# Growth for Algebras Satisfying Polynomial Identities

Authors: Amitai Regev
Journal: ISRN Algebra
 (2012)

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## Abstract

Thenth codimension cn(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 cn(A) as n goes to infinity. Since in most cases there is no hope in finding nice closed formula for cn(A), we study its asymptotics. We review here such results about cn(A), when A is an associative PI algebra. We start with the exponential bound on cn(A) then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of cn(A). The representation theory of the symmetric group Sn is an important tool in this theory.


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## Body

## 1. Introduction

We study algebrasA 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 byId(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

Givenn, we let Vn denote the multilinear polynomials of degree n in x1,…,xn, so in the associative case dimVn=n!. Identities can always be multilinearized, hence the subset Id(A)∩Vn 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)∩Vn}n≥1, but we make no use of that remark in the sequel.To studydim(Id(A)∩Vn), we introduce the quotient space Vn/(Id(A)∩Vn) and its dimension
(1.1)cn(A)=dim(VnId(A)∩Vn).
The integer cn(A) is the nth codimension of A. Clearly cn(A) determines dim(Id(A)∩Vn) since dimVn is known.The study of growth for PI algebraA is mostly the study of the rate of growth of the sequence cn(A) of its codimensions, as n goes to infinity. We have the following basic property.Theorem 1.1 (see [1]).
In the associative case,cn(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 forcn(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,an~bn if limn→∞an/bn=1.(1) The asymptotics for thek×k matrices Mk(F), see Section 6.Theorem 1.2 (see [2]).
Whenn goes to infinity,
(1.2)cn(Mk(F))~[(12π)k-1(12)(k2-1)/2·1!2!⋯(k-1)!·k(k2/2)]·(1n)(k2-1)/2·k2(n+1).(2) The integrality theorem of Giambruno-Zaicev, see Section10.Theorem 1.3 (see [3]).
LetA be an associative PI F-algebra with char(F)=0, then the limit
(1.3)limn→∞(cn(A))1/n
exists and is an integer. We denote exp(A)=limn→∞(cn(A))1/n, so exp(A)∈ℕ.(3) The “1/2” theorem of Berele, see Section11.Theorem 1.4 (see [4, 5]).
LetA be a PI algebra with 1∈A. Then as n goes to infinity, cn(A)~a·nt·hn, 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 primeT-ideals. After introducing the codimensions, two proofs of their exponential bound are reviewed. The Sn character of that space, χSn(Vn/(Id(A)∩Vn)) is denoted as follows:
(1.4)χn(A)=χSn(VnId(A)∩Vn)
and is called the nth cocharacter of A. Since cn(A)=degχn(A), cocharacters are refinement of codimensions, and are important tool in their study. By a theorem of Amitsur-Regev and of Kemer, χn(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∑n‍cn(A)·xn is examined in Section 12.In the last section, we review a construction of nonassociative algebras where the integrality property of the exponent fails.


## 1.1. Growth for PI Algebras

Givenn, we let Vn denote the multilinear polynomials of degree n in x1,…,xn, so in the associative case dimVn=n!. Identities can always be multilinearized, hence the subset Id(A)∩Vn 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)∩Vn}n≥1, but we make no use of that remark in the sequel.To studydim(Id(A)∩Vn), we introduce the quotient space Vn/(Id(A)∩Vn) and its dimension
(1.1)cn(A)=dim(VnId(A)∩Vn).
The integer cn(A) is the nth codimension of A. Clearly cn(A) determines dim(Id(A)∩Vn) since dimVn is known.The study of growth for PI algebraA is mostly the study of the rate of growth of the sequence cn(A) of its codimensions, as n goes to infinity. We have the following basic property.Theorem 1.1 (see [1]).
In the associative case,cn(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 forcn(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,an~bn if limn→∞an/bn=1.(1) The asymptotics for thek×k matrices Mk(F), see Section 6.Theorem 1.2 (see [2]).
Whenn goes to infinity,
(1.2)cn(Mk(F))~[(12π)k-1(12)(k2-1)/2·1!2!⋯(k-1)!·k(k2/2)]·(1n)(k2-1)/2·k2(n+1).(2) The integrality theorem of Giambruno-Zaicev, see Section10.Theorem 1.3 (see [3]).
LetA be an associative PI F-algebra with char(F)=0, then the limit
(1.3)limn→∞(cn(A))1/n
exists and is an integer. We denote exp(A)=limn→∞(cn(A))1/n, so exp(A)∈ℕ.(3) The “1/2” theorem of Berele, see Section11.Theorem 1.4 (see [4, 5]).
LetA be a PI algebra with 1∈A. Then as n goes to infinity, cn(A)~a·nt·hn, 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 primeT-ideals. After introducing the codimensions, two proofs of their exponential bound are reviewed. The Sn character of that space, χSn(Vn/(Id(A)∩Vn)) is denoted as follows:
(1.4)χn(A)=χSn(VnId(A)∩Vn)
and is called the nth cocharacter of A. Since cn(A)=degχn(A), cocharacters are refinement of codimensions, and are important tool in their study. By a theorem of Amitsur-Regev and of Kemer, χn(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∑n‍cn(A)·xn 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 andT Ideals

### 2.1.T-ideals

LetF 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,…,xn)∈F{x} is an identity of the F-algebra A if f(a1,…,an)=0 for every a1,…,an∈A. The algebra A satisfies polynomial identities, or in short is PI, if there exist a nonzero polynomial 0≠f(x)=f(x1,…,xn)∈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 Mk(F) of the k×k matrices over F plays a basic role in PI theory.Definition 2.1.
(1) LetId(A)⊆F{x} be the subset of the identities of the algebra A.
(2.1)Id(A)={f∈F{x}∣f(x)=0isanidentityofA}.
(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 forT-Ideals (See [6])

The Specht Problem
One of the main problems in PI theory is the Specht problem: AreT 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(Mk(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(Mk(F)).Kemer introduced the notion ofverbally prime ideals as follows.Definition 2.2.
TheT ideal I is verbally prime if it satisfies the following condition: Let f(x1,…,xm) and g(xm+1,…,xm+n) be polynomials in disjoint sets of variables. If
(2.2)f(x1,…,xm)·g(xm+1,…,xm+n)∈Ithenf∈Iorg∈I.
Kemer then classified the verbally prime ideals in F{x}, see Theorem 2.3 below.


#### 2.2.1. The AlgebrasMk(F), Mk(G), and Mk,ℓ

LetU=spanF{u1,u2,…} be an infinite dimensional vector space, and let G=G(U) be the corresponding infinite dimensional Grassmann (Exterior) algebra. Then
(2.3)G=span{ui1⋯uir∣r=1,2,…and1≤i1<⋯<ir},
and G=G0⊕G1, where
(2.4)G0=span{ui1⋯uir∣reven},G1=span{ui1⋯uir∣rodd}.
Now Mk(F) are the k×k matrices over F, while Mk(G) are the k×k matrices over G.The AlgebraMk,ℓ
We haveMk,ℓ⊆Mk+ℓ(G). The elements of Mk,ℓ are block matrices
(2.5)(ABCD),
where A∈Mk(G0), D∈Mℓ(G0), B is k×ℓ and C is ℓ×k, both with entries from G1. For example:
(2.6)M1,1=(G0G1G1G0).Theorem 2.3 (see [6]).
The following are the three families of the verbally primeT ideals: Id(Mk(F)), Id(Mk(G)), and Id(Mk,ℓ).The importance of the verbally prime ideals is demonstrated in the following theorem.Theorem 2.4 (see [6]).
LetI⊂F{x} be a T ideal. Then there exist verbally prime T ideals J1,…,Jr such that
(2.7)J1⋯Jr⊆I⊆J1∩⋯∩Jr.


## 2.1.T-ideals

LetF 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,…,xn)∈F{x} is an identity of the F-algebra A if f(a1,…,an)=0 for every a1,…,an∈A. The algebra A satisfies polynomial identities, or in short is PI, if there exist a nonzero polynomial 0≠f(x)=f(x1,…,xn)∈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 Mk(F) of the k×k matrices over F plays a basic role in PI theory.Definition 2.1.
(1) LetId(A)⊆F{x} be the subset of the identities of the algebra A.
(2.1)Id(A)={f∈F{x}∣f(x)=0isanidentityofA}.
(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 forT-Ideals (See [6])

The Specht Problem
One of the main problems in PI theory is the Specht problem: AreT 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(Mk(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(Mk(F)).Kemer introduced the notion ofverbally prime ideals as follows.Definition 2.2.
TheT ideal I is verbally prime if it satisfies the following condition: Let f(x1,…,xm) and g(xm+1,…,xm+n) be polynomials in disjoint sets of variables. If
(2.2)f(x1,…,xm)·g(xm+1,…,xm+n)∈Ithenf∈Iorg∈I.
Kemer then classified the verbally prime ideals in F{x}, see Theorem 2.3 below.


### 2.2.1. The AlgebrasMk(F), Mk(G), and Mk,ℓ

LetU=spanF{u1,u2,…} be an infinite dimensional vector space, and let G=G(U) be the corresponding infinite dimensional Grassmann (Exterior) algebra. Then
(2.3)G=span{ui1⋯uir∣r=1,2,…and1≤i1<⋯<ir},
and G=G0⊕G1, where
(2.4)G0=span{ui1⋯uir∣reven},G1=span{ui1⋯uir∣rodd}.
Now Mk(F) are the k×k matrices over F, while Mk(G) are the k×k matrices over G.The AlgebraMk,ℓ
We haveMk,ℓ⊆Mk+ℓ(G). The elements of Mk,ℓ are block matrices
(2.5)(ABCD),
where A∈Mk(G0), D∈Mℓ(G0), B is k×ℓ and C is ℓ×k, both with entries from G1. For example:
(2.6)M1,1=(G0G1G1G0).Theorem 2.3 (see [6]).
The following are the three families of the verbally primeT ideals: Id(Mk(F)), Id(Mk(G)), and Id(Mk,ℓ).The importance of the verbally prime ideals is demonstrated in the following theorem.Theorem 2.4 (see [6]).
LetI⊂F{x} be a T ideal. Then there exist verbally prime T ideals J1,…,Jr such that
(2.7)J1⋯Jr⊆I⊆J1∩⋯∩Jr.


## 2.2.1. The AlgebrasMk(F), Mk(G), and Mk,ℓ

LetU=spanF{u1,u2,…} be an infinite dimensional vector space, and let G=G(U) be the corresponding infinite dimensional Grassmann (Exterior) algebra. Then
(2.3)G=span{ui1⋯uir∣r=1,2,…and1≤i1<⋯<ir},
and G=G0⊕G1, where
(2.4)G0=span{ui1⋯uir∣reven},G1=span{ui1⋯uir∣rodd}.
Now Mk(F) are the k×k matrices over F, while Mk(G) are the k×k matrices over G.The AlgebraMk,ℓ
We haveMk,ℓ⊆Mk+ℓ(G). The elements of Mk,ℓ are block matrices
(2.5)(ABCD),
where A∈Mk(G0), D∈Mℓ(G0), B is k×ℓ and C is ℓ×k, both with entries from G1. For example:
(2.6)M1,1=(G0G1G1G0).Theorem 2.3 (see [6]).
The following are the three families of the verbally primeT ideals: Id(Mk(F)), Id(Mk(G)), and Id(Mk,ℓ).The importance of the verbally prime ideals is demonstrated in the following theorem.Theorem 2.4 (see [6]).
LetI⊂F{x} be a T ideal. Then there exist verbally prime T ideals J1,…,Jr such that
(2.7)J1⋯Jr⊆I⊆J1∩⋯∩Jr.


## 3. Multilinear Polynomials

As usual,Sn denotes the nth symmetric group.Definition 3.1.
The polynomialf(x1,…,xn)∈F{x} is multilinear (in x1,…,xn) if
(3.1)f(x1,…,xn)=∑σ∈Sn‍ασxσ(1)⋯xσ(n)
for some coefficients ασ∈F. Let Vn=Vn(x1,…,xn) denote the vector space of multilinear polynomials in x1,…,xn, so dimVn=n!. Extending the map σ→xσ(1)⋯xσ(n) by linearity yields the vector space isomorphism Vn≅FSn, where FSn is the group algebra of Sn. We will identify
(3.2)FSn≡Vn.
By the process of multilinearization [8, 9] one proves the following theorem.Theorem 3.2.
Let the PI algebraA 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 ifA satisfies an identity of degree d then A satisfies an identity of the form
(3.3)y1⋯yd-∑1≠π∈Sd‍απyπ(1)⋯yπ(d),απ∈F.
This fact is applied in Section 4.1 in proving the exponential bound for the codimensions.


### 3.1. Example: Standard and Capelli Identities

The polynomialf(x1,…,xn) is alternating in x1,…,xn if for every permutation π∈Sn, f(xπ(1),…,xπ(n))=sgn(π)f(x1,…,xn). For example, the polynomial
(3.4)Stn[x]=Stn[x1,…,xn]=∑σ∈Sn‍sgn(σ)·xσ(1)·xσ(2)⋯xσ(n)
is alternating. It is called the nth Standard polynomial.Similarly the polynomial(3.5)Capn[x;y]=∑σ∈Sn‍sgn(σ)·xσ(1)·y1·xσ(2)·y2⋯yn-1·xσ(n),
which is called the (nth) Capelli polynomial, is multilinear of degree 2n-1 and is alternating in x1,…,xn.It is rather easy to show that ifdimA=d<∞ then A satisfies both Capd+1[x] and Std+1[x], and hence every finite dimensional algebra is PI. And the same argument applies in the nonassociative case. In particular the algebra Mk(F) of the k×k matrices satisfies Stk2+1[x1,x2,…,xk2+1]. The celebrated Amitsur-Levitzki Theorem [10–12] states that Mk(F) satisfies the standard identity St2k[x1,…,x2k]=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 algebraF{x} itself is not PI.


## 3.1. Example: Standard and Capelli Identities

The polynomialf(x1,…,xn) is alternating in x1,…,xn if for every permutation π∈Sn, f(xπ(1),…,xπ(n))=sgn(π)f(x1,…,xn). For example, the polynomial
(3.4)Stn[x]=Stn[x1,…,xn]=∑σ∈Sn‍sgn(σ)·xσ(1)·xσ(2)⋯xσ(n)
is alternating. It is called the nth Standard polynomial.Similarly the polynomial(3.5)Capn[x;y]=∑σ∈Sn‍sgn(σ)·xσ(1)·y1·xσ(2)·y2⋯yn-1·xσ(n),
which is called the (nth) Capelli polynomial, is multilinear of degree 2n-1 and is alternating in x1,…,xn.It is rather easy to show that ifdimA=d<∞ then A satisfies both Capd+1[x] and Std+1[x], and hence every finite dimensional algebra is PI. And the same argument applies in the nonassociative case. In particular the algebra Mk(F) of the k×k matrices satisfies Stk2+1[x1,x2,…,xk2+1]. The celebrated Amitsur-Levitzki Theorem [10–12] states that Mk(F) satisfies the standard identity St2k[x1,…,x2k]=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 algebraF{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 theT-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
(4.1)Id(A)∩Vn(x1,…,xn)=Id(A)∩Vn,Vn as in (3.2). Note that if y1,…,yn are any n variables then Id(A)∩Vn(x)≅Id(A)∩Vn(y). A first step is the study of the dimensions dim(Id(A)∩Vn). Now (Id(A)∩Vn)⊆Vn, and since dimVn=n!, clearly
(4.2)dim(Id(A)∩Vn)=n!-dim(VnId(A)∩Vn).
Thus, knowing dim(Id(A)∩Vn) is equivalent to knowing dim(Vn/(Id(A)∩Vn)). This leads us to introduce the codimensions cn(A) as follows.Definition 4.1.
LetA be an F-algebra, then
(4.3)cn(A)=dim(VnId(A)∩Vn)
is called the nth codimension of A, and {cn(A)}n=1∞ is the sequence of codimensions of A.For example,A satisfies An=0 (i.e., is nilpotent) if and only if cn(A)=0. And A is commutative if and only if cn(A)≤1 for all n.Remark 4.2.
Note that ifA is not PI then Id(A)=0, hence cn(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 cn(A)<n!. This follows directly from the definition.


### 4.1. Exponential Bound for the Codimensions

A basic property of the codimensions sequencecn(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) algebraA satisfies an identity of degree d, then
(4.4)cn(A)≤((d-1)2)n
for all n.Proof.
We sketch two proofs (both different from the original proof). Both proofs apply the notion of ad-good permutation, which we now review.
Callσ∈Snd-bad if there are indices 1≤i1<⋯<id≤n with σ(i1)>⋯>σ(id). Otherwise σ is d-good, and we denote
(4.5)gd(n)=|{σ∈Sn∣σisd-good}|.
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 cn(A)≤gd(n).
By an argument based on Dilworth Theorem in Combinatorics [15], Latyshev then showed that gd(n)≤((d-1)2)n, thus completing the (first) proof.
A second proof of the boundgd(n)≤((d-1)2)n goes as follows [16]. First, by the RSK correspondence [17],
(4.6)gd(n)=∑λ⊢n,ℓ(λ)≤d-1‍(fλ)2,
where fλ 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,
(4.7)∑λ⊢n,ℓ(λ)≤d-1‍(fλ)2=dim(B(d-1,n)),
where dimU=d-1 and B(d-1,n)⊆EndF(U⊗n). The (second) proof now follows since dim(EndF(U⊗n))=(d-1)2n.


#### 4.1.1. Application: TheA⊗B Theorem

Codimensions where introduced, in [1], in order to prove the following theorem.Theorem 4.4 (see [1]).
IfA and B are (associative) PI algebras then A⊗B is PI.Proof.
It is not too difficult to show thatcn(A⊗B)≤cn(A)·cn(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
(4.8)cn(A⊗B)≤(d-1)2n(h-1)2n,
hence for a large enough n, cn(A⊗B)<n!. Finally by Remark 4.2,  R is PI if and only if there exist n such that cn(R)<n!, and the proof follows.Remark 4.5.
For explicit identities forA⊗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 Theorems4.3 and 4.4 fail in some non-associative cases.The above results motivate the study of the following problem.Question 1.
Given a PI algebraA, find a formula for the sequence cn(A). In most cases this seems to be too difficult, and one tries to, at least, get a “nice” asymptotic approximation of cn(A).In Section6 we determine such asymptotics for the algebras Mk(F). We also give such partial results for the other verbally prime algebras Mk,ℓ and Mk(G).


## 4.1. Exponential Bound for the Codimensions

A basic property of the codimensions sequencecn(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) algebraA satisfies an identity of degree d, then
(4.4)cn(A)≤((d-1)2)n
for all n.Proof.
We sketch two proofs (both different from the original proof). Both proofs apply the notion of ad-good permutation, which we now review.
Callσ∈Snd-bad if there are indices 1≤i1<⋯<id≤n with σ(i1)>⋯>σ(id). Otherwise σ is d-good, and we denote
(4.5)gd(n)=|{σ∈Sn∣σisd-good}|.
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 cn(A)≤gd(n).
By an argument based on Dilworth Theorem in Combinatorics [15], Latyshev then showed that gd(n)≤((d-1)2)n, thus completing the (first) proof.
A second proof of the boundgd(n)≤((d-1)2)n goes as follows [16]. First, by the RSK correspondence [17],
(4.6)gd(n)=∑λ⊢n,ℓ(λ)≤d-1‍(fλ)2,
where fλ 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,
(4.7)∑λ⊢n,ℓ(λ)≤d-1‍(fλ)2=dim(B(d-1,n)),
where dimU=d-1 and B(d-1,n)⊆EndF(U⊗n). The (second) proof now follows since dim(EndF(U⊗n))=(d-1)2n.


### 4.1.1. Application: TheA⊗B Theorem

Codimensions where introduced, in [1], in order to prove the following theorem.Theorem 4.4 (see [1]).
IfA and B are (associative) PI algebras then A⊗B is PI.Proof.
It is not too difficult to show thatcn(A⊗B)≤cn(A)·cn(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
(4.8)cn(A⊗B)≤(d-1)2n(h-1)2n,
hence for a large enough n, cn(A⊗B)<n!. Finally by Remark 4.2,  R is PI if and only if there exist n such that cn(R)<n!, and the proof follows.Remark 4.5.
For explicit identities forA⊗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 Theorems4.3 and 4.4 fail in some non-associative cases.The above results motivate the study of the following problem.Question 1.
Given a PI algebraA, find a formula for the sequence cn(A). In most cases this seems to be too difficult, and one tries to, at least, get a “nice” asymptotic approximation of cn(A).In Section6 we determine such asymptotics for the algebras Mk(F). We also give such partial results for the other verbally prime algebras Mk,ℓ and Mk(G).


## 4.1.1. Application: TheA⊗B Theorem

Codimensions where introduced, in [1], in order to prove the following theorem.Theorem 4.4 (see [1]).
IfA and B are (associative) PI algebras then A⊗B is PI.Proof.
It is not too difficult to show thatcn(A⊗B)≤cn(A)·cn(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
(4.8)cn(A⊗B)≤(d-1)2n(h-1)2n,
hence for a large enough n, cn(A⊗B)<n!. Finally by Remark 4.2,  R is PI if and only if there exist n such that cn(R)<n!, and the proof follows.Remark 4.5.
For explicit identities forA⊗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 Theorems4.3 and 4.4 fail in some non-associative cases.The above results motivate the study of the following problem.Question 1.
Given a PI algebraA, find a formula for the sequence cn(A). In most cases this seems to be too difficult, and one tries to, at least, get a “nice” asymptotic approximation of cn(A).In Section6 we determine such asymptotics for the algebras Mk(F). We also give such partial results for the other verbally prime algebras Mk,ℓ and Mk(G).


## 5. Cocharacters

Recall the identificationFSn≡Vn (see (3.2)) and let A be a PI algebra. The regular left action of Sn on Vn is as follows. Let σ∈Sn and g(x1,…,xn)∈Vn, then
(5.1)σg(x1,…,xn)=g(xσ(1),…,xσ(n)).
This makes Vn into a left Sn module, with Id(A)∩Vn a submodule of Vn.Definition 5.1.
TheSn-character
(5.2)χn(A)∶=χSn(VnId(A)∩Vn)
of the quotient module Vn/(Id(A)∩Vn) is called the nth cocharacter of A.Cocharacters where introduced in [18] in the study of standard identities.Sincechar(F)=0, from the ordinary representation theory of Sn [19, 20] it is well known that the irreducible Sn characters are parametrized by the partitions λ on n:
(5.3)Irred(Sn)={χλ∣λ⊢n}.
Also, FSn decomposes as follows:
(5.4)FSn=⨁λ⊢nIλ,
where the Iλ are the minimal two sided ideals of FSn, and there are natural bijections
(5.5)χλ⟷λ⟷Iλ.
Here degχλ=fλ and dimIλ=(fλ)2 where, again, fλ is the number of standard Young taableaux of shape λ.Given a tableauTλ of shape λ⊢n, it determines the semi-idempotent eTλ=RTλ+CTλ-, and FSneTλ⊆Iλ is a minimal left ideal.TheSn-character of the regular representation FSn≡Vn is χSn(Vn)=∑λ⊢n‍fλχλ. It follows that the nth cocharacter of A can be written as
(5.6)χn(A)=∑λ⊢n‍mλ(A)χλ
for some multiplicities mλ(A), and mλ(A)≤fλ.Clearly, the degree ofχn(A) is the corresponding codimension cn(A), and since deg(χλ)=fλ, hence
(5.7)cn(A)=deg(χn(A))=∑λ⊢n‍mλ(A)fλ.
For example [21] let G be the infinite dimensional Grassmann (Exterior) algebra, then
(5.8)mλ(G)={1ifλ=(m-r,1r)0otherwise.Remark 5.2.
Properties of the identificationFSn≡Vn imply the following characterization of Capelli identities (Section 3.1) by cocharacters [22]. Let A be a PI algebra with χn(A)=∑λ⊢n‍mλ(A)χλ its cocharacter. Then A satisfies Capd+1[x;y] if and only if mλ(A)=0 whenever ℓ(λ)≥d+1.As an application of cocharacters one can prove the following.Proposition 5.3 (see [23]).
IfA satisfies Capd+1 and B satisfies Caph+1 then A⊗B satisfies Capdh+1.We remark that a proof of this result without applying cocharacters is yet unknown.Question 2.
Given a PI algebraA, find a formula for the multiplicities mλ(A). In most cases this is too difficult, and one tries to at least get some approximate description of mλ(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 the2×2 matrices M2(F) there is the following nice formula for the multiplicities mλ(M2(F)) of χn(M2(F)).Example 5.5 (see [24], [25, Theorem 12.6.5], [26]).
Denotemλ(M2(F))=mλ. First, if ℓ(λ)>4 then mλ=0. So let λ=(λ1,…,λ4).Ifλ=(n) then m(n)=1.Ifλ=(λ1,λ2) with λ2>0 then m(λ1,λ2)=(λ1-λ2+1)·λ2.Ifλ=(λ1,1,1,λ4) (so λ4≤1) then m(λ1,1,1,λ4)=λ1·(2-λ4)-1.Andmλ=(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 mλ(Mk(F)). A somewhat similar phenomena is discussed in Section 12.


### 5.2. Trace Identities, Codimensions, and Cocharacters

In the case ofk×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 examplex1·x2·tr(x3·x4) is a (mixed) trace polynomial. These trace polynomials can be evaluated on the algebra Mk(F) (or on any algebra with a trace) hence yielding trace identities. For example, it can be proved that the trace polynomial
(5.9)g(x1,x2)=tr(x1)tr(x2)-x1tr(x2)-x2tr(x1)-tr(x1x2)+x1x2+x2x1
is a trace identity of M2(F). This is an example of a mixed trace polynomial, while the polynomial
(5.10)p(x1,x2,x3)=tr(x1)tr(x2)tr(x3)-tr(x1x3)tr(x2)-tr(x2x3)tr(x1)-tr(x1x2)tr(x3)+tr(x1x2x3)+tr(x2x1x3)
is a pure trace polynomial, which is also an identity of M2(F). We then have trace identities of Mk(F), “pure” and “mixed,” hence trace codimensions cnptr(Mk(F)) and cnmtr(Mk(F)), and trace cocharacters χnptr(Mk(F)) and χnmtr(Mk(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 χnptr(Mk(F)) of Mk(F).Theorem 5.6.
We have(5.11)χnptr(Mk(F))=∑λ⊢n,ℓ(λ)≤k‍χλ⊗χλ.This implies that the trace codimensions are given by the following formula:(5.12)cnptr(Mk(F))=∑λ⊢n,ℓ(λ)≤k‍(fλ)2.
This formula is the starting point for computing the asymptotic formula of cn(Mk(F)) given in Section 6.


## 5.1. The Cocharacters of Matrix Algebras

In the case of the2×2 matrices M2(F) there is the following nice formula for the multiplicities mλ(M2(F)) of χn(M2(F)).Example 5.5 (see [24], [25, Theorem 12.6.5], [26]).
Denotemλ(M2(F))=mλ. First, if ℓ(λ)>4 then mλ=0. So let λ=(λ1,…,λ4).Ifλ=(n) then m(n)=1.Ifλ=(λ1,λ2) with λ2>0 then m(λ1,λ2)=(λ1-λ2+1)·λ2.Ifλ=(λ1,1,1,λ4) (so λ4≤1) then m(λ1,1,1,λ4)=λ1·(2-λ4)-1.Andmλ=(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 mλ(Mk(F)). A somewhat similar phenomena is discussed in Section 12.


## 5.2. Trace Identities, Codimensions, and Cocharacters

In the case ofk×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 examplex1·x2·tr(x3·x4) is a (mixed) trace polynomial. These trace polynomials can be evaluated on the algebra Mk(F) (or on any algebra with a trace) hence yielding trace identities. For example, it can be proved that the trace polynomial
(5.9)g(x1,x2)=tr(x1)tr(x2)-x1tr(x2)-x2tr(x1)-tr(x1x2)+x1x2+x2x1
is a trace identity of M2(F). This is an example of a mixed trace polynomial, while the polynomial
(5.10)p(x1,x2,x3)=tr(x1)tr(x2)tr(x3)-tr(x1x3)tr(x2)-tr(x2x3)tr(x1)-tr(x1x2)tr(x3)+tr(x1x2x3)+tr(x2x1x3)
is a pure trace polynomial, which is also an identity of M2(F). We then have trace identities of Mk(F), “pure” and “mixed,” hence trace codimensions cnptr(Mk(F)) and cnmtr(Mk(F)), and trace cocharacters χnptr(Mk(F)) and χnmtr(Mk(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 χnptr(Mk(F)) of Mk(F).Theorem 5.6.
We have(5.11)χnptr(Mk(F))=∑λ⊢n,ℓ(λ)≤k‍χλ⊗χλ.This implies that the trace codimensions are given by the following formula:(5.12)cnptr(Mk(F))=∑λ⊢n,ℓ(λ)≤k‍(fλ)2.
This formula is the starting point for computing the asymptotic formula of cn(Mk(F)) given in Section 6.


## 6. Asymptotics of the Codimensionscn(Mk(F))

For the2×2 matrices, Procesi [34] proved the following formula for cn(M2(F)).Theorem 6.1 (see [34]).
We have(6.1)cn(M2(F))=1n+1(2n+2n+1)-(n3)+1-2n.It was already mentioned that whenk≥3, it most likely is impossible to find an exact formula for the multiplicities mλ(Mk(F)), and the same is probably true about cn(Mk(F)). Instead of giving up, one looks for the asymptotic of cn(Mk(F)).Here we have the folowing theorem.Theorem 6.2 (see [2]).
Whenn goes to infinity,
(6.2)cn(Mk(F))~[(12π)k-1(12)(k2-1)/2·1!2!⋯(k-1)!·k(k2/2)]·(1n)(k2-1)/2·k2(n+1).For example, whenk=2 both Theorems 6.1 and 6.2 give the same asymptotic value
(6.3)cn(M2(F))~4π·1n·4n,
compare with (12.13).We review the major steps toward the proof of Theorem6.2. 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:
(6.4)cn(Mk(F))~cn+1ptr(Mk(F)).
We saw that by Theorem 5.6,
(6.5)cnptr(Mk(F))=∑λ⊢n,ℓ(λ)≤k2‍(fλ)2.
Thus,
(6.6)cn(Mk(F))~∑λ⊢n+1,ℓ(λ)≤k2‍(fλ)2.
The last major step here is the computation of the asymptotic behavior of the following sum:
(6.7)Sk2(2)(n)=∑λ⊢n,ℓ(λ)≤k2‍(fλ)2.
The asymptotics, as n→∞, of the more general sums
(6.8)Sh(β)(n)=∑λ⊢n,ℓ(λ)≤h‍(fλ)β
is given in [16]. That asymptotic is of the form Sh(β)(n)~a·nb·rn, where a=a(β,h), b=b(β,h) and r=r(β,h)=hβ, 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(6.9)∫01‍⋯∫01‍∏i=1k‍uix-1(1-ui)y-1∏1≤i<j≤k‍|ui-uj|2zdu1⋯duk=∏j=0k-1‍Γ(x+jz)Γ(y+jz)Γ(1+(j+1)z)Γ(x+y+(k+j-1)z)Γ(1+z).Together, the above steps yield the asymptotic value of Theorem6.2.


### 6.1. The Other Verbally Prime Algebras

For the other verbally prime algebras (see Section2.2.1), we quote the following partial asymptotic results.Theorem 6.4.
(1) [37, Theorem 7]
(6.10)cn(Mk,ℓ)~a·(1n)(k2+ℓ2-1)/2·(k+ℓ)2n.
The constant a is yet unknown.
(2) [37, Theorem 8] Let G be the infinite dimensional Grassmann algebra, then
(6.11)cn(Mk(G))~b·(1n)g·(2k2)n,
where the constants b and g are yet unknown, and (k2-1)/2≤g≤(2k2-1)/2.


## 6.1. The Other Verbally Prime Algebras

For the other verbally prime algebras (see Section2.2.1), we quote the following partial asymptotic results.Theorem 6.4.
(1) [37, Theorem 7]
(6.10)cn(Mk,ℓ)~a·(1n)(k2+ℓ2-1)/2·(k+ℓ)2n.
The constant a is yet unknown.
(2) [37, Theorem 8] Let G be the infinite dimensional Grassmann algebra, then
(6.11)cn(Mk(G))~b·(1n)g·(2k2)n,
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 alphabetx1,…,xℓ of ℓ letters; W(x1,…,xℓ) is the set of all words (i.e., monomials) in x1,…,xℓ; U(d,ℓ) the subset of the words of length ≤d. We consider finitely generated PI algebra A=F{a1,…,aℓ}.Theorem 7.1 (Shirshov's Height Theorem).
Consider a PI algebra satisfying the identity (3.3):
(7.1)y1⋯yd-∑1≠π∈Sd‍απyπ(1)⋯yπ(d).
There exists h=h(d,ℓ) large enough such that any finitely generated algebra A=F{a1,…,aℓ} that satisfies the identity (3.3), satisfies the following condition.
ModuloId(A), F{x1,…,xℓ} is spanned by the elements
(7.2){u1k1⋯uhkh∣ui∈U(d,ℓ),anykj}.


### 7.2. Application: Bounds on the Cocharacters

#### 7.2.1. The(k,ℓ) Hook Theorem

Denote byH(k,ℓ;n) the partitions of n in the (k,ℓ) hook:
(7.3)H(k,ℓ;n)={(λ1,λ2,…)⊢n∣λk+1≤ℓ},H(k,ℓ)=∪n(H(k,ℓ;n)).
Let χn be Sn characters, n=1,2,…. We say that χn is supported on H(k,ℓ), and denote χn⊆H(k,ℓ), if for all n,
(7.4)χn=∑λ∈H(k,ℓ;n)‍mλχλ.
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]).
LetA be any (associative) PI algebra, then there exist k,ℓ such that its cocharacters χn(A) are supported on the (k,ℓ) hook H(k,ℓ).
Explicitly, letA 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 χn(A)⊆H(k,ℓ), namely,
(7.5)χn(A)=∑λ∈H(k,ℓ;n)‍mλ(A)·χλ.


#### 7.2.2. An Application of Shirshov's Theorem

We remark that the proof of Theorem7.2 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 algebraA, all its cocharacter-multiplicities mλ(A) are polynomially bounded. There exist a constant C and a power p such that for all n and λ⊢n, mλ(A)≤C·np.The proof of Theorem7.3 also applies Shirshov's Height Theorem 7.1, as well as a ℤ2 version of the Schur-Weyl theory [32, 33, 41].


## 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 alphabetx1,…,xℓ of ℓ letters; W(x1,…,xℓ) is the set of all words (i.e., monomials) in x1,…,xℓ; U(d,ℓ) the subset of the words of length ≤d. We consider finitely generated PI algebra A=F{a1,…,aℓ}.Theorem 7.1 (Shirshov's Height Theorem).
Consider a PI algebra satisfying the identity (3.3):
(7.1)y1⋯yd-∑1≠π∈Sd‍απyπ(1)⋯yπ(d).
There exists h=h(d,ℓ) large enough such that any finitely generated algebra A=F{a1,…,aℓ} that satisfies the identity (3.3), satisfies the following condition.
ModuloId(A), F{x1,…,xℓ} is spanned by the elements
(7.2){u1k1⋯uhkh∣ui∈U(d,ℓ),anykj}.


## 7.2. Application: Bounds on the Cocharacters

### 7.2.1. The(k,ℓ) Hook Theorem

Denote byH(k,ℓ;n) the partitions of n in the (k,ℓ) hook:
(7.3)H(k,ℓ;n)={(λ1,λ2,…)⊢n∣λk+1≤ℓ},H(k,ℓ)=∪n(H(k,ℓ;n)).
Let χn be Sn characters, n=1,2,…. We say that χn is supported on H(k,ℓ), and denote χn⊆H(k,ℓ), if for all n,
(7.4)χn=∑λ∈H(k,ℓ;n)‍mλχλ.
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]).
LetA be any (associative) PI algebra, then there exist k,ℓ such that its cocharacters χn(A) are supported on the (k,ℓ) hook H(k,ℓ).
Explicitly, letA 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 χn(A)⊆H(k,ℓ), namely,
(7.5)χn(A)=∑λ∈H(k,ℓ;n)‍mλ(A)·χλ.


### 7.2.2. An Application of Shirshov's Theorem

We remark that the proof of Theorem7.2 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 algebraA, all its cocharacter-multiplicities mλ(A) are polynomially bounded. There exist a constant C and a power p such that for all n and λ⊢n, mλ(A)≤C·np.The proof of Theorem7.3 also applies Shirshov's Height Theorem 7.1, as well as a ℤ2 version of the Schur-Weyl theory [32, 33, 41].


## 7.2.1. The(k,ℓ) Hook Theorem

Denote byH(k,ℓ;n) the partitions of n in the (k,ℓ) hook:
(7.3)H(k,ℓ;n)={(λ1,λ2,…)⊢n∣λk+1≤ℓ},H(k,ℓ)=∪n(H(k,ℓ;n)).
Let χn be Sn characters, n=1,2,…. We say that χn is supported on H(k,ℓ), and denote χn⊆H(k,ℓ), if for all n,
(7.4)χn=∑λ∈H(k,ℓ;n)‍mλχλ.
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]).
LetA be any (associative) PI algebra, then there exist k,ℓ such that its cocharacters χn(A) are supported on the (k,ℓ) hook H(k,ℓ).
Explicitly, letA 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 χn(A)⊆H(k,ℓ), namely,
(7.5)χn(A)=∑λ∈H(k,ℓ;n)‍mλ(A)·χλ.


## 7.2.2. An Application of Shirshov's Theorem

We remark that the proof of Theorem7.2 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 algebraA, all its cocharacter-multiplicities mλ(A) are polynomially bounded. There exist a constant C and a power p such that for all n and λ⊢n, mλ(A)≤C·np.The proof of Theorem7.3 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 Stuv[x]=(Stu[x])v, where Stu[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 FSn≡Vn, together with the exponential bound on cn(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]).
LetA be a PI algebra satisfying cn(A)≤αn, and let u and v be integers satisfying u,v≥e·α2, then Stuv[x1,…,xu]∈Id(A). In particular if A satisfies an identity of degree d, and u,v≥e·(d-1)4, then Stuv[x1,…,xu]∈Id(A) (of degree uv which is about e2(d-1)8.)
In fact, with theseu and vA satisfies the power of the Capelli identity: (Capu[x;y])v∈Id(A).Remark 8.2.
LetA satisfy an identity of degree d and B an identity of degree h. Denote α=(d-1)2(h-1)2 then cn(A⊗B)≤αn. Thus, if u,v≥e·((d-1)(h-1))4 then A⊗B satisfies the identities Stuv[x1,…,xu] and (Capu[x;y])v. Note that here the degree of Stuv[x1,…,xu], 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 αn<n!. Thus, if e(d-1)2(h-1)2<n then, for that n,
(8.1)cn(A⊗B)≤((d-1)2(h-1)2)n<n!
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 PolynomialLk[x;y]

Usually, lower bounds for codimensions are harder to obtain than upper bounds. Given a PI algebraA, a lower bound for cn(A) can be obtained by the following technique. Find a polynomial p=p(x1,…,xn)∈Vn which is a nonidentity of A, namely, p∉Id(A). In addition, with the identification Vn≡FSn=⊕λ⊢nIλ (see (5.4)), verify that p∈Iμ for some μ⊢n. Then cn(A)≥fμ. This follows since 0≠FSnp⊆Iμ, so Jμ⊆FSnp for some minimal left ideal Jμ⊆Iμ with Jμ∩Id(A)=0, hence fμ=dimJμ≤cn(A). We now construct such a polynomial p via the polynomial Lk(x;y), when A=Mk(F) [44, Definition  2.3].Corresponding to the sum1+3+⋯+(2k-1)=k2, construct the monomial Nk(x;y) of degree 2k2:
(9.1)Nk(x;y)=(x1)(y1)(x2x3x4)(y2y3y4)(x5⋯x9)(y5⋯y9)⋯.
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 Lk(x;y):
(9.2)Lk(x;y)=∑σ,π∈Sk2‍sgn(σ)sgn(π)Nk(xσ(1),…,xσ(k2);yπ(1),…,yπ(k2)).
Let Tμ′ be the conjugate tableau of Tμ, and
(9.3)Tμ′=13459⋯13⋯267814⋯18⋯.
Then Lk(x;y)=CTμ- and in that sense Lk(x;y) corresponds to the tableau Tμ where μ=(2k2). It is not difficult to show that Lk(x;y) takes central values on Mk(F).Fork=2 and k=3 it was verified that Lk(x;y)∉Id(Mk(F)), and it was conjectured that for all kLk(x;y)∉Id(Mk(F)) namely, that Lk(x;y) is a nonidentity of Mk(F) [44]. This conjecture was verified by Formanek [35].Theorem 9.1 (see [35]).
The plolynomialLk(x;y), which corresponds to the rectangle μ=(2k2), is a nonidentity of Mk(F). Hence Lk(x;y) is a central polynomial.By Young's rule it follows that(9.4)Lk(x;y)=pμ+∑λ⊢2k2,ℓ(λ)≥k2+1‍pλ,
where pμ∈Iμ and pλ∈Iλ, see (5.4). Since Mk(F) satisfies the Capelli identity Capk2+1, it follows that for all λ with ℓ(λ)≥k2+1, pλ∈Id(Mk(F)). And since Lk(x;y)∉Id(Mk(F)), hence pμ∉Id(Mk(F)), where μ is of the 2×k2 rectangular shape μ=(2k2). Thus, also pμ is a central polynomial for Mk(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.Forcn(Mk(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 cn(Mk(F)). Also, the fact that Lk(x;y) is a nonidentity of Mk(F) has applications in proving lower bounds for various other types of codimensions.


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

For most PI algebrasA it seems hopeless to find a precise, or even asymptotic, formula for the codimensions cn(A). We therefore ask a much more restricted question.Question 3.
Given the associative PI algebraA, what can be said about the asymptotic behavior of the codimensions cn(A)?As a first step we have the remarkable integrality property given by Theorem10.1 below. See also Theorem 11.1 and its relation to Theorem 10.1.Theorem 10.1 (see [3]).
LetA be an associative PI algebra (with char(F)=0), then the limit
(10.1)limn→∞(cn(A))1/n
exists and is an integer.
We denoteexp(A)=limn→∞(cn(A))1/n, so exp(A)∈ℕ.


### 10.1. Review of the Proof Whendim(A)<∞

WhenA 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⊕⋯⊕Br where Bj are simple, namely, Bj≅Mkj(F). Consider now all possible nonzero products of the type
(10.2)Bi1JBi2J⋯JBiq≠0(q≥1),
where the Bij are distinct, and for such nonzero products let
(10.3)h∶=maxdim(Bi1⊕⋯⊕Biq).
Then h=ki12+⋯+kiq2, and h is the limit in Theorem 10.1:
(10.4)limn→∞(cn(A))1/n=h,
Namely, exp(A)=ki12+⋯+kiq2.For example, consider the algebra of upper block triangular matrices(10.5)A=UT(k1,k2,k3)=(B1J1J20B2J300B3),
where for 1≤i≤3, Bi≅Mki(F) and Ji 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 ei,j in A satisfy
(10.6)0≠e1,1·e1,k1+1·ek1+1,k1+1·ek1+1,k1+k2+1·ek1+k2+1,k1+k2+1∈B1J1B2J3B3⊆B1JB2JB3,
so B1JB2JB3≠0 and also, this is the maximal such nonzero product. It follows that here
(10.7)h=h(A)=dim(B1⊕B2⊕B3)=k12+k22+k32.


#### 10.1.1. The Upper Bound

In the general case the exponenth=exp(A)=limn→∞(cn(A))1/n is given by (10.3). To prove this, one first proves the following upper bound.Lemma 10.2.
There exist constantsa1, g1 such that for all n, cn(A)≤a1·ng1·hn.LetPar(n) denote the partitions of n. Given h∈ℕ, define the subsets NSK(h,n)⊆Par(n) by
(10.8)NSK(h,n)={λ=(λ1,λ2,…)⊢n∣∑j>h‍λj≤K},
so NSK(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 χn(A) are supported on such nearly a strip NSK(h,n), and by the polynomial bound mλ(A)≤a·nb on the multiplicities in the cocharacters, see Theorem 7.3. Thus cn(A)≤anb∑‍fλ, where the sum ∑‍fλ is supported on such nearly a strip NSK(h,n). Similar to the estimates in [16], such sum
(10.9)∑λ∈NSK(h,n)‍fλ
is bounded by a-·nb-·hn, 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 constantsa2>0 and g2 such that for all n, a2·ng2·hn≤cn(A).A key ingredient in proving the lower bound is the polynomialLk(x;y) which is a nonidentity for Mk(F), see Section 9. We already noted that Lk(x;y) has a component pμ∈Iμ which is central (nonidentity) for Mk(F), where μ is the 2×k2 rectangle (2k2), see Theorem 9.1.We start with a single matrix algebraB=Mk(F), with the corresponding central polynomials Lk(x;y) and pμ(1). Rectangles of the same height (width) can be glued horizontally (vertically). Gluing horizontally (2k2) to itself w times yields the 2w×k2 rectangle μ(w)=((2w)k2), with a corresponding w-power (pμ(1))w=pμ(w), which is central and nonidentity for Mk(F), and pμ(w)∈Iμ(w). Thus fμ(w)≤cn(Mk(F)). For that μ(w)=((2w)k2)⊢n, n=2wk2, the asymptotic of fμ(w) then yields the lower bound
(10.10)a-2·ng-2·(k2)n≤cn(Mk(F))
for some constants a-2>0 and g-2.In the general case we are givenBi1JBi2J⋯JBiq≠0 as in (10.2). To each Bij corresponds the nonidentity polynomial Lkij(x;y), with the corresponding rectangular tableaux (2kij2), all with the same width =2. These tableaux can be glued vertically, thus yielding the rectangular tableau ρ=(2h), where h=∑j‍(kij)2. To that tableau there corresponds a polynomial which is essentially the product of the polynomials Lkij(x;y), with the corresponding component which is the product of the corresponding polynomials pμ, 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=Mk(F), the asymptotic of fρ(w)=f((2w)h) then yields the lower bound
(10.11)a2·ng2·hn≤cn(A)
for some constants 0<a2 and g2. This proves the lower bound.Corollary 10.4.
Putting together the lower and the upper bounds, Theorem10.1 then follows.This completes our review of the proof of Theorem10.1. For extensions of this theorem, see Remark 11.2.


## 10.1. Review of the Proof Whendim(A)<∞

WhenA 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⊕⋯⊕Br where Bj are simple, namely, Bj≅Mkj(F). Consider now all possible nonzero products of the type
(10.2)Bi1JBi2J⋯JBiq≠0(q≥1),
where the Bij are distinct, and for such nonzero products let
(10.3)h∶=maxdim(Bi1⊕⋯⊕Biq).
Then h=ki12+⋯+kiq2, and h is the limit in Theorem 10.1:
(10.4)limn→∞(cn(A))1/n=h,
Namely, exp(A)=ki12+⋯+kiq2.For example, consider the algebra of upper block triangular matrices(10.5)A=UT(k1,k2,k3)=(B1J1J20B2J300B3),
where for 1≤i≤3, Bi≅Mki(F) and Ji 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 ei,j in A satisfy
(10.6)0≠e1,1·e1,k1+1·ek1+1,k1+1·ek1+1,k1+k2+1·ek1+k2+1,k1+k2+1∈B1J1B2J3B3⊆B1JB2JB3,
so B1JB2JB3≠0 and also, this is the maximal such nonzero product. It follows that here
(10.7)h=h(A)=dim(B1⊕B2⊕B3)=k12+k22+k32.


### 10.1.1. The Upper Bound

In the general case the exponenth=exp(A)=limn→∞(cn(A))1/n is given by (10.3). To prove this, one first proves the following upper bound.Lemma 10.2.
There exist constantsa1, g1 such that for all n, cn(A)≤a1·ng1·hn.LetPar(n) denote the partitions of n. Given h∈ℕ, define the subsets NSK(h,n)⊆Par(n) by
(10.8)NSK(h,n)={λ=(λ1,λ2,…)⊢n∣∑j>h‍λj≤K},
so NSK(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 χn(A) are supported on such nearly a strip NSK(h,n), and by the polynomial bound mλ(A)≤a·nb on the multiplicities in the cocharacters, see Theorem 7.3. Thus cn(A)≤anb∑‍fλ, where the sum ∑‍fλ is supported on such nearly a strip NSK(h,n). Similar to the estimates in [16], such sum
(10.9)∑λ∈NSK(h,n)‍fλ
is bounded by a-·nb-·hn, 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 constantsa2>0 and g2 such that for all n, a2·ng2·hn≤cn(A).A key ingredient in proving the lower bound is the polynomialLk(x;y) which is a nonidentity for Mk(F), see Section 9. We already noted that Lk(x;y) has a component pμ∈Iμ which is central (nonidentity) for Mk(F), where μ is the 2×k2 rectangle (2k2), see Theorem 9.1.We start with a single matrix algebraB=Mk(F), with the corresponding central polynomials Lk(x;y) and pμ(1). Rectangles of the same height (width) can be glued horizontally (vertically). Gluing horizontally (2k2) to itself w times yields the 2w×k2 rectangle μ(w)=((2w)k2), with a corresponding w-power (pμ(1))w=pμ(w), which is central and nonidentity for Mk(F), and pμ(w)∈Iμ(w). Thus fμ(w)≤cn(Mk(F)). For that μ(w)=((2w)k2)⊢n, n=2wk2, the asymptotic of fμ(w) then yields the lower bound
(10.10)a-2·ng-2·(k2)n≤cn(Mk(F))
for some constants a-2>0 and g-2.In the general case we are givenBi1JBi2J⋯JBiq≠0 as in (10.2). To each Bij corresponds the nonidentity polynomial Lkij(x;y), with the corresponding rectangular tableaux (2kij2), all with the same width =2. These tableaux can be glued vertically, thus yielding the rectangular tableau ρ=(2h), where h=∑j‍(kij)2. To that tableau there corresponds a polynomial which is essentially the product of the polynomials Lkij(x;y), with the corresponding component which is the product of the corresponding polynomials pμ, 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=Mk(F), the asymptotic of fρ(w)=f((2w)h) then yields the lower bound
(10.11)a2·ng2·hn≤cn(A)
for some constants 0<a2 and g2. This proves the lower bound.Corollary 10.4.
Putting together the lower and the upper bounds, Theorem10.1 then follows.This completes our review of the proof of Theorem10.1. For extensions of this theorem, see Remark 11.2.


## 10.1.1. The Upper Bound

In the general case the exponenth=exp(A)=limn→∞(cn(A))1/n is given by (10.3). To prove this, one first proves the following upper bound.Lemma 10.2.
There exist constantsa1, g1 such that for all n, cn(A)≤a1·ng1·hn.LetPar(n) denote the partitions of n. Given h∈ℕ, define the subsets NSK(h,n)⊆Par(n) by
(10.8)NSK(h,n)={λ=(λ1,λ2,…)⊢n∣∑j>h‍λj≤K},
so NSK(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 χn(A) are supported on such nearly a strip NSK(h,n), and by the polynomial bound mλ(A)≤a·nb on the multiplicities in the cocharacters, see Theorem 7.3. Thus cn(A)≤anb∑‍fλ, where the sum ∑‍fλ is supported on such nearly a strip NSK(h,n). Similar to the estimates in [16], such sum
(10.9)∑λ∈NSK(h,n)‍fλ
is bounded by a-·nb-·hn, 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 constantsa2>0 and g2 such that for all n, a2·ng2·hn≤cn(A).A key ingredient in proving the lower bound is the polynomialLk(x;y) which is a nonidentity for Mk(F), see Section 9. We already noted that Lk(x;y) has a component pμ∈Iμ which is central (nonidentity) for Mk(F), where μ is the 2×k2 rectangle (2k2), see Theorem 9.1.We start with a single matrix algebraB=Mk(F), with the corresponding central polynomials Lk(x;y) and pμ(1). Rectangles of the same height (width) can be glued horizontally (vertically). Gluing horizontally (2k2) to itself w times yields the 2w×k2 rectangle μ(w)=((2w)k2), with a corresponding w-power (pμ(1))w=pμ(w), which is central and nonidentity for Mk(F), and pμ(w)∈Iμ(w). Thus fμ(w)≤cn(Mk(F)). For that μ(w)=((2w)k2)⊢n, n=2wk2, the asymptotic of fμ(w) then yields the lower bound
(10.10)a-2·ng-2·(k2)n≤cn(Mk(F))
for some constants a-2>0 and g-2.In the general case we are givenBi1JBi2J⋯JBiq≠0 as in (10.2). To each Bij corresponds the nonidentity polynomial Lkij(x;y), with the corresponding rectangular tableaux (2kij2), all with the same width =2. These tableaux can be glued vertically, thus yielding the rectangular tableau ρ=(2h), where h=∑j‍(kij)2. To that tableau there corresponds a polynomial which is essentially the product of the polynomials Lkij(x;y), with the corresponding component which is the product of the corresponding polynomials pμ, 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=Mk(F), the asymptotic of fρ(w)=f((2w)h) then yields the lower bound
(10.11)a2·ng2·hn≤cn(A)
for some constants 0<a2 and g2. This proves the lower bound.Corollary 10.4.
Putting together the lower and the upper bounds, Theorem10.1 then follows.This completes our review of the proof of Theorem10.1. For extensions of this theorem, see Remark 11.2.


## 11. Berele's “1/2” Theorem (See [4])

Applying Theorem10.1, Berele proved the following remarkable theorem.Theorem 11.1 (see [4], see also [5]).
LetA be a PI algebra with 1∈A. Then as n goes to infinity, cn(A)~a·nt·hn, 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 thatcn(A)=∑‍mλ(A)fλ, where mλ(A) is the multiplicity of χλ. If A satisfies a Capelli identity then there exists an integer k such that mλ(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 mλ(A) and fλ, and we proceed to do so.
Let(11.1)Uk(A)=F{x1,…,xk}Id(A)∩F{x1,…,xk}
be the universal PI algebra for A in k generators. This algebra has an ℕ grading by degree and a corresponding Poincaré series Pk(t). It also has a finer ℕk grading by multidegree with corresponding Poincaré series P(t1,…,tk). Belov [46] proved that Pk(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,…,tk) 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-t1a1⋯tkak). We call such a rational function “nice.”
The Poincaré seriesP(t1,…,tk) is related to the cocharacters χn(A)=∑‍mλ(A)χλ via
(11.2)P(t1,…,tk)=∑n=0∞‍∑λ⊢n‍mλ(A)Sλ(t1,…,tk),
where Sλ 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,…,tk) is any nice rational function with Taylor series ∑‍b(n1,…,nk)t1n1⋯tknk, then the coefficients b(n1,…,nk) can be described using a finite set of polynomials. Namely, ℕk can be partitioned into regions R1,…,Rm with corresponding polynomials q1,…,qm such that for each i,
(11.3)c(n1,…,nk)=qi(n1,…,nk)for(n1,…,nk)∈Ri.
Using properties of Schur functions it can be shown that an analogue formula holds for the mλ(A). Hence we may write
(11.4)cn(A)=∑i‍{qi(λ)fλ∣λ∈Ri}.
The regions Ri, 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-t12)-1(1-t2)-1(1-t1t2)-1. Then b(n1,n2) is given by
(11.5)b(n1,n2)={n22+1ifn1≥n2,n2iseven,(n2+1)2ifn1≥n2,n2isodd,n12+1ifn1<n2,n1iseven,(n1+1)2ifn1<n2,n1isodd.
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 mλ(A) in which λ=(λ1,…,λk) and λh can be taken arbitrarily large. Giambruno and Zaicev proved that h=limn→∞(cn(A))1/n. This number, denoted exp(A), is also important for the proof of Theorem 11.1. Let v1 be the vector in ℕk whose coordinates are h ones followed by k-h zeros. Then in (11.4) certain summands will be on regions of the form
(11.6){v0+α1v1+⋯+αdvd∣αi,…,αd∈ℕ}∩L,
where L is an integer lattice. Let R1′,…,Rp′ be the regions of this form. Then in the computation of cn(A) these terms dominate and we can refine (11.4) to
(11.7)cn(A)=∑i‍{qi(λ)fλ∣λ∈Ri′}.
The rest of the proof closely immitates the computation of [16]. The main theorem is that if R' is as above and q(x1,…,xk) is a polynomial then ∑‍q(λ)fλ summed over partitions of n in R' will be asymptotic to a constant—times n to an integer power—times hn. Hence, the cocharacter cn(A) will be a sum of such terms.
There remains the problem that the powers ofn 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,…,tk) can be written as (1-t1)-1⋯(1-tk)-1g(t1,…,tk), 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 Sections10 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, Theorems10.1 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 sequencean, then F(x)=∑n≥0‍anxn is its corresponding  ordinary generating function. In particular CA(x)=∑n≥0‍cn(A)xn is the generating function of the codimensions of the algebra A.
(2) The functionF(x) is  algebraic if there exist polynomials P0(x),…,Pr(x) such that
(12.1)Pr(x)Fr(x)+⋯+P1(x)F(x)+P0(x)=0.
Algebraicity or nonalgebraicity of the generating function F(x)=∑n≥0‍anxn is an indication of the complexity of the sequence an.Example 12.2 (see [34], [25, Theorem 12.6.8]).
For the2×2 matrices M2(F) we have
(12.2)cn(M2(F))=1n+2(2n+2n+1)-(n3)+1-2n.
This implies that
(12.3)CM2(F)(x)=1x2(1-2x-1-4x)-x3(1-x)4+11-x-11-2x,
which is clearly algebraic.Note that whenk=1, M1(F)=F and cn(F)=1, hence
(12.4)CF(x)=11-x
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]).
Letf:ℕ→ℂ,  F(x)=∑n≥0‍f(n)xn, and assume that as n goes to infinity,
(12.5)f(n)~b·n-g·an,
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(12.6)CMk(F)(x)=∑n≥0‍cn(Mk(F))·xn
be the generating functions of the (ordinary) codimensions of Mk(F). If k≥3 and k is odd then CMk(F)(x) is not algebraic.Proof.
By Theorem6.2(12.7)cn(Mk(F))~b·n-g·k2n,
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, Example12.2 and Theorem 12.4 motivate the following conjecture.Conjecture 12.5.
Ifk≥3 then the generating function
(12.8)CMk(F)(x)=∑n≥0‍cn(Mk(F))·xn
is not algebraic.For the other verbally prime algebras (see Section2.2.1), recall from Theorem 6.4 the following partial asymptotic results:
(12.9)cn(Mk,ℓ)~a·(1n)(k2+ℓ2-1)/2·(k+ℓ)2n,
where the constant a is yet unknown. Also,
(12.10)cn(Mk(G))~b·(1n)g·(2k2)n,
where the constants b and g are yet unknown, and (k2-1)/2≤g≤(2k2-1)/2.Corollary 12.6.
Ifk≢ℓ(mod2) then the generating function CMk,ℓ(x) of the codimensions cn(Mk,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 M1,1 have the same identities, hence the same codimensions. Now
(12.11)c0(M1,1)=1,cn(M1,1)=12(2nn)+n+1-2n,n=1,2,…
see [65], and hence
(12.12)CM1,1(x)=12+121-4x+x(1-x)2+11-x+11-2x,
which is clearly algebraic. Note that
(12.13)cn(M1,1)~12π·1n·4n,
compare with (6.3) of Section 6. See also [64].Conjecture 12.8.
Ifk,ℓ≥1 and (k,ℓ)≠(1,1) then CMk,ℓ(x) is not 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]).
Letf:ℕ→ℂ,  F(x)=∑n≥0‍f(n)xn, and assume that as n goes to infinity,
(12.5)f(n)~b·n-g·an,
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(12.6)CMk(F)(x)=∑n≥0‍cn(Mk(F))·xn
be the generating functions of the (ordinary) codimensions of Mk(F). If k≥3 and k is odd then CMk(F)(x) is not algebraic.Proof.
By Theorem6.2(12.7)cn(Mk(F))~b·n-g·k2n,
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, Example12.2 and Theorem 12.4 motivate the following conjecture.Conjecture 12.5.
Ifk≥3 then the generating function
(12.8)CMk(F)(x)=∑n≥0‍cn(Mk(F))·xn
is not algebraic.For the other verbally prime algebras (see Section2.2.1), recall from Theorem 6.4 the following partial asymptotic results:
(12.9)cn(Mk,ℓ)~a·(1n)(k2+ℓ2-1)/2·(k+ℓ)2n,
where the constant a is yet unknown. Also,
(12.10)cn(Mk(G))~b·(1n)g·(2k2)n,
where the constants b and g are yet unknown, and (k2-1)/2≤g≤(2k2-1)/2.Corollary 12.6.
Ifk≢ℓ(mod2) then the generating function CMk,ℓ(x) of the codimensions cn(Mk,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 M1,1 have the same identities, hence the same codimensions. Now
(12.11)c0(M1,1)=1,cn(M1,1)=12(2nn)+n+1-2n,n=1,2,…
see [65], and hence
(12.12)CM1,1(x)=12+121-4x+x(1-x)2+11-x+11-2x,
which is clearly algebraic. Note that
(12.13)cn(M1,1)~12π·1n·4n,
compare with (6.3) of Section 6. See also [64].Conjecture 12.8.
Ifk,ℓ≥1 and (k,ℓ)≠(1,1) then CMk,ℓ(x) is not algebraic.


## 13. NonassociativeA 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 algebraF{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 Vn of the multilinear polynomials of degree n is now of dimension  dimVn=Cn·n!, where Cn is the nth Catalan number (which counts the number of different parenthesizes of a monomial of degree n). The definition of the codimensions cn(A) is, formally, the same as that in the associative case.Definition 13.1.
(13.1)
c
n
(
A
)
=
dim
(
V
n
Id
(
A
)
∩
V
n
)
,
compare with Definition4.1.Similarly for the cocharacters, the action ofSn on Vn 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 Theorem10.1 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 limn→∞(cn(A))1/n exists and is equal to α. We briefly describe that construction.


### 13.1. The AlgebraA(m,w)

Letw=w1w2⋯ be an infinite word on the alphabet {0,1}. Given an integer m≥2, define the sequence {ki}i≥1=Km,w by
(13.2)ki=wi+m.
Construct the algebra A=A(m,w)=A(Km,w) as follows. Its basis is
(13.3){a,b}∪Z1∪Z2∪⋯,
where
(13.4)Zi={zj(i)∣1≤j≤ki},i=1,2,…,
and multiplication is given as follows:
(13.5)z1(i)a=z2(i),z2(i)a=z3(i),…,zki-1(i)a=zki(i),i=1,2,…,zki(i)b=z1(i+1),i=1,2,…
and all other products are zero.Letw=w1w2⋯ be an infinite word. The notion of the complexity of w is classical. For each n, complexityw(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]).
Given1≤α≤2, we can choose m≥2 and a word w such that
(13.6)limn→∞(cn(A(m,w)))1/nexistsandisequaltoα.


## 13.1. The AlgebraA(m,w)

Letw=w1w2⋯ be an infinite word on the alphabet {0,1}. Given an integer m≥2, define the sequence {ki}i≥1=Km,w by
(13.2)ki=wi+m.
Construct the algebra A=A(m,w)=A(Km,w) as follows. Its basis is
(13.3){a,b}∪Z1∪Z2∪⋯,
where
(13.4)Zi={zj(i)∣1≤j≤ki},i=1,2,…,
and multiplication is given as follows:
(13.5)z1(i)a=z2(i),z2(i)a=z3(i),…,zki-1(i)a=zki(i),i=1,2,…,zki(i)b=z1(i+1),i=1,2,…
and all other products are zero.Letw=w1w2⋯ be an infinite word. The notion of the complexity of w is classical. For each n, complexityw(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]).
Given1≤α≤2, we can choose m≥2 and a word w such that
(13.6)limn→∞(cn(A(m,w)))1/nexistsandisequaltoα.



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*Source: 170697-2012-11-21.xml*