Algebraic Properties of Codimension Series of PI-Algebras

TL;DR

This paper investigates the algebraic nature of the codimension series of PI-algebras, demonstrating how their rationality or algebraicity is preserved under certain algebraic operations involving T-ideals.

Contribution

It establishes that the rationality or algebraicity of codimension series is maintained when combining PI-algebras with specific T-ideal factorizations, using properties of exponential generating functions.

Findings

If c(A,t) and c(B,t) are rational, then c(R,t) is rational.

If c(A,t) is rational and c(B,t) is algebraic, then c(R,t) is algebraic.

The proof uses properties of exponential generating functions related to symmetric group representations.

Abstract

For a PI-algebra R over a field of characteristic 0 let T(R) be the T-ideal of the polynomial identities of R and let c(R,t) be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let A, B and R be PI-algebras such that T(R)=T(A)T(B). We show that if c(A,t) and c(B,t) are rational functions, then c(R,t) is also rational. If c(A,t) is rational and c(B,t) is algebraic, then c(R,t) is also algebraic. The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of symmetric groups.