Categories of two-colored pair partitions part I: categories indexed by cyclic groups

Abstract

Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to "half-liberated easy" interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.

Quantum groups under very strong axioms

“…Skipping the details here, let us mention that the problem is solved by [26] for the upper and lower faces, is elementary as well for the front and right face, using the results from [12], [26], and is still in need of some non-trivial combinatorial work, based on the results in [23], [24], [25], in what regards the left face and the bottom face. This is another interesting direction.…”

Non-hyperoctahedral Categories of Two-Colored Partitions Part II: All Possible Parameter Values

“…We introduce an equivalence relation on pairs of partitions and consecutive sets therein by which to compare partitions locally (cf. [3,Definition 6.2]). Definition 3.1 For all i ∈ {1, 2}, let P p i denote the set of all points of p i ∈ P •• and let S i ⊆ P p i be consecutive.…”

“…In truth, of course, for any consecutive set S in p ∈ P •• the projection P( p, S) depends only on the equivalence class of ( p, S). The following lemma constitutes a generalization of [3,Lemma 6.4].…”

“…Step 1.1: Recall from [3,Definition 4.1] that by S 0 we denote the set of all p ∈ P •• 2 with σ p (B) = 0 and δ p (α, β) = 0 for all blocks B of p and all α, β ∈ B. We justify that it suffices to prove…”

“…Since the classification program for categories of two-colored partitions was begun, different subclasses have been indexed by various contributors (see [2][3][4]6]). The present article is the second part of a series aiming to determine and describe all so-called non-hyperoctahedral categories, i.e., all categories C ⊆ P •• with ⊗ ∈ C or / ∈ C. In this regard the first article [5] and the present one pursue complementary approaches to detecting whether a given set of partitions is a non-hyperoctahedral category: Part I gave sufficient conditions for being a non-hyperoctahedral category, Part II now provides necessary ones.…”