Asymptotic behavior of modular representations over abelian $p$-groups

TL;DR

This paper investigates the asymptotic properties of modular representations over abelian p-groups, revealing non-recursive behaviors and introducing methods to compute core series.

Contribution

It provides new insights into the asymptotic dimension growth, answers a question by Benson and Symonds, and develops systematic computation techniques for core series.

Findings

The dimension of the core of tensor powers grows as Cγ^n n^α.

The core dimension is not eventually recursive for certain modules.

Existence of a transcendental core series from iterated syzygies.

Abstract

In this paper, we prove some results on the asymptotic behavior arising in modular representation theory over abelian $p$-groups. First, we embed the representation ring of a cyclic $p$-group into a real algebra of functions. Second, we calculate the asymptotic order of the dimension of the core of $n$-th tensor power of a direct sum of syzygies and cosyzygies of the trivial module, which is of the form $C\gamma^nn^\alpha$. This result leads to a negative answer to a question by Benson and Symonds, that is, the dimension of the core of $M^{\otimes n}$ for certain $\Omega$-algebraic module $M$ is not eventually recursive. Third, we give a systematic way of computing the core series of $\Omega$-algebraic modules. Finally, we show the existence of a transcendental core series, which comes from iterated syzygy modules of the trivial representation.