A type B analogue of the category of finite sets with surjections

TL;DR

This paper introduces a type B analogue of the finite sets with surjections category, explores its representation theory, and reveals structural properties and restrictions on modules related to type B Coxeter groups.

Contribution

It defines a new category analogous to finite sets with surjections for type B and analyzes its representation theory, including properties of finitely generated modules.

Findings

Opposite category is quasi-Grobner, ensuring submodules of finitely generated modules are finitely generated.

Generating functions of finitely generated modules have specific pole structures.

Restrictions on type B Coxeter group representations in these modules.

Abstract

We define a type B analogue of the category of finite sets with surjections, and we study the representation theory of this category. We show that the opposite category is quasi-Grobner, which implies that submodules of finitely generated modules are again finitely generated. We prove that the generating functions of finitely generated modules have certain prescribed poles, and we obtain restrictions on the representations of type B Coxeter groups that can appear in such modules. Our main example is a module that categorifies the degree i Kazhdan-Lusztig coefficients of type B Coxeter arrangements.