# The Hopf algebra of skew shapes, torsion sheaves on A^n/F_1, and ideals in Hall algebras of monoid representations

## TL;DR

This paper explores the structure of Hall algebras associated with monoid representations, leading to new combinatorial Hopf algebras of skew shapes and their geometric interpretation as torsion sheaves on affine space over the field of one element.

## Contribution

It introduces a new class of combinatorial Hopf algebras derived from monoid representation ideals, generalizing symmetric functions to higher dimensions.

## Key findings

- Defined ideals in Hall algebras of monoid representations.
- Constructed a co-commutative Hopf algebra of skew shapes.
- Connected primitive elements to a graded Lie algebra.

## Abstract

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on $n$-dimensional skew shapes, whose underlying associative product amounts to a "stacking" operation on the skew shapes. The primitive elements of this Hopf algebra correspond to connected skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative product. We interpret this Hopf algebra as the Hall algebra of a certain category of coherent torsion sheaves on $\mathbb{A}_{/ \mathbb{F}_1}^n$ supported at the origin, where $\mathbb{F}_1$ denotes the field of one element. This Hopf algebra may be viewed as an $n$-dimensional generalization of the Hopf algebra of symmetric functions, which corresponds to the case $n=1$.