On homological properties of strict polynomial functors of degree p

TL;DR

This paper explores the homological algebra of strict polynomial functors of degree p, revealing their Ext-structures, decomposition matrices, and categorical properties, including formality and derived equivalences.

Contribution

It provides explicit computations of Ext-groups, establishes the Kazhdan-Lusztig theory for the category, and generalizes results to blocks of p-weight 1, advancing understanding of polynomial functor categories.

Findings

Determined the decomposition matrix of the category

Computed Ext-algebras of simple and Schur functors

Established the formality of DG algebras and derived categories

Abstract

We study the homological algebra in the category $\mathcal{P}_p$ of strict polynomial functors of degree $p$ over a field of positive characteristic $p$. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category $\mathcal{P}_p$ has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of $\mathcal{P}_p$ as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of $p$-weight $1$ in $\mathcal{P}_e$ for $e>p.$