Generic extensions of nilpotent $k[T]$-modules, monoids of partitions and constant terms of Hall polynomials

TL;DR

This paper establishes an isomorphism between the monoid of generic extensions of nilpotent modules over a polynomial ring and the monoid of partitions, and provides a combinatorial algorithm for Hall polynomial constants.

Contribution

It introduces a new isomorphism linking module extensions to partitions and offers an algorithm for computing Hall polynomial constants.

Findings

Monoid of generic extensions is isomorphic to partitions monoid.

A combinatorial algorithm for Hall polynomial constants is developed.

Provides new insights into the structure of nilpotent modules and Hall polynomials.

Abstract

We prove that the monoid of generic extensions of finite dimensional nilpotent $k[T]$-modules is isomorphic to the monoid of partitions (with addition of partitions). Moreover we give a combinatorial algorithm that calculates constant terms of classical Hall polynomials.