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The Image of Functor Morphing

Ehud Meir

Abstract

Functor morphing provides a method to translate complex representations of automorphism groups of finite modules over finite rings to representations of automorphism groups of functors in some abelian category. In this paper we give an explicit criterion for a representation to be in the image of functor morphing using the action of parabolic subgroups. We then demonstrate this criterion on Borel groups of finite fields.

The Image of Functor Morphing

Abstract

Functor morphing provides a method to translate complex representations of automorphism groups of finite modules over finite rings to representations of automorphism groups of functors in some abelian category. In this paper we give an explicit criterion for a representation to be in the image of functor morphing using the action of parabolic subgroups. We then demonstrate this criterion on Borel groups of finite fields.

Paper Structure

This paper contains 9 sections, 32 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functor morphing
  4. Hom-spaces
  5. linear relations in hom-spaces
  6. concrete linear relations from $M$-redundant functors and a proof of the main theorem
  7. Examples
  8. The groups $\mathop{\mathrm{GL}}\nolimits_n(\mathbb{F}_q)$
  9. The groups $B_n(\mathbb{F}_q)$

Key Result

Lemma 1.1

A necessary condition for representations of $\text{Aut}_{\mathcal{C}_{+1}}(F)$ to appear as functor morphing of representations of $\mathop{\mathrm{Aut}}\nolimits_R(M)$ is that if $(X\stackrel{f}{\to} Y)$ is a minimal presentation of $F$ then $X$ is a direct summand of $M$.

Theorems & Definitions (66)

  • Lemma 1.1: See Lemma \ref{['lem:XM']}
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Theorem A, CMO4
  • Theorem 2.2: Theorem B, CMO4
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.8
  • ...and 56 more