On strict polynomial functors with bounded domain

$\def\Pdn\{\mathcal{P}_{d,n}}$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

Full Text (PDF format)

Received 16 August 2022

Received revised 22 December 2022

Accepted 24 January 2023

Published 21 February 2024