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Limit representation theory on some classes of representations of abelian groups

Cheng Meng

Abstract

In this paper, we prove some results on the asymptotic behavior arising in the representation theory of modular representations of cyclic groups and syzygies and cosyzygies of the trivial module over abelian $p$-groups. We show that the representation ring of a cyclic $p$-group can be embedded into a real algebra of functions, and that the dimension of the non-projective part of $n$-th tensor power of a direct sum of syzygies and cosyzygies of the trivial module is approximately $Cγ^nn^α$. Using the existence of a non-integer exponent $α$, we answer a question of Benson and Symonds in negative, that is, there is an $Ω$-algebraic module where the dimension of the core of $M^{\otimes n}$ is not eventually recursive.

Limit representation theory on some classes of representations of abelian groups

Abstract

In this paper, we prove some results on the asymptotic behavior arising in the representation theory of modular representations of cyclic groups and syzygies and cosyzygies of the trivial module over abelian -groups. We show that the representation ring of a cyclic -group can be embedded into a real algebra of functions, and that the dimension of the non-projective part of -th tensor power of a direct sum of syzygies and cosyzygies of the trivial module is approximately . Using the existence of a non-integer exponent , we answer a question of Benson and Symonds in negative, that is, there is an -algebraic module where the dimension of the core of is not eventually recursive.
Paper Structure (23 sections, 37 theorems, 111 equations)

This paper contains 23 sections, 37 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The idea of limit representation theory
  4. Results
  5. Representation theory of $k$-objects
  6. $k$-objects and the additive structure on classes of $k$-objects
  7. Multilinear forms on $\Gamma$
  8. Tensor products and modular representations of cyclic groups
  9. Operations on representations
  10. $\Gamma$ and axioms for a representation ring
  11. The real algebra $\mathcal{F}$
  12. The definitions of $\mathcal{F}$ and $*$-product
  13. The associativity of $*$-product
  14. The functional counterpart of the induction and the restriction of representations.
  15. A norm on the real algebra $\mathcal{F}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Under the above notations,

Theorems & Definitions (88)

  • Theorem 1.1
  • Theorem 1.2: See \ref{['4.16']}
  • Theorem 1.3: See \ref{['4.22']}
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 78 more