On Extensions of Polynomial Functors

TL;DR

This paper develops a formula for calculating extensions of polynomial functors, especially integral ones, using projective resolutions, and provides explicit examples including a non-trivial extension of Divided Cubes by Symmetric Cubes.

Contribution

It introduces a new formula for extensions of polynomial functors and explicitly describes complex extensions like Divided Cubes by Symmetric Cubes.

Findings

Established a formula for extensions based on projective resolutions

Computed explicit examples demonstrating non-trivial extensions

Provided an explicit description of the extension of Divided Cubes by Symmetric Cubes

Abstract

A formula for calculating Extensions of (mainly integral) Polynomial Functors is established, based upon projective resolutions. Sample computations are performed, which, in particular, exhibit a surprising non-trivial extension of Divided Cubes by Symmetric Cubes. An explicit description of the latter is given.