English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study ideals in Hall algebras of monoid representations on pointed setscorresponding to certain conditions on the representations. These conditionsinclude the property that the monoid act via partial permutations, that therepresentation possess a compatible grading, and conditions on the support ofthe module. Quotients by these ideals lead to combinatorial Hopf algebras whichcan be interpreted as Hall algebras of certain sub-categories of modules. Inthe case of the free commutative monoid on n generators, we obtain aco-commutative Hopf algebra structure on $n$-dimensional skew shapes, whoseunderlying associative product amounts to a "stacking" operation on the skewshapes. The primitive elements of this Hopf algebra correspond to connectedskew shapes, and form a graded Lie algebra by anti-symmetrizing the associativeproduct. We interpret this Hopf algebra as the Hall algebra of a certaincategory 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 theHopf algebra of symmetric functions, which corresponds to the case $n=1$.

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