑛𝑥₁,…,𝑥_𝑛=Id(𝐴)∩𝑉_𝑛, (4.1)𝑉_𝑛 as in (3.2). Note that if 𝑦₁,…,𝑦_𝑛 are any 𝑛 variables then Id(𝐴)∩𝑉_𝑛(𝑥)≅Id(𝐴)∩𝑉_𝑛(𝑦). A first step is the study of the dimensions dim(Id(𝐴)∩𝑉_𝑛). Now (Id(𝐴)∩𝑉_𝑛)⊆𝑉_𝑛, and since dim𝑉_𝑛=𝑛!, clearly dimId(𝐴)∩𝑉_𝑛𝑉=𝑛!−dim^𝑛Id(𝐴)∩𝑉_𝑛. (4.2) Thus, knowing dim(Id(𝐴)∩𝑉_𝑛) is equivalent to knowing dim(𝑉_𝑛/(Id(𝐴)∩𝑉_𝑛)). This leads us to introduce the codimensions 𝑐_𝑛(𝐴) as follows.

Definition 4.1. Let 𝐴 be an 𝐹-algebra, then 𝑐_𝑛(𝐴)𝑉=dim^𝑛Id(𝐴)∩𝑉_𝑛 (4.3) is called the 𝑛th codimension of 𝐴, and {𝑐_𝑛(𝐴)}^∞_𝑛=1 is the sequence of codimensions of 𝐴.

For example, 𝐴 satisfies 𝐴^𝑛=0 (i.e., is nilpotent) if and only if 𝑐_𝑛(𝐴)=0. And 𝐴 is commutative if and only if 𝑐_𝑛(𝐴)≤1 for all 𝑛.

Remark 4.2. Note that if 𝐴 is not PI then Id(𝐴)=0, hence 𝑐_𝑛(𝐴)=𝑛! for all 𝑛. In fact, the algebra 𝐴 is PI (i.e., Id(𝐴)≠0) if and only if there exist 𝑛 such that 𝑐_𝑛(𝐴)<𝑛!. This follows directly from the definition.

4.1. Exponential Bound for the Codimensions

A basic property of the codimensions sequence 𝑐_𝑛(𝐴) 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 [1]). Assume the (associative) algebra 𝐴 satisfies an identity of degree 𝑑, then 𝑐_𝑛(𝐴)≤(𝑑−1)^2𝑛 (4.4) for all 𝑛.

Proof. We sketch two proofs (both different from the original proof). Both proofs apply the notion of a 𝑑-good permutation, which we now review.
Call 𝜎∈𝑆_𝑛𝑑-bad if there are indices 1≤𝑖₁<⋯<𝑖_𝑑≤𝑛 with 𝜎(𝑖₁)>⋯>𝜎(𝑖_𝑑). Otherwise 𝜎 is 𝑑-good, and we denote 𝑔_𝑑(𝑛)=𝜎∈𝑆_𝑛∣𝜎is𝑑-good. (4.5) By a Shirshov-Latyshev argument [13, 14], if 𝐴 satisfies an identity of degree 𝑑, then 𝐴 satisfies an identity (3.3) which, by a certain inductive argument implies that 𝑐_𝑛(𝐴)≤𝑔_𝑑(𝑛).
By an argument based on Dilworth Theorem in Combinatorics [15], Latyshev then showed that 𝑔_𝑑(𝑛)≤((𝑑−1)²)^𝑛, thus completing the (first) proof.
A second proof of the bound 𝑔_𝑑(𝑛)≤((𝑑−1)²)^𝑛 goes as follows [16]. First, by the RSK correspondence [17], 𝑔_𝑑(𝑛)=_{𝜆⊢𝑛,ℓ(𝜆)≤𝑑−1}𝑓^𝜆2, (4.6) where 𝑓^𝜆 is the number of standard Young taableaux of shape 𝜆, and ℓ(𝜆) is the number of parts of 𝜆. Then by the Schur-Weyl theory, _{(𝜆)𝜆⊢𝑛,ℓ≤𝑑−1}𝑓^𝜆2=dim(𝐵(𝑑−1,𝑛)), (4.7) where dim𝑈=𝑑−1 and 𝐵(𝑑−1,𝑛)⊆End_𝐹(𝑈^⊗𝑛). The (second) proof now follows since dim(End_𝐹(𝑈^⊗𝑛))=(𝑑−1)^2𝑛.

4.1.1. Application: The 𝐴⊗𝐵 Theorem

Codimensions where introduced, in [1], in order to prove the following theorem.

Theorem 4.4 (see [1]). If 𝐴 and 𝐵 are (associative) PI algebras then 𝐴⊗𝐵 is PI.

Proof. It is not too difficult to show that 𝑐_𝑛(𝐴⊗𝐵)≤𝑐_𝑛(𝐴)⋅𝑐_𝑛(𝐵). Assume now that 𝐴 satisfies an identity of degree 𝑑, and 𝐵 satisfies an identity of degree ℎ. Together with Theorem 4.3, this implies that 𝑐_𝑛(𝐴⊗𝐵)≤(𝑑−1)^2𝑛(ℎ−1)^2𝑛, (4.8) hence for a large enough 𝑛, 𝑐_𝑛(𝐴⊗𝐵)<𝑛!. Finally by Remark 4.2,  𝑅 is PI if and only if there exist 𝑛 such that 𝑐_𝑛(𝑅)<𝑛!, and the proof follows.

Remark 4.5. For explicit identities for 𝐴⊗𝐵 see Remark 8.2 below, which also implies the following. Again let 𝐴 satisfy an identity of degree 𝑑, and 𝐵 an identity of degree ℎ. Then 𝐴⊗𝐵 satisfies an identity of degree about 𝑒(𝑑−1)²(ℎ−1)², where 𝑒=2.718….

We remark that both Theorems 4.3 and 4.4 fail in some non-associative cases.

The above results motivate the study of the following problem.

Question 1. Given a PI algebra 𝐴, find a formula for the sequence 𝑐_𝑛(𝐴). In most cases this seems to be too difficult, and one tries to, at least, get a “nice” asymptotic approximation of 𝑐_𝑛(𝐴).

In Section 6 we determine such asymptotics for the algebras 𝑀_𝑘(𝐹). We also give such partial results for the other verbally prime algebras 𝑀_𝑘,ℓ and 𝑀_𝑘(𝐺).

5. Cocharacters

Recall the identification 𝐹𝑆_𝑛≡𝑉_𝑛 (see (3.2)) and let 𝐴 be a PI algebra. The regular left action of 𝑆_𝑛 on 𝑉_𝑛 is as follows. Let 𝜎∈𝑆_𝑛 and 𝑔(𝑥₁,…,𝑥_𝑛)∈𝑉_𝑛, then 𝜎𝑔𝑥₁,…,𝑥_𝑛=𝑔𝑥_𝜎(1),…,𝑥_𝜎(𝑛). (5.1) This makes 𝑉_𝑛 into a left 𝑆_𝑛 module, with Id(𝐴)∩𝑉_𝑛 a submodule of 𝑉_𝑛.

Definition 5.1. The 𝑆_𝑛-character 𝜒_𝑛(𝐴)∶=𝜒_𝑆𝑛𝑉_𝑛Id(𝐴)∩𝑉_𝑛 (5.2) of the quotient module 𝑉_𝑛/(Id(𝐴)∩𝑉_𝑛) is called the 𝑛th cocharacter of 𝐴.

Cocharacters where introduced in [18] in the study of standard identities.

Since char(𝐹)=0, from the ordinary representation theory of 𝑆_𝑛 [19, 20] it is well known that the irreducible 𝑆_𝑛 characters are parametrized by the partitions 𝜆 on 𝑛: Irred𝑆_𝑛=𝜒^𝜆∣𝜆⊢𝑛. (5.3) Also, 𝐹𝑆_𝑛 decomposes as follows: 𝐹𝑆_𝑛=_𝜆⊢𝑛𝐼_𝜆, (5.4) where the 𝐼_𝜆 are the minimal two sided ideals of 𝐹𝑆_𝑛, and there are natural bijections 𝜒^𝜆⟷𝜆⟷𝐼_𝜆. (5.5) Here deg𝜒^𝜆=𝑓^𝜆 and dim𝐼_𝜆=(𝑓^𝜆)² where, again, 𝑓^𝜆 is the number of standard Young taableaux of shape 𝜆.

Given a tableau 𝑇_𝜆 of shape 𝜆⊢𝑛, it determines the semi-idempotent 𝑒_𝑇𝜆=𝑅⁺_𝑇𝜆𝐶⁻_𝑇𝜆, and 𝐹𝑆_𝑛𝑒_𝑇𝜆⊆𝐼_𝜆 is a minimal left ideal.

The 𝑆_𝑛-character of the regular representation 𝐹𝑆_𝑛≡𝑉_𝑛 is 𝜒_𝑆𝑛(𝑉_𝑛)=∑_𝜆⊢𝑛𝑓^𝜆𝜒^𝜆. It follows that the 𝑛th cocharacter of 𝐴 can be written as 𝜒_𝑛(𝐴)=_𝜆⊢𝑛𝑚_𝜆(𝐴)𝜒^𝜆 (5.6) for some multiplicities 𝑚_𝜆(𝐴), and 𝑚_𝜆(𝐴)≤𝑓^𝜆.

Clearly, the degree of 𝜒_𝑛(𝐴) is the corresponding codimension 𝑐_𝑛(𝐴), and since deg(𝜒^𝜆)=𝑓^𝜆, hence 𝑐_𝑛(𝐴)=deg𝜒_𝑛(𝐴)=_𝜆⊢𝑛𝑚_𝜆(𝐴)𝑓^𝜆. (5.7) For example [21] let 𝐺 be the infinite dimensional Grassmann (Exterior) algebra, then 𝑚_𝜆(𝐺)=1if𝜆=𝑚−𝑟,1^𝑟0otherwise. (5.8)

Remark 5.2. Properties of the identification 𝐹𝑆_𝑛≡𝑉_𝑛 imply the following characterization of Capelli identities (Section 3.1) by cocharacters [22]. Let 𝐴 be a PI algebra with 𝜒_𝑛(𝐴)=∑_𝜆⊢𝑛𝑚_𝜆(𝐴)𝜒^𝜆 its cocharacter. Then 𝐴 satisfies Cap_𝑑+1[𝑥;𝑦] if and only if 𝑚_𝜆(𝐴)=0 whenever ℓ(𝜆)≥𝑑+1.

As an application of cocharacters one can prove the following.

Proposition 5.3 (see [23]). If 𝐴 satisfies Cap_𝑑+1 and 𝐵 satisfies Cap_ℎ+1 then 𝐴⊗𝐵 satisfies Cap_𝑑ℎ+1.

We remark that a proof of this result without applying cocharacters is yet unknown.

Question 2. Given a PI algebra 𝐴, find a formula for the multiplicities 𝑚_𝜆(𝐴). In most cases this is too difficult, and one tries to at least get some approximate description of 𝑚_𝜆(𝐴).

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 2×2 matrices 𝑀₂(𝐹) there is the following nice formula for the multiplicities 𝑚_𝜆(𝑀₂(𝐹)) of 𝜒_𝑛(𝑀₂(𝐹)).

Example 5.5 (see [24], [25, Theorem 12.6.5], [26]). Denote 𝑚_𝜆(𝑀₂(𝐹))=𝑚_𝜆. First, if ℓ(𝜆)>4 then 𝑚_𝜆=0. So let 𝜆=(𝜆₁,…,𝜆₄). If 𝜆=(𝑛) then 𝑚_(𝑛)=1. If 𝜆=(𝜆₁,𝜆₂) with 𝜆₂>0 then 𝑚_{(𝜆1,𝜆2)}=(𝜆₁−𝜆₂+1)⋅𝜆₂. If 𝜆=(𝜆₁,1,1,𝜆₄) (so 𝜆₄≤1) then 𝑚_{(𝜆1,1,1,𝜆4)}=𝜆₁⋅(2−𝜆₄)−1. And 𝑚_𝜆=(𝜆₁−𝜆₂+1)⋅(𝜆₂−𝜆₃+1)⋅(𝜆₃−𝜆₄+1) in all other cases. Recent results of Berele [27] and of Drensky and Genov [28] indicate that when 𝑘≥3, there is no hope in getting nice formulas for the cocharacter-multiplicities 𝑚_𝜆(𝑀_𝑘(𝐹)). A somewhat similar phenomena is discussed in Section 12.

5.2. Trace Identities, Codimensions, and Cocharacters

In the case of 𝑘×𝑘 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 𝑥₁⋅𝑥₂⋅tr(𝑥₃⋅𝑥₄) is a (mixed) trace polynomial. These trace polynomials can be evaluated on the algebra 𝑀_𝑘(𝐹) (or on any algebra with a trace) hence yielding trace identities. For example, it can be proved that the trace polynomial 𝑔𝑥₁,𝑥₂=tr𝑥₁tr𝑥₂−𝑥₁tr𝑥₂−𝑥₂tr𝑥₁−tr𝑥₁𝑥₂+𝑥₁𝑥₂+𝑥₂𝑥₁ (5.9) is a trace identity of 𝑀₂(𝐹). This is an example of a mixed trace polynomial, while the polynomial 𝑝𝑥₁,𝑥₂,𝑥₃=tr𝑥₁tr𝑥₂tr𝑥₃−tr𝑥₁𝑥₃tr𝑥₂−tr𝑥₂𝑥₃tr𝑥₁−tr𝑥₁𝑥₂tr𝑥₃+tr𝑥₁𝑥₂𝑥₃+tr𝑥₂𝑥₁𝑥₃ (5.10) is a pure trace polynomial, which is also an identity of 𝑀₂(𝐹). We then have trace identities of 𝑀_𝑘(𝐹), “pure” and “mixed,” hence trace codimensions 𝑐^ptr_𝑛(𝑀_𝑘(𝐹)) and 𝑐^mtr_𝑛(𝑀_𝑘(𝐹)), and trace cocharacters 𝜒^ptr_𝑛(𝑀_𝑘(𝐹)) and 𝜒^mtr_𝑛(𝑀_𝑘(𝐹)).

The Procesi-Razmyslov theory of trace identities [29–31], together with the Schur-Weyl theory [32, 33], imply the following formula for the pure trace cocharacters 𝜒^ptr_𝑛(𝑀_𝑘(𝐹)) of 𝑀_𝑘(𝐹).

Theorem 5.6. We have 𝜒^ptr_𝑛𝑀_𝑘(𝐹)=_{𝜆⊢𝑛,ℓ(𝜆)≤𝑘}𝜒^𝜆⊗𝜒^𝜆. (5.11)

This implies that the trace codimensions are given by the following formula: 𝑐^ptr_𝑛𝑀_𝑘(𝐹)=_{𝜆⊢𝑛,ℓ(𝜆)≤𝑘}𝑓^𝜆2. (5.12) This formula is the starting point for computing the asymptotic formula of 𝑐_𝑛(𝑀_𝑘(𝐹)) given in Section 6.

6. Asymptotics of the Codimensions 𝑐_𝑛(𝑀_𝑘(𝐹))

For the 2×2 matrices, Procesi [34] proved the following formula for 𝑐_𝑛(𝑀₂(𝐹)).

Theorem 6.1 (see [34]). We have 𝑐_𝑛𝑀₂(𝐹)=1−𝑛𝑛+132𝑛+2𝑛+1+1−2^𝑛. (6.1)

It was already mentioned that when 𝑘≥3, it most likely is impossible to find an exact formula for the multiplicities 𝑚_𝜆(𝑀_𝑘(𝐹)), and the same is probably true about 𝑐_𝑛(𝑀_𝑘(𝐹)). Instead of giving up, one looks for the asymptotic of 𝑐_𝑛(𝑀_𝑘(𝐹)).

Here we have the folowing theorem.

Theorem 6.2 (see [2]). When 𝑛 goes to infinity, 𝑐_𝑛𝑀_𝑘(𝐹∼1)√2𝜋^𝑘−112^{(𝑘2−1)/2}⋅1!2!⋯(𝑘−1)!⋅𝑘^(𝑘2/2)⋅1𝑛^{(𝑘2−1)/2}⋅𝑘^2(𝑛+1). (6.2)

For example, when 𝑘=2 both Theorems 6.1 and 6.2 give the same asymptotic value 𝑐_𝑛𝑀₂(𝐹∼4)√𝜋⋅1√𝑛⋅4^𝑛, (6.3) compare with (12.13).

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 𝑛th codimensions and the 𝑛+1-st pure trace codimensions are asymptotically equal: 𝑐_𝑛𝑀_𝑘(𝐹)∼𝑐^ptr_𝑛+1𝑀_𝑘(𝐹). (6.4) We saw that by Theorem 5.6, 𝑐^ptr_𝑛𝑀_𝑘(𝐹)=_{𝜆⊢𝑛,ℓ(𝜆)≤𝑘2}𝑓^𝜆2. (6.5) Thus, 𝑐_𝑛𝑀_𝑘(𝐹)∼_{𝜆⊢𝑛+1,ℓ(𝜆)≤𝑘2}𝑓^𝜆2. (6.6) The last major step here is the computation of the asymptotic behavior of the following sum: 𝑆_𝑘⁽²⁾₂(𝑛)=_{𝜆⊢𝑛,ℓ(𝜆)≤𝑘2}𝑓^𝜆2. (6.7) The asymptotics, as 𝑛→∞, of the more general sums 𝑆_ℎ^(𝛽)(𝑛)=_{𝜆⊢𝑛,ℓ(𝜆)≤ℎ}𝑓^𝜆𝛽 (6.8) is given in [16]. That asymptotic is of the form 𝑆_ℎ^(𝛽)(𝑛)∼𝑎⋅𝑛^𝑏⋅𝑟^𝑛, where 𝑎=𝑎(𝛽,ℎ), 𝑏=𝑏(𝛽,ℎ) and 𝑟=𝑟(𝛽,ℎ)=ℎ^𝛽, all given explicitly in [16]. We remark that the constant term 𝑎 is evaluated by applying the Selberg integral (6.9).

Theorem 6.3 (see [36]). The Selberg integral ¹₀⋯¹₀^𝑘_𝑖=1𝑢_𝑖^𝑥−11−𝑢_𝑖^𝑦−1_{1≤𝑖<𝑗≤𝑘}𝑢_𝑖−𝑢_𝑗^2𝑧𝑑𝑢₁⋯𝑑𝑢_𝑘=^𝑘−1_𝑗=0Γ𝑥+𝑗𝑧Γ𝑦+𝑗𝑧Γ1+𝑗+1𝑧Γ𝑧Γ().𝑥+𝑦+𝑘+𝑗−11+𝑧 (6.9)

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.

Theorem 6.4. (1) [37, Theorem 7] 𝑐_𝑛𝑀_𝑘,ℓ1∼𝑎⋅𝑛^{(𝑘2+ℓ2−1)/2}⋅(𝑘+ℓ)^2𝑛. (6.10) The constant 𝑎 is yet unknown.
(2) [37, Theorem 8] Let 𝐺 be the infinite dimensional Grassmann algebra, then 𝑐_𝑛𝑀_𝑘(𝐺)1∼𝑏⋅𝑛^𝑔⋅2𝑘^2𝑛, (6.11) where the constants 𝑏 and 𝑔 are yet unknown, and (𝑘²−1)/2≤𝑔≤(2𝑘²−1)/2.

7. Shirshov's Height Theorem (See [13])

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 𝑥₁,…,𝑥_ℓ of ℓ letters; 𝑊(𝑥₁,…,𝑥_ℓ) is the set of all words (i.e., monomials) in 𝑥₁,…,𝑥_ℓ; 𝑈(𝑑,ℓ) the subset of the words of length ≤𝑑. We consider finitely generated PI algebra 𝐴=𝐹{𝑎₁,…,𝑎_ℓ}.

Theorem 7.1 (Shirshov's Height Theorem). Consider a PI algebra satisfying the identity (3.3): 𝑦₁⋯𝑦_𝑑−_{1≠𝜋∈𝑆𝑑}𝛼_𝜋𝑦_𝜋(1)⋯𝑦_𝜋(𝑑). (7.1) There exists ℎ=ℎ(𝑑,ℓ) large enough such that any finitely generated algebra 𝐴=𝐹{𝑎₁,…,𝑎_ℓ} that satisfies the identity (3.3), satisfies the following condition.
Modulo Id(𝐴), 𝐹{𝑥₁,…,𝑥_ℓ} is spanned by the elements 𝑢^𝑘1₁⋯𝑢^𝑘ℎ_ℎ∣𝑢_𝑖∈𝑈(𝑑,ℓ),any𝑘_𝑗. (7.2)

7.2. Application: Bounds on the Cocharacters

7.2.1. The (𝑘,ℓ) Hook Theorem

Denote by 𝐻(𝑘,ℓ;𝑛) the partitions of 𝑛 in the (𝑘,ℓ) hook: 𝐻(𝑘,ℓ;𝑛)=𝜆₁,𝜆₂,…⊢𝑛∣𝜆_𝑘+1≤ℓ,𝐻(𝑘,ℓ)=∪_𝑛(𝐻(𝑘,ℓ;𝑛)). (7.3) Let 𝜒_𝑛 be 𝑆_𝑛 characters, 𝑛=1,2,…. We say that 𝜒_𝑛 is supported on 𝐻(𝑘,ℓ), and denote 𝜒_𝑛⊆𝐻(𝑘,ℓ), if for all 𝑛, 𝜒_𝑛=_{𝜆∈𝐻(𝑘,ℓ;𝑛)}𝑚_𝜆𝜒^𝜆. (7.4) Similar terminology applies when 𝐻(𝑘,ℓ) is replaced by another family of subsets of partitions. We have the following theorem.

Theorem 7.2 (see [6, 38]). Let 𝐴 be any (associative) PI algebra, then there exist 𝑘,ℓ such that its cocharacters 𝜒_𝑛(𝐴) are supported on the (𝑘,ℓ) hook 𝐻(𝑘,ℓ).
Explicitly, let 𝐴 satisfy an identity of degree 𝑑, and let 𝑘,ℓ≥𝑒⋅(𝑑−1)⁴−1, where 𝑒=2.718… is the base of the natural logarithms. Then 𝜒_𝑛(𝐴)⊆𝐻(𝑘,ℓ), namely, 𝜒_𝑛(𝐴)=_{()𝜆∈𝐻𝑘,ℓ;𝑛}𝑚_𝜆(𝐴)⋅𝜒^𝜆. (7.5)

7.2.2. An Application of Shirshov's Theorem

We remark that the proof of Theorem 7.2 applies the exponential-bound of Theorem 4.3. Theorem 7.2 is the first step towards proving the following polynomial bound.

Theorem 7.3 (see [39, 40]). For any PI algebra 𝐴, all its cocharacter-multiplicities 𝑚_𝜆(𝐴) are polynomially bounded. There exist a constant 𝐶 and a power 𝑝 such that for all 𝑛 and 𝜆⊢𝑛, 𝑚_𝜆(𝐴)≤𝐶⋅𝑛^𝑝.

The proof of Theorem 7.3 also applies Shirshov's Height Theorem 7.1, as well as a ℤ₂ version of the Schur-Weyl theory [32, 33, 41].

8. Explicit Identities

Amitsur [42] proved that any PI algebra satisfies a power of a standard identity, namely, an identity of the form St^𝑣_𝑢[𝑥]=(St_𝑢[𝑥])^𝑣, where St_𝑢[𝑥] is the 𝑢th standard polynomial. Amitsur's proof, which applies Structure Theory of Rings [8], yields a bound on the index 𝑢 but not on 𝑣. A recent proof of Amitsur's theorem [43] applies the identification 𝐹𝑆_𝑛≡𝑉_𝑛, together with the exponential bound on 𝑐_𝑛(𝐴) and yields a combinatorial proof of that theorem, a proof which gives bounds on both 𝑢 and 𝑣. Moreover, the same arguments yield explicit identities in various other cases, for example in the 𝐴⊗𝐵 case.

Theorem 8.1 (see [38, 43]). Let 𝐴 be a PI algebra satisfying 𝑐_𝑛(𝐴)≤𝛼^𝑛, and let 𝑢 and 𝑣 be integers satisfying 𝑢,𝑣≥𝑒⋅𝛼², then St^𝑣_𝑢[𝑥₁,…,𝑥_𝑢]∈Id(𝐴). In particular if 𝐴 satisfies an identity of degree 𝑑, and 𝑢,𝑣≥𝑒⋅(𝑑−1)⁴, then St^𝑣_𝑢[𝑥₁,…,𝑥_𝑢]∈Id(𝐴) (of degree 𝑢𝑣 which is about 𝑒²(𝑑−1)⁸.)
In fact, with these 𝑢 and 𝑣𝐴 satisfies the power of the Capelli identity: (Cap_𝑢[𝑥;𝑦])^𝑣∈Id(𝐴).

Remark 8.2. Let 𝐴 satisfy an identity of degree 𝑑 and 𝐵 an identity of degree ℎ. Denote 𝛼=(𝑑−1)²(ℎ−1)² then 𝑐_𝑛(𝐴⊗𝐵)≤𝛼^𝑛. Thus, if 𝑢,𝑣≥𝑒⋅((𝑑−1)(ℎ−1))⁴ then 𝐴⊗𝐵 satisfies the identities St^𝑣_𝑢[𝑥₁,…,𝑥_𝑢] and (Cap_𝑢[𝑥;𝑦])^𝑣. Note that here the degree of St^𝑣_𝑢[𝑥₁,…,𝑥_𝑢], for example, is about 𝑒²⋅((𝑑−1)(ℎ−1))⁸.

Also, 𝐴⊗𝐵 satisfies some identity of degree 𝑛, where 𝑛 is about 𝑒𝛼=𝑒((𝑑−1)(ℎ−1))². Indeed, let 𝑒𝛼<𝑛, then the classical inequality (𝑛/𝑒)^𝑛<𝑛! implies that 𝛼^𝑛<𝑛!. Thus, if 𝑒(𝑑−1)²(ℎ−1)²<𝑛 then, for that 𝑛, 𝑐_𝑛(𝐴⊗𝐵)≤(𝑑−1)²(ℎ−1)^2𝑛<𝑛! (8.1) and by Remark 4.2  𝐴⊗𝐵 satisfies an identity of degree 𝑛 where 𝑛 is about 𝑒((𝑑−1)(ℎ−1))².

9. Nonidentities for Matrices: The Polynomial 𝐿_𝑘[𝑥;𝑦]

Usually, lower bounds for codimensions are harder to obtain than upper bounds. Given a PI algebra 𝐴, a lower bound for 𝑐_𝑛(𝐴) can be obtained by the following technique. Find a polynomial 𝑝=𝑝(𝑥₁,…,𝑥_𝑛)∈𝑉_𝑛 which is a nonidentity of 𝐴, namely, 𝑝∉Id(𝐴). In addition, with the identification 𝑉_𝑛≡𝐹𝑆_𝑛=⊕_𝜆⊢𝑛𝐼_𝜆 (see (5.4)), verify that 𝑝∈𝐼_𝜇 for some 𝜇⊢𝑛. Then 𝑐_𝑛(𝐴)≥𝑓^𝜇. This follows since 0≠𝐹𝑆_𝑛𝑝⊆𝐼_𝜇, so 𝐽_𝜇⊆𝐹𝑆_𝑛𝑝 for some minimal left ideal 𝐽_𝜇⊆𝐼_𝜇 with 𝐽_𝜇∩Id(𝐴)=0, hence 𝑓^𝜇=dim𝐽_𝜇≤𝑐_𝑛(𝐴). We now construct such a polynomial 𝑝 via the polynomial 𝐿_𝑘(𝑥;𝑦), when 𝐴=𝑀_𝑘(𝐹) [44, Definition  2.3].

Corresponding to the sum 1+3+⋯+(2𝑘−1)=𝑘², construct the monomial 𝑁_𝑘(𝑥;𝑦) of degree 2𝑘²: 𝑁_𝑘𝑥;𝑦=𝑥₁𝑦₁𝑥₂𝑥₃𝑥₄𝑦₂𝑦₃𝑦₄𝑥₅⋯𝑥₉𝑦₅⋯𝑦₉⋯. (9.1) For example, 𝑁₃(𝑥;𝑦)=(𝑥₁)(𝑦₁)(𝑥₂𝑥₃𝑥₄)(𝑦₂𝑦₃𝑦₄)(𝑥₅⋯𝑥₉)(𝑦₅⋯𝑦₉). Now alternate the 𝑥's and alternate the 𝑦's to obtain 𝐿_𝑘(𝑥;𝑦): 𝐿_𝑘𝑥;𝑦=_{𝜎,𝜋∈𝑆𝑘2}sgn(𝜎)sgn(𝜋)𝑁_𝑘𝑥_𝜎(1),…,𝑥_𝜎(𝑘²₎;𝑦_𝜋(1),…,𝑦_𝜋(𝑘²₎. (9.2) Let 𝑇^′_𝜇 be the conjugate tableau of 𝑇_𝜇, and 𝑇^′_𝜇=13459⋯13⋯267814⋯18⋯. (9.3) Then 𝐿_𝑘(𝑥;𝑦)=𝐶⁻_𝑇𝜇 and in that sense 𝐿_𝑘(𝑥;𝑦) corresponds to the tableau 𝑇_𝜇 where 𝜇=(2^𝑘2). It is not difficult to show that 𝐿_𝑘(𝑥;𝑦) takes central values on 𝑀_𝑘(𝐹).

For 𝑘=2 and 𝑘=3 it was verified that 𝐿_𝑘(𝑥;𝑦)∉Id(𝑀_𝑘(𝐹)), and it was conjectured that for all 𝑘𝐿_𝑘(𝑥;𝑦)∉Id(𝑀_𝑘(𝐹)) namely, that 𝐿_𝑘(𝑥;𝑦) is a nonidentity of 𝑀_𝑘(𝐹) [44]. This conjecture was verified by Formanek [35].

Theorem 9.1 (see [35]). The plolynomial 𝐿_𝑘(𝑥;𝑦), which corresponds to the rectangle 𝜇=(2^𝑘2), is a nonidentity of 𝑀_𝑘(𝐹). Hence 𝐿_𝑘(𝑥;𝑦) is a central polynomial.

By Young's rule it follows that 𝐿_𝑘𝑥;𝑦=𝑝_𝜇+_{𝜆⊢2𝑘2,ℓ(𝜆)≥𝑘2+1}𝑝_𝜆, (9.4) where 𝑝_𝜇∈𝐼_𝜇 and 𝑝_𝜆∈𝐼_𝜆, see (5.4). Since 𝑀_𝑘(𝐹) satisfies the Capelli identity Cap_𝑘²₊₁, it follows that for all 𝜆 with ℓ(𝜆)≥𝑘²+1, 𝑝_𝜆∈Id(𝑀_𝑘(𝐹)). And since 𝐿_𝑘(𝑥;𝑦)∉Id(𝑀_𝑘(𝐹)), hence 𝑝_𝜇∉Id(𝑀_𝑘(𝐹)), where 𝜇 is of the 2×𝑘² rectangular shape 𝜇=(2^𝑘2). Thus, also 𝑝_𝜇 is a central polynomial for 𝑀_𝑘(𝐹)). The fact that 𝜇 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 𝑐_𝑛(𝑀_𝑘(𝐹)), 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 𝑐_𝑛(𝑀_𝑘(𝐹)). Also, the fact that 𝐿_𝑘(𝑥;𝑦) is a nonidentity of 𝑀_𝑘(𝐹) has applications in proving lower bounds for various other types of codimensions.

10. The Giambruno-Zaicev Theorem: exp(𝐴)∈ℤ

For most PI algebras 𝐴 it seems hopeless to find a precise, or even asymptotic, formula for the codimensions 𝑐_𝑛(𝐴). We therefore ask a much more restricted question.

Question 3. Given the associative PI algebra 𝐴, what can be said about the asymptotic behavior of the codimensions 𝑐_𝑛(𝐴)?

As a first step we have the remarkable integrality property given by Theorem 10.1 below. See also Theorem 11.1 and its relation to Theorem 10.1.

Theorem 10.1 (see [3]). Let 𝐴 be an associative PI algebra (with char(𝐹)=0), then the limit lim_𝑛→∞𝑐_𝑛(𝐴)^1/𝑛 (10.1) exists and is an integer.
We denote exp(𝐴)=lim_𝑛→∞(𝑐_𝑛(𝐴))^1/𝑛, so exp(𝐴)∈ℕ.

10.1. Review of the Proof When dim(𝐴)<∞

When 𝐴 is finite dimensional, the number exp(𝐴) can be calculated as follows. We may assume that 𝐹 is algebraically closed. First, by a classical theorem of Wedderburn and Malcev [45, Theorem 3.4.3], 𝐴=𝐵⊕𝐽, where 𝐽=𝐽(𝐴) is the Jacobson radical of 𝐴, and 𝐵 is semisimple. Thus 𝐵=𝐵₁⊕⋯⊕𝐵_𝑟 where 𝐵_𝑗 are simple, namely, 𝐵_𝑗≅𝑀_𝑘𝑗(𝐹). Consider now all possible nonzero products of the type 𝐵_𝑖1𝐽𝐵_𝑖2𝐽⋯𝐽𝐵_𝑖𝑞≠0𝑞≥1, (10.2) where the 𝐵_𝑖𝑗 are distinct, and for such nonzero products let ℎ∶=maxdim𝐵_𝑖1⊕⋯⊕𝐵_𝑖𝑞. (10.3) Then ℎ=𝑘²_𝑖1+⋯+𝑘²_𝑖𝑞, and ℎ is the limit in Theorem 10.1: lim_𝑛→∞𝑐_𝑛(𝐴)^1/𝑛=ℎ, (10.4) Namely, exp(𝐴)=𝑘²_𝑖1+⋯+𝑘²_𝑖𝑞.

For example, consider the algebra of upper block triangular matrices 𝐴=𝑈𝑇𝑘₁,𝑘₂,𝑘₃=𝐵₁𝐽₁𝐽₂0𝐵₂𝐽₃00𝐵₃, (10.5) where for 1≤𝑖≤3, 𝐵_𝑖≅𝑀_𝑘𝑖(𝐹) and 𝐽_𝑖 are rectangular matrices of the corresponding sizes. Then the Wedderburn-Malcev decomposition of 𝐴 is 𝐴=𝐵₁⊕𝐵₂⊕𝐵₃+𝐽 where 𝐵₁⊕𝐵₂⊕𝐵₃ is the semisimple part and 𝐽=𝐽₁+𝐽₂+𝐽₃ is the Jacobson radical. Notice that the matrix units 𝑒_𝑖,𝑗 in 𝐴 satisfy 0≠𝑒_1,1⋅𝑒_1,𝑘1+1⋅𝑒_{𝑘1+1,𝑘1+1}⋅𝑒_{𝑘1+1,𝑘1+𝑘2+1}⋅𝑒_{𝑘1+𝑘2+1,𝑘1+𝑘2+1}∈𝐵₁𝐽₁𝐵₂𝐽₃𝐵₃⊆𝐵₁𝐽𝐵₂𝐽𝐵₃, (10.6) so 𝐵₁𝐽𝐵₂𝐽𝐵₃≠0 and also, this is the maximal such nonzero product. It follows that here ℎ=ℎ(𝐴)=dim𝐵₁⊕𝐵₂⊕𝐵₃=𝑘²₁+𝑘²₂+𝑘²₃. (10.7)

10.1.1. The Upper Bound

In the general case the exponent ℎ=exp(𝐴)=lim_𝑛→∞(𝑐_𝑛(𝐴))^1/𝑛 is given by (10.3). To prove this, one first proves the following upper bound.

Lemma 10.2. There exist constants 𝑎₁, 𝑔₁ such that for all 𝑛, 𝑐_𝑛(𝐴)≤𝑎₁⋅𝑛^𝑔1⋅ℎ^𝑛.

Let Par(𝑛) denote the partitions of 𝑛. Given ℎ∈ℕ, define the subsets 𝑁𝑆_𝐾(ℎ,𝑛)⊆Par(𝑛) by 𝑁𝑆_𝐾(ℎ,𝑛)=𝜆=𝜆₁,𝜆₂,…⊢𝑛∣_𝑗>ℎ𝜆_𝑗≤𝐾, (10.8) so 𝑁𝑆_𝐾(ℎ,𝑛) is nearly a strip of height ℎ (with at most 𝐾 cells below the ℎ row). The proof of Lemma 10.2 follows by showing that the cocharacters 𝜒_𝑛(𝐴) are supported on such nearly a strip 𝑁𝑆_𝐾(ℎ,𝑛), and by the polynomial bound 𝑚_𝜆(𝐴)≤𝑎⋅𝑛^𝑏 on the multiplicities in the cocharacters, see Theorem 7.3. Thus 𝑐_𝑛(𝐴)≤𝑎𝑛^𝑏∑𝑓^𝜆, where the sum ∑𝑓^𝜆 is supported on such nearly a strip 𝑁𝑆_𝐾(ℎ,𝑛). Similar to the estimates in [16], such sum _{𝜆∈𝑁𝑆𝐾(ℎ,𝑛)}𝑓^𝜆 (10.9) is bounded by 𝑎⋅𝑛^𝑏⋅ℎ^𝑛, 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 𝑎₂>0 and 𝑔₂ such that for all 𝑛, 𝑎₂⋅𝑛^𝑔2⋅ℎ^𝑛≤𝑐_𝑛(𝐴).

A key ingredient in proving the lower bound is the polynomial 𝐿_𝑘(𝑥;𝑦) which is a nonidentity for 𝑀_𝑘(𝐹), see Section 9. We already noted that 𝐿_𝑘(𝑥;𝑦) has a component 𝑝_𝜇∈𝐼_𝜇 which is central (nonidentity) for 𝑀_𝑘(𝐹), where 𝜇 is the 2×𝑘² rectangle (2^𝑘2), see Theorem 9.1.

We start with a single matrix algebra 𝐵=𝑀_𝑘(𝐹), with the corresponding central polynomials 𝐿_𝑘(𝑥;𝑦) and 𝑝_𝜇(1). Rectangles of the same height (width) can be glued horizontally (vertically). Gluing horizontally (2^𝑘2) to itself 𝑤 times yields the 2𝑤×𝑘² rectangle 𝜇^(𝑤)=((2𝑤)^𝑘2), with a corresponding 𝑤-power (𝑝_𝜇(1))^𝑤=𝑝_𝜇(𝑤), which is central and nonidentity for 𝑀_𝑘(𝐹), and 𝑝_𝜇(𝑤)∈𝐼_𝜇(𝑤). Thus 𝑓^𝜇(𝑤)≤𝑐_𝑛(𝑀_𝑘(𝐹)). For that 𝜇^(𝑤)=((2𝑤)^𝑘2)⊢𝑛, 𝑛=2𝑤𝑘², the asymptotic of 𝑓^𝜇(𝑤) then yields the lower bound 𝑎₂⋅𝑛^𝑔2⋅𝑘^2𝑛≤𝑐_𝑛𝑀_𝑘(𝐹) (10.10) for some constants 𝑎₂>0 and 𝑔₂.

In the general case we are given 𝐵_𝑖1𝐽𝐵_𝑖2𝐽⋯𝐽𝐵_𝑖𝑞≠0 as in (10.2). To each 𝐵_𝑖𝑗 corresponds the nonidentity polynomial 𝐿_𝑘𝑖𝑗(𝑥;𝑦), with the corresponding rectangular tableaux (2^{𝑘𝑖𝑗2}), all with the same width =2. These tableaux can be glued vertically, thus yielding the rectangular tableau 𝜌=(2^ℎ), where ℎ=∑_𝑗(𝑘_𝑖𝑗)². To that tableau there corresponds a polynomial which is essentially the product of the polynomials 𝐿_𝑘𝑖𝑗(𝑥;𝑦), with the corresponding component which is the product of the corresponding polynomials 𝑝_𝜇, hence that polynomial is central nonidentity of 𝐴, and similarly for powers of these polynomials. Similar to the above case of a single matrix algebra 𝐵=𝑀_𝑘(𝐹), the asymptotic of 𝑓^𝜌(𝑤)=𝑓^((2𝑤)ℎ) then yields the lower bound 𝑎₂⋅𝑛^𝑔2⋅ℎ^𝑛≤𝑐_𝑛(𝐴) (10.11) for some constants 0<𝑎₂ and 𝑔₂. This proves the lower bound.

Corollary 10.4. Putting together the lower and the upper bounds, Theorem 10.1 then follows.

This completes our review of the proof of Theorem 10.1. For extensions of this theorem, see Remark 11.2.

11. Berele's “1/2” Theorem (See [4])

Applying Theorem 10.1, Berele proved the following remarkable theorem.

Theorem 11.1 (see [4], see also [5]). Let 𝐴 be a PI algebra with 1∈𝐴. Then as 𝑛 goes to infinity, 𝑐_𝑛(𝐴)∼𝑎⋅𝑛^𝑡⋅ℎ^𝑛, where, ℎ∈ℕ (given by Theorem 10.1) and 𝑡∈(1/2)ℤ, namely, 𝑡 is an integer or a half integer.

Based on various examples, that theorem was conjectured for some time. It was first proved in [5] under the hypothesis that 𝐴 satisfies a Capelli identity, and later in general in [4]. Here is an outline of the proof in the Capelli case.

Proof . Recall that 𝑐_𝑛(𝐴)=∑𝑚_𝜆(𝐴)𝑓^𝜆, where 𝑚_𝜆(𝐴) is the multiplicity of 𝜒^𝜆. If 𝐴 satisfies a Capelli identity then there exists an integer 𝑘 such that 𝑚_𝜆(𝐴)≠0 only for partitions 𝜆 with at most 𝑘 parts (see Remark 5.2). The proof is now based on investigating the asymptotics of the 𝑚_𝜆(𝐴) and 𝑓^𝜆, and we proceed to do so.
Let 𝑈_𝑘(𝐴)=𝐹𝑥₁,…,𝑥_𝑘Id(𝐴)∩𝐹𝑥₁,…,𝑥_𝑘 (11.1) be the universal PI algebra for 𝐴 in 𝑘 generators. This algebra has an ℕ grading by degree and a corresponding Poincaré series 𝑃_𝑘(𝑡). It also has a finer ℕ^𝑘 grading by multidegree with corresponding Poincaré series 𝑃(𝑡₁,…,𝑡_𝑘). Belov [46] proved that 𝑃_𝑘(𝑡) is (the Taylor series of) a rational function with coefficients in ℤ (see also [47], Theorem 9.44). It is not difficult to adapt that proof to show that 𝑃(𝑡₁,…,𝑡_𝑘) 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 (1−𝑡^𝑎1₁⋯𝑡^𝑎𝑘_𝑘). We call such a rational function “nice.”
The Poincaré series 𝑃(𝑡₁,…,𝑡_𝑘) is related to the cocharacters 𝜒_𝑛(𝐴)=∑𝑚_𝜆(𝐴)𝜒^𝜆 via 𝑃𝑡₁,…,𝑡_𝑘=^∞_{𝑛=0𝜆⊢𝑛}𝑚_𝜆(𝐴)𝑆_𝜆𝑡₁,…,𝑡_𝑘, (11.2) where S_𝜆 is the Schur function of 𝜆.
Quite a lot is known about Taylor series of nice rational functions, see for example [48]. If 𝐹(𝑡₁,…,𝑡_𝑘) is any nice rational function with Taylor series ∑𝑏(𝑛₁,…,𝑛_𝑘)𝑡^𝑛1₁⋯𝑡^𝑛𝑘_𝑘, then the coefficients 𝑏(𝑛₁,…,𝑛_𝑘) can be described using a finite set of polynomials. Namely, ℕ^𝑘 can be partitioned into regions 𝑅₁,…,𝑅_𝑚 with corresponding polynomials 𝑞₁,…,𝑞_𝑚 such that for each 𝑖, 𝑐𝑛₁,…,𝑛_𝑘=𝑞_𝑖𝑛₁,…,𝑛_𝑘for𝑛₁,…,𝑛_𝑘∈𝑅_𝑖. (11.3) Using properties of Schur functions it can be shown that an analogue formula holds for the 𝑚_𝜆(𝐴). Hence we may write 𝑐_𝑛(𝐴)=_𝑖𝑞_𝑖(𝜆)𝑓^𝜆∣𝜆∈𝑅_𝑖. (11.4) The regions 𝑅_𝑖, in general, are defined by linear inequalities and modular linear equations. Here is a simple example to make this more clear. Let 𝐹(𝑡₁,𝑡₂)=(1−𝑡²₁)⁻¹(1−𝑡₂)⁻¹(1−𝑡₁𝑡₂)⁻¹. Then 𝑏(𝑛₁,𝑛₂) is given by 𝑏𝑛₁,𝑛₂=𝑛₂2+1if𝑛₁≥𝑛₂,𝑛₂𝑛iseven,₂+12if𝑛₁≥𝑛₂,𝑛₂𝑛isodd,₁2+1if𝑛₁<𝑛₂,𝑛₁𝑛iseven,₁+12if𝑛₁<𝑛₂,𝑛₁isodd. (11.5) Now back to the general case. There exists a number ℎ≤𝑘 called the essential height of the cocharacter defined to be the largest number such that there exist nonzero multiplicities 𝑚_𝜆(𝐴) in which 𝜆=(𝜆₁,…,𝜆_𝑘) and 𝜆_ℎ can be taken arbitrarily large. Giambruno and Zaicev proved that ℎ=lim_𝑛→∞(𝑐_𝑛(𝐴))^1/𝑛. This number, denoted exp(𝐴), is also important for the proof of Theorem 11.1. Let 𝑣₁ be the vector in ℕ^𝑘 whose coordinates are ℎ ones followed by 𝑘−ℎ zeros. Then in (11.4) certain summands will be on regions of the form 𝑣₀+𝛼₁𝑣₁+⋯+𝛼_𝑑𝑣_𝑑∣𝛼_𝑖,…,𝛼_𝑑∈ℕ∩𝐿, (11.6) where 𝐿 is an integer lattice. Let 𝑅^′₁,…,𝑅^′_𝑝 be the regions of this form. Then in the computation of 𝑐_𝑛(𝐴) these terms dominate and we can refine (11.4) to 𝑐_𝑛(𝐴)=_𝑖𝑞_𝑖(𝜆)𝑓^𝜆∣𝜆∈𝑅^′_𝑖. (11.7) The rest of the proof closely immitates the computation of [16]. The main theorem is that if 𝑅^′ is as above and 𝑞(𝑥₁,…,𝑥_𝑘) is a polynomial then ∑𝑞(𝜆)𝑓^𝜆 summed over partitions of 𝑛 in 𝑅^′ will be asymptotic to a constant—times √𝑛 to an integer power—times ℎ^𝑛. Hence, the cocharacter 𝑐_𝑛(𝐴) will be a sum of such terms.
There remains the problem that the powers of 𝑛 might not be equal, and the way around this difficulty is to use the fact that if 1∈𝐴 then the cocharacter sequence is Young derived, see [49]. Namely, the Poincaré series 𝑃(𝑡₁,…,𝑡_𝑘) can be written as (1−𝑡₁)⁻¹⋯(1−𝑡_𝑘)⁻¹𝑔(𝑡₁,…,𝑡_𝑘), where 𝑔 has all the nice properties of 𝑃.

We conclude this section with the following general remark.

Remark 11.2. Recent works extended the above theorems of Sections 10 and 11 to graded polynomial identities and to PI algebras with the action of Hopf algebra, see, for example, [50–59].

12. Algebraicity of Some Generating Functions

As we show below, Theorems 10.1 and 11.1 are related to the question of whether or not the generating function of the codimensions is algebraic. We begin with the following definition.

Definition 12.1 (see [60, 61]). (1) Given the sequence 𝑎_𝑛, then 𝐹(𝑥)=∑_𝑛≥0𝑎_𝑛𝑥^𝑛 is its corresponding ordinary generating function. In particular 𝐶_𝐴(𝑥)=∑_𝑛≥0𝑐_𝑛(𝐴)𝑥^𝑛 is the generating function of the codimensions of the algebra 𝐴.
(2) The function 𝐹(𝑥) is algebraic if there exist polynomials 𝑃₀(𝑥),…,𝑃_𝑟(𝑥) such that 𝑃_𝑟(𝑥)𝐹^𝑟(𝑥)+⋯+𝑃₁(𝑥)𝐹(𝑥)+𝑃₀(𝑥)=0. (12.1) Algebraicity or nonalgebraicity of the generating function 𝐹(𝑥)=∑_𝑛≥0𝑎_𝑛𝑥^𝑛 is an indication of the complexity of the sequence 𝑎_𝑛.

Example 12.2 (see [34], [25, Theorem 12.6.8]). For the 2×2 matrices 𝑀₂(𝐹) we have 𝑐_𝑛𝑀₂(𝐹)=1−𝑛𝑛+232𝑛+2𝑛+1+1−2^𝑛. (12.2) This implies that 𝐶_𝑀2(𝐹)(𝑥)=1𝑥²1−2𝑥−√1−4𝑥−𝑥³(1−𝑥)⁴+1−11−𝑥,1−2𝑥 (12.3) which is clearly algebraic.

Note that when 𝑘=1, 𝑀₁(𝐹)=𝐹 and 𝑐_𝑛(𝐹)=1, hence 𝐶_𝐹(𝑥)=11−𝑥 (12.4) which is algebraic.

12.1. Nonalgebraicity of Some Generating Functions

We quote here a classical theorem of Jungen [62], see also [63].

Theorem 12.3 (see [62]). Let 𝑓∶ℕ→ℂ,  𝐹(𝑥)=∑_𝑛≥0𝑓(𝑛)𝑥^𝑛, and assume that as 𝑛 goes to infinity, 𝑓(𝑛)∼𝑏⋅𝑛^−𝑔⋅𝑎^𝑛, (12.5) where 𝑏 and 𝑎 are complex constants and 𝑔 is a real number. For 𝐹(𝑥) to be algebraic it is necessary that 𝑔 be rational; and if 𝑔>0 then 𝑔 must also be non-integral.

Applying this theorem to the codimensions of matrices we deduce the following theorem.

Theorem 12.4. Let 𝐶_{𝑀𝑘(𝐹)}(𝑥)=_𝑛≥0𝑐_𝑛𝑀_𝑘(𝐹)⋅𝑥^𝑛 (12.6) be the generating functions of the (ordinary) codimensions of 𝑀_𝑘(𝐹). If 𝑘≥3 and 𝑘 is odd then 𝐶_{𝑀𝑘(𝐹)}(𝑥) is not algebraic.

Proof. By Theorem 6.2c_𝑛𝑀_𝑘(𝐹)∼𝑏⋅𝑛^−𝑔⋅𝑘^2𝑛, (12.7) where 𝑔=(𝑘²−1)/2. The proof now follows by Theorem 12.3, since 𝑔=(𝑘²−1)/2 is an integer when 𝑘 is odd.

Obviously, Example 12.2 and Theorem 12.4 motivate the following conjecture.

Conjecture 12.5. If 𝑘≥3 then the generating function 𝐶_{𝑀𝑘(𝐹)}(𝑥)=_𝑛≥0𝑐_𝑛𝑀_𝑘(𝐹)⋅𝑥^𝑛 (12.8) is not algebraic.

For the other verbally prime algebras (see Section 2.2.1), recall from Theorem 6.4 the following partial asymptotic results: 𝑐_𝑛𝑀_𝑘,ℓ1∼𝑎⋅𝑛^{(𝑘2+ℓ2−1)/2}⋅(𝑘+ℓ)^2𝑛, (12.9) where the constant 𝑎 is yet unknown. Also, 𝑐_𝑛𝑀_𝑘(𝐺)1∼𝑏⋅𝑛^𝑔⋅2𝑘^2𝑛, (12.10) where the constants 𝑏 and 𝑔 are yet unknown, and (𝑘²−1)/2≤𝑔≤(2𝑘²−1)/2.

Corollary 12.6. If 𝑘≢ℓ(mod2) then the generating function 𝐶_𝑀𝑘,ℓ(𝑥) of the codimensions 𝑐_𝑛(𝑀_𝑘,ℓ) in not algebraic.

Proof . Indeed, 𝑘≢ℓ(mod2) implies that 𝑘²+ℓ²−1 is even, and the proof follows from Theorems 12.3 and 6.4.

Example 12.7 (see [64]). We apply here a theorem due to Kemer [6] which says that the algebras 𝐺⊗𝐺 and 𝑀_1,1 have the same identities, hence the same codimensions. Now 𝑐₀𝑀_1,1=1,𝑐_𝑛𝑀_1,1=12𝑛2𝑛+𝑛+1−2^𝑛,𝑛=1,2,… (12.11) see [65], and hence 𝐶_𝑀1,1(𝑥)=12+12√+𝑥1−4𝑥(1−𝑥)²+1+11−𝑥,1−2𝑥 (12.12) which is clearly algebraic. Note that 𝑐_𝑛𝑀_1,1∼12√𝜋⋅1√𝑛⋅4^𝑛, (12.13) compare with (6.3) of Section 6. See also [64].

Conjecture 12.8. If 𝑘,ℓ≥1 and (𝑘,ℓ)≠(1,1) then 𝐶_𝑀𝑘,ℓ(𝑥) is not algebraic.

13. Nonassociative 𝐴 with exp(𝐴) 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 𝐹{𝑥}=𝐹{𝑥₁,𝑥₂,…} is replaced by the free algebra 𝐹⟨𝑥₁,𝑥₂,…⟩=𝐹⟨𝑥⟩. Now 𝐴 is a nonassociative PI algebra and again, Id(𝐴)⊆𝐹⟨𝑥⟩ are the identities of 𝐴. 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 𝑉_𝑛 of the multilinear polynomials of degree 𝑛 is now of dimension  dim𝑉_𝑛=𝐶_𝑛⋅𝑛!, where 𝐶_𝑛 is the 𝑛th Catalan number (which counts the number of different parenthesizes of a monomial of degree 𝑛). The definition of the codimensions 𝑐_𝑛(𝐴) is, formally, the same as that in the associative case.

Definition 13.1. 𝑐_𝑛(𝐴)𝑉=dim^𝑛Id(𝐴)∩𝑉_𝑛, (13.1)
compare with Definition 4.1.

Similarly for the cocharacters, the action of 𝑆_𝑛 on 𝑉_𝑛 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 [66].

A counter-example to Theorem 10.1 in the nonassociative case was first constructed in [67]. Recently, Giambruno et al. [68] constructed a family of nonassociative PI algebras 𝐴 such that for every 1<𝛼<2 there is such an algebra 𝐴 for which the limit lim_𝑛→∞(𝑐_𝑛(𝐴))^1/𝑛 exists and is equal to 𝛼. We briefly describe that construction.

13.1. The Algebra 𝐴(𝑚,𝑤)

Let 𝑤=𝑤₁𝑤₂⋯ be an infinite word on the alphabet {0,1}. Given an integer 𝑚≥2, define the sequence {𝑘_𝑖}_𝑖≥1=𝐾_𝑚,𝑤 by 𝑘_𝑖=𝑤_𝑖+𝑚. (13.2) Construct the algebra 𝐴=𝐴(𝑚,𝑤)=𝐴(𝐾_𝑚,𝑤) as follows. Its basis is {𝑎,𝑏}∪𝑍₁∪𝑍₂∪⋯, (13.3) where 𝑍_𝑖=𝑧_𝑗^(𝑖)∣1≤𝑗≤𝑘_𝑖,𝑖=1,2,…, (13.4) and multiplication is given as follows: 𝑧₁^(𝑖)𝑎=𝑧₂^(𝑖),𝑧₂^(𝑖)𝑎=𝑧₃^(𝑖),…,𝑧_𝑘^(𝑖)_𝑖−1𝑎=𝑧_𝑘^(𝑖)_𝑖𝑧,𝑖=1,2,…,_{𝑘(𝑖)𝑖}𝑏=𝑧₁^(𝑖+1),𝑖=1,2,… (13.5) and all other products are zero.

Let 𝑤=𝑤₁𝑤₂⋯ be an infinite word. The notion of the complexity of 𝑤 is classical. For each 𝑛, complexity_𝑤(𝑛) is the number of distinct subwords of 𝑤 of length 𝑛. The algebra 𝐴=𝐴(𝑚,𝑤) depends on the integer 𝑚 and on the complexity of the word 𝑤, and the following theorem is proved.

Theorem 13.2 (see [68]). Given 1≤𝛼≤2, we can choose 𝑚≥2 and a word 𝑤 such that lim_𝑛→∞𝑐_𝑛(𝐴(𝑚,𝑤))^1/𝑛existsandisequalto𝛼. (13.6)