The

We study algebras

Denote by

Given

To study

The study of growth for PI algebra

In the associative case,

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

Recall that for two sequences of numbers,

(1) The asymptotics for the

When

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

Let

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

Let

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

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

In the last section, we review a construction of nonassociative algebras where the integrality property of the exponent fails.

Let

(1) Let

(2)

(3)

One of the main problems in PI theory is the Specht problem: Are

Kemer [

Amitsur [

Kemer introduced the notion of

The

Let

We have

The following are the three families of the verbally prime

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

Let

As usual,

The polynomial

Let the PI algebra

It follows that if

The polynomial

Similarly the polynomial

It is rather easy to show that if

Many infinite dimensional algebras are PI. For example, any infinite dimensional commutative algebra is PI. We remark that obviously, the free algebra

Computing dimensions might seem useless at first sight, since if the

Let

For example,

Note that if

A basic property of the codimensions sequence

Assume the (associative) algebra

We sketch two proofs (both different from the original proof). Both proofs apply the notion of a

Call

By an argument based on Dilworth Theorem in Combinatorics [

A second proof of the bound

Codimensions where introduced, in [

If

It is not too difficult to show that

For explicit identities for

We remark that both Theorems

The above results motivate the study of the following problem.

Given a PI algebra

In Section

Recall the identification

The

Cocharacters where introduced in [

Since

Given a tableau

The

Clearly, the degree of

Properties of the identification

As an application of cocharacters one can prove the following.

If

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

Given a PI algebra

The approach of codimensions and of cocharacters in the study of PI algebras applies also in the nonassociative case (though with different phenomena).

In the case of the

Denote

If

If

If

And

In the case of

Instead of ordinary polynomials we can consider trace polynomials, namely, polynomials involving variables and traces, for example

The Procesi-Razmyslov theory of trace identities [

We have

This implies that the trace codimensions are given by the following formula:

For the

We have

It was already mentioned that when

Here we have the folowing theorem.

When

For example, when

We review the major steps toward the proof of Theorem

The Selberg integral

Together, the above steps yield the asymptotic value of Theorem

For the other verbally prime algebras (see Section

(1) [

(2) [

A powerful tool in the study of PI algebras is Shirshov's Height Theorem, which we now quote. We consider the alphabet

Consider a PI algebra satisfying the identity (

Modulo

Denote by

Let

Explicitly, let

We remark that the proof of Theorem

For any PI algebra

The proof of Theorem

Amitsur [

Let

In fact, with these

Let

Also,

Usually, lower bounds for codimensions are harder to obtain than upper bounds. Given a PI algebra

Corresponding to the sum

For

The plolynomial

By Young's rule it follows that

For

For most PI algebras

Given the associative PI algebra

As a first step we have the remarkable integrality property given by Theorem

Let

We denote

When

For example, consider the algebra of upper block triangular matrices

In the general case the exponent

There exist constants

Let

Here we prove the folowing lemma.

There exist constants

A key ingredient in proving the lower bound is the polynomial

We start with a single matrix algebra

In the general case we are given

Putting together the lower and the upper bounds, Theorem

This completes our review of the proof of Theorem

Applying Theorem

Let

Based on various examples, that theorem was conjectured for some time. It was first proved in [

Recall that

Let

The Poincaré series

Quite a lot is known about Taylor series of nice rational functions, see for example [

There remains the problem that the powers of

We conclude this section with the following general remark.

Recent works extended the above theorems of Sections

As we show below, Theorems

(1) Given the sequence

(2) The function

For the

Note that when

We quote here a classical theorem of Jungen [

Let

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

Let

By Theorem

Obviously, Example

If

For the other verbally prime algebras (see Section

If

Indeed,

We apply here a theorem due to Kemer [

If

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

compare with Definition

Similarly for the cocharacters, the action of

A counter-example to Theorem

Let

Let

Given