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Polynomial functors on flags

Teresa Yu

Abstract

We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite rank groups, and we apply these descriptions to study polynomial representations and representation stability of parabolic subgroups of general linear groups.

Polynomial functors on flags

Abstract

We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite rank groups, and we apply these descriptions to study polynomial representations and representation stability of parabolic subgroups of general linear groups.
Paper Structure (28 sections, 23 theorems, 43 equations)

This paper contains 28 sections, 23 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Relation to other work
  4. The infinite symmetric group
  5. Representation stability
  6. Brauer categories and their representations
  7. Twisted commutative algebras and their applications
  8. Outline
  9. Notation and conventions
  10. Representations of categories of weighted finite sets
  11. Preliminaries
  12. Duality
  13. Simple, projective, and injective modules
  14. Tensor products
  15. Structured $\text{Hom}$ spaces and tensor products
  16. ...and 13 more sections

Key Result

Theorem 1.1

There is an equivalence among the following tensor categories:

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • ...and 33 more