Non-Hyperoctahedral Categories of Two-Colored Partitions, Part I: New Categories

TL;DR

This paper explores new combinatorial categories of two-colored partitions related to quantum groups, focusing on non-hyperoctahedral classes, and aims to classify all such categories through combinatorial methods.

Contribution

It introduces numerous new non-hyperoctahedral categories of two-colored partitions defined by block size, coloring, and non-crossing conditions, advancing classification efforts.

Findings

Many new categories of two-colored partitions are constructed.

The categories are characterized by combinatorial properties like block size and coloring.

The work sets the stage for a complete classification of non-hyperoctahedral categories.

Abstract

Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum groups these categories arise from a combinatorial structure: Rows of two-colored points form the objects, partitions of two such rows the morphisms; vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes $\mathcal{O}$, $\mathcal{B}$, $\mathcal{S}$ and $\mathcal{H}$ of such categories (inspired respectively by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups) we treat the first three -- the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. The article is purely combinatorial in nature; The quantum group aspects are left out.