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# Homology, Homotopy and Applications

## Volume 26 (2024)

### Number 1

### On strict polynomial functors with bounded domain

Pages: 87 – 104

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n1.a6

#### Authors

#### Abstract

$\def\Pdn\{\mathcal{P}_{d,n}}$We introduce a new functor category: the category $\Pdn$ of strict polynomial functors of degree $d$ with domain of dimension bounded by $n$. It is equivalent to the category of finite dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to apply the tools available in functor categories to representations of the algebraic group $\mathrm{GL}_n$. We investigate in detail the homological algebra in $\Pdn$ for $d = p$, where $p \gt 0$ is the characteristic of a ground field. We also establish equivalences between certain subcategories of $\Pdn\textrm{’s}$ which resemble the Spanier–Whitehead duality in stable homotopy theory.

#### Keywords

block, Ext-group, polynomial representation, Schur algebra, Schur functor, strict polynomial functor, Spanier–Whitehead duality

#### 2010 Mathematics Subject Classification

16E30, 16E35, 18A25, 20G15

Received 16 August 2022

Received revised 22 December 2022

Accepted 24 January 2023

Published 21 February 2024