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

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.


Introduction
In Woronowicz's approach ( [Wor87a], [Wor87b], [Wor98]), (compact) "quantum groups" are understood as certain non-commutative spaces, the formal duals of C *algebras, carrying a special Hopf algebra structure, for which a non-commutative version of Pontryagin duality can be proven. Reminiscent of the theorems of Tannaka-Krein and Schur-Weyl, a duality exists between the class of compact quantum groups and a particular class of involutive monoidal linear categories. The finite-dimensional unitary co-representations of a given compact quantum group form such a category, and, conversely, a unique maximal compact quantum group can be reconstructed from any such tensor category ( [Wor87b]).
Banica and Speicher ( [BS09]) showed that sets of points as objects and partitions of finite sets as morphisms with vertical concatenation as composition, horizontal concatenation as monoidal product and reflection as involution provide concrete, combinatorial initial data for such representation categories. Their construction yields compact quantum subgroups of the free orthogonal quantum group O + n introduced by Wang ([Wan95]) as a non-commutative counterpart of the classical group O n of orthogonal real-valued matrices. All examples of compact quantum groups arising in this fashion, the so-called "easy" quantum groups, have since been classified ( [BS09], [BCS09], [Web13], [RW13], and [RW16]).
As Freslon, Tarrago and the second author demonstrated ( [FW16], [TW17a], [TW17b]), Banica and Speicher's approach can be generalized to categories of partitions of sets of two-colored points. In contrast to the uncolored case, here, vertical concatenation of partitions, i.e. the composition of morphisms, is restricted to such partitions with matching colorings of their points. This construction yields combinatorial compact quantum subgroups of the free unitary quantum group U + n , a quantum analogue of the classical unitary group U n also introduced by Wang ( [Wan95]). A collective endeavour to find all such "unitary easy" quantum groups was initiated by Tarrago and the second author in [TW17a] and has since been advanced by Gromada in [Gro18] as well as by the authors in [MW18] and [MW19].
The classification of all unitary easy quantum groups has been approached from several different angles of attack. Tarrago and the second author classified in [TW17a] all non-crossing categories C ⊆ P ○• of two-colored partitions, i.e., C ⊆ N C ○• , and all categories C of two-colored partitions with ∈ C, the so-called group case. In contrast, in [Gro18] Gromada determined all categories C with the property of being globally colorized, meaning ⊗ ∈ C. Lastly, the authors of the present article found all categories C with ⟨∅⟩ ⊆ C ⊆ ⟨ ⟩, i.e. categories of neutral pair partitions, corresponding to easy compact quantum groups G with U + n ⊇ G ⊇ U n , the "unitary half-liberations", in [MW18] and [MW19].
The present article is concerned with non-hyperoctahedral categories of twocolored partitions, i.e. categories C ⊆ P ○• with ∈ C or ∉ C. We define explicitly certain sets of partitions and show that each of them constitutes a nonhyperoctahedral category. See the next section for an overview.
This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. In subsequent articles it will be shown that the categories found in the present article are pairwise distinct and actually constitute all possible non-hyperoctahedral categories. Furthermore, a set of generating partitions for each non-hyperoctahedral category will be determined.
About hyperoctahedral categories of two-colored partitions very little is known for the moment. Note that in the uncolored case categories C ⊆ P with ⊗ ∉ C and ∈ C give rise to quantum subgroups of the free hyperoctahedral quantum group H + n ( [Bic04]), hence the name. Amongst others, Laura Maaßen is currently doing research on the hyperoctahedral case.

Main Result
Many new examples of non-hyperoctahedral categories of two-colored partitions are provided. Roughly, they are determined by combinations of constraints on the (i) sizes of blocks, (ii) coloring of the points, (iii) allowed crossings between blocks of their partitions.
More precisely: The coloring of any two-colored partition p ∈ P ○• induces on the set of points a measure-like structure, the color sum σ p , and a metric-like one, the color distance δ p . Measuring the set of all points yields the total color sum Σ(p).
Let now S ⊆ P ○• be an arbitrary set of two-colored partitions and consider the following data: (1) The set of block sizes: F (S) ∶= { B p ∈ S, B block of p}.
p B α 1 α 2 (6) The set of color distances between any two legs belonging to two crossing blocks It is a subtle question which combinations of conditions on these six quantities define categories of partitions. We clarify it and we obtain a huge variety of new categories of partitions.

Basic Definitions: Partitions
Details on, examples and illustrations of two-colored partitions and their categories can be found in [TW17a]. However, we quickly recall the basics. In addition, certain definitions from [MW18] and [MW19] are given here in greater generality.
After revisiting the fundamental definition of two-colored partitions, specialized language is introduced to describe them, in particular the concepts of orientation, normalized color, color sum and color distance.
3.1. Two-Colored Partitions. A (two-colored) partition is specified by the following three data: 1) two disjoint (possibly empty) finite totally ordered sets R L , the lower row comprising the lower points, and R U , the upper row comprising the upper points, 2) an exhaustive decomposition of the union R L ∪R U , the set of points, into disjoint subsets, the blocks, and 3) a two-valued mapping on R L ∪ R U , the coloration, assigning to every point its color, either • (black ) or ○ (white). We represent partitions pictorially as follows: Moreover, we say that the two colors • and ○ are inverse to each other. A point α which is less than another point β with respect to the total order of that row is said to lie left of β. And, if so, β lies right of α.
The elements of a block are called its legs. If a block contains points from both R L and R U , we speak of a through block. If on the other hand a block is contained in one row, either R L or R U , it is said to be non-through, speaking of lower and upper non-through blocks respectively.
We call a block with just one leg a singleton. Blocks with exactly two legs are pairs. The number of legs of a block is its size. Likewise, the total number of points of R L ∪ R U is the size of the partition. The number of lower points is the length of the lower row and we adopt the same terminology for the upper row. A partition with only pairs is called a pair partition.
We say that the upper and the lower rows are opposite each other. To address points in a specified row we sometimes use the rank of the point in the total order of that row, which we count up from the least element, which ranks 1. To refer to the lower point of rank i, we write ◾ i, and, similarly, ◾ i for the upper point of rank i. The points ◾ i and ◾ i are said to be counterparts of each other. Likewise, a set of points of one given row is said to be the counterpart of the set comprising exactly the points of the same ranks of the opposite row. A through block B is called straight if L ∩ B and L ∩ U are counterparts of each other.
The set of all two-colored partitions is denoted P ○• . The empty partition ∅ is the unique element with empty rows. We denote by P ○• 2 the set of all pair partitions and by P ○• ≤2 the set of all partitions all of whose blocks have sizes one or two. Whenever we deal with multiple partitions at a time, we identify all their points via the total orderings to the extent that this is possible. For example, given p, p ′ ∈ P ○• and a point ◾ i in p, we do not distinguish between the point ◾ i in p and the point ◾ i in p ′ (if p ′ has at least i lower points). And the same goes for subsets of points.
Here is an overview of the graphical representation of partitions: singleton two pairs crossing   1  2  3  4  5  6  7  8  9  10   1  2  3  4  5  6  7  8  9  10 11 four-legged blocks each just for readability two separate blocks 3.2. Orientation and Intervals. While any p ∈ P ○• already comes with total orders ≤ L on its lower row R L and ≤ U on its upper row R U , in many situations it is more advantageous to consider a cyclic order on the entire set R L ∪ R U of its points. This orientation is uniquely determined by the following four conditions: It concurs with ≤ L on R L , but with the inverse of ≤ U on R U , and the maximum of ≤ U succeeds the maximum of ≤ L and likewise the minimum of ≤ L succeeds the minimum of ≤ U .
That means p carries the counter-clockwise orientation.
The terms successor, predecessor and neighbor always refer to the cyclic order. If necessary to avoid confusion between this cyclic order and the total orders ≤ L and ≤ U , the latter two will be referred to as the native orderings.
With respect to the cyclic order it makes sense to speak of intervals ]α, β[ p , ]α, β] p , [α, β[ p and [α, β] p for any two points α, β ∈ R L ∪ R U of p with α ≠ β. Note that intervals of any kind are only defined for distinct limits.
We call a set S of points consecutive if S is empty, an interval or of the forms {α} or (R L ∪ R U ) {α} for some point α.
3.3. Ordered Tuples and Crossings. We can extend the notion of intervals to tuples of more than two points: For n ≥ 3 pairwise distinct points α 1 , . . . , α n in p ∈ P ○• we say that the tuple (α 1 , . . . , α n ) is ordered in p if for all i, j, k ∈ N with i, j, k ≤ n and i < j the set ]α i , α j [ p contains α k if and only if i < k < j. In fact, we can even talk about tuples of pairwise disjoint consecutive sets being ordered.
If p has no crossing blocks, we call p a non-crossing partition. The set of all non-crossing partitions is denoted by N C ○• .
3.4. Normalized Color and Color Sum. Just like the cyclic order is often more convenient than the total orderings of the rows, it is useful to consider besides the original, native coloring of the points a second one: By the normalized color of a point α in p ∈ P ○• we mean simply its native color in case α is a lower point, but the inverse of its native color if α is an upper point.
We call the signed measure σ p on the set P p of all points of p which assigns 1 to normalized ○ and −1 to normalized • the color sum of p. Null sets of σ p are also referred to as neutral.
The color sum Σ(p) ∶= σ p (P p ) of the set P p of all points of p is called the total color sum of p.
3.5. Color Distance. Besides the color sum, a measure-like structure on the points of a partition, the coloring of the partition also induces a metric-like one: Given two points α and β in p ∈ P ○• we call if α ≠ β and α, β have different normalized colors, σ p (]α, β] p ) if α ≠ β and α, β have the same normalized color, the color distance from α to β in p. The map δ p indeed has properties of a "distance".
(a) The first claim is part of the definition of δ p . (b) We can assume α ≠ β. Rewrite the definition of δ p as (c) Again, we can suppose α, β and γ are pairwise different. Otherwise Parts (a) and (b) already prove the claim. Now compute, employing the formula for δ p from the proof of Claim (b), ) . From σ p (]α, β] p ) + σ p (]β, γ] p ) ≡ σ p (]α, γ] p ) mod Σ(p) now follows the claim.

Basic Definitions: Categories of Partitions
The definition of two-colored partitions recalled, we recapitulate the definitions of operations for partitions and of categories. Again, see [TW17a] for more.

Fundamental Operations on
Partitions. For all p, p ′ ∈ P ○• we call the partition which is obtained by appending the rows of p ′ (left to right) to the right of the respective rows of p the tensor product p ⊗ p ′ of (p, p ′ ). Especially, we can write tensor powers like p ⊗n given by p ⊗ . . . ⊗ p with n factors. And we define p ⊗0 ∶= ∅.
The partition which is obtained from p by switching the roles of upper and lower row is called the involution p * of p.
We say that the pairing (p, p ′ ) is composable if the upper row of p and the lower row of p ′ agree in size and coloration if compared according to their total orders both left to right.
For all x ∈ {p, p ′ } let R x,U denote the upper and R x,L the lower row of x. If (p, p ′ ) is composable, then the composition pp ′ of (p, p ′ ) is defined as follows: Its lower row is given by R p,L and its upper row by R p ′ ,U . The blocks of p contained in R p,L and those of p ′ contained in R p ′ ,U are also blocks of pp ′ . Once we identify the points of R p,U and R p ′ ,L we can form the join s of the partitions induced there by p and p ′ . For every block D of s the set is a block of pp ′ unless it is empty. For every set G ⊆ P ○• we write ⟨G⟩ for the smallest category (with respect to ⊆) which contains G. We say that G generates ⟨G⟩. If G = {p} for some p ∈ P ○• , we slightly abuse notation by writing ⟨p⟩ instead of ⟨{p}⟩ for ⟨G⟩. Also, we mix the two, writing, ⟨G, q⟩ for q ∈ P ○• instead of ⟨G ∪ {p}⟩.
In this series of articles, we are only interested in the first three cases; Compare with the classification in the uncolored case, where also the corresponding Case H, i.e. ⊗ ∉ C or ∈ C for categories C ⊆ P, is most complex ( [RW13] and [RW16]).
Definition 4.2. Let C ⊆ P ○• be a category.
(a) Equivalently to saying C is case H, we also call C hyperoctahedral.
The set of all non-hyperoctahedral categories is denoted by PCat ○• NHO . 4.3. Composite Category Operations. The basic category operations can be utilized to construct further generic transformations.
Categories are closed under four basic rotations: We obtain the partition p ⤹ from p ∈ P ○• by removing the leftmost upper point α and adding a new point β to the left of the leftmost lower point, assigning to β the inverse color of α and also replacing α by β as far as the blocks are concerned. We say that α is rotated down. Similarly, we can rotate down the rightmost upper point by inverting its color and appending it to the lower row, producing p ⤸ . And the rotations p and p result form reversing these procedures and rotating up the leftmost respectively rightmost point. We call p ↻ ∶= (p ) ⤸ the clockwise and p ↺ ∶= (p ⤹ ) the counter-clockwise cyclic rotation of p.
From p we obtain the reflectionp by reversing the native total orders on both rows. The color inversion p of p is constructed by inverting the native coloring. And the verticolor reflectionp is the color inversion of the reflection of p. Categories are closed under verticolor reflection but generally neither under reflection nor color inversion.

p ↦p
Lastly, categories are closed under erasing turns: A turn is a consecutive neutral set of size two. The erasing of a set S of points from p is the partition E(p, S) obtained by removing S and combining all the points whose blocks had a non-empty intersection with S into one block.  Proof. Suppose C satisfies (i) and (ii). Since ⤸ = , since = and since ⤸ = and because C is closed under rotations, we find , , ∈ C. Erasing the only turn in ∈ C, under which C is invariant, produces ∅ ∈ C.
For every partition p ∈ C with k upper and m lower points, the identity p * = ((p) k ) ⤸l and the assumptions that C is stable under rotations and verticolor reflection proves that p * ∈ C. Hence, C is also involution-invariant.
Lastly, suppose that (p, q) is a composable pairing from C. We want to show r ∶= pq ∈ C. Let q have k upper and l lower and let p have l upper and m lower points. Since C is closed under rotations, it suffices to prove r m ∈ C. Let (c 1 , . . . , c l ) for c 1 , . . . , c l ∈ {○, •} be the coloring of the lower row of q left to right. Let s be the tensor product of partitions from { , } with lower row of coloring (c 1 , . . . , c l ). Then, (p, s) and (s, q) are composable and psq = pq = r. The diagram below illustrates that the pairing (s l , q ⊗ ((p ⤹l ) m )) is composable as well and that its composition yields the partition r m .
Our assumptions guarantee e 0 ∶= q ⊗((p ⤹l ) m ) ∈ C. Because C is assumed invariant under erasing turns, if we define the turn T 0 ∶= { ◾ l, ◾ (l + 1)} in e 0 ∈ C and then for every j = 1, . . . , l − 1 the turn then the partition e l ∶= E(e l−1 , T l−1 ) ∈ C is identical with r m as the diagram below shows.
Thus we have deduced r m ∈ C as claimed. This proves that C is closed under composition of composable pairs and thus a category.

The Sets R Q
In the following, we will define in several steps an index set Q and for each Q ∈ Q a set R Q ⊆ P ○• of partitions. The aim will be to show that each of these constitutes a non-hyperoctahedral category (see Theorem 6.16, the main result of this article). Auxiliary objects L and Z aid in defining Q and (R Q ) Q∈Q .
Notation 5.1. For every set S denote its power set by P(S).
Definition 5.2. We define the parameter domain L as the six-fold Cartesian product of P(Z): is the set of color distances between any two subsequent legs of the same block having the same normalized color, is the set of color distances between any two subsequent legs of the same block having different normalized colors and is the set of color distances between any two legs belonging to two crossing blocks.
The parameter domain L and the analyzer Z allow us to now define the following map which will later induce the announced family (R Q ) Q∈Q .
Notation 5.4. Given a family (S i ) i∈I of sets, we write ≤ for the product order on the Cartesian product ⨉ i∈I P(S i ) induced by the partial orders ⊆ on the factors.
Definition 5.5. Define the parametrization as the mapping With R we can single out sets of partitions by placing restrictions on the six aggregated combinatorial features of partitions listed above. Definition 5.7. Define the parameter range Q as the subset of L comprising exactly all tuples With Q and R defined, so has been the family (R Q ) Q∈Q . The characterizing conditions Z({ ⋅ }) ≤ Q of the sets R Q , Q ∈ Q, will be successively explained in Section 6 in the process of proving their invariance under the category operations.

Invariance of R Q under the Category Operations
The strategy for proving that the sets defined in the preceding Section 5 are actually categories of partitions is the following: We choose the most convenient elements Q of Q, the ones for which it is easiest to prove that R Q is a category. Once we have verified that these sets R Q are categories, we show that every other Q ∈ Q can be written as a meet of a suitable family Q ′ ⊆ Q of convenient ones in the complete lattice L. Then, the following Lemma 6.2 will allow us to conclude that R Q is a category for every Q ∈ Q.
Notation 6.1. Given a family (S i ) i∈I of sets, we use the symbols ⋂ × for the meet and ⋃ × for the join operator of the product order ≤ with respect to ⊆ on ⨉ i∈I P(S i ).
Lemma 6.2. The mapping R ∶ L → P(P ○• ) is monotonic and preserves meets.
where we have only used the definition of R.
Proof. Let ρ be the map that rotates the points of p to those of p r .
Step 1: Relating σ p r to σ p . Because the normalized color is defined specially to compensate for the color change involved in rotations, for all sets S of points of p r .
Step 2: Relating δ p r to δ p . The cyclic order is also defined exactly to be respected by rotations: For all points α and β of p r , the identity ]α, β] p r = ρ(]ρ −1 (α), ρ −1 (β)] p ), Equation (1) and the definition of δ p imply (2) Step 3: Implications for Z. By definition, the blocks of p r are precisely the sets ρ(B) for blocks B of p. Hence, F ({p r }) = F ({p}) as ρ is bijective. But also therefore, V ({p r }) = V ({p}) by Equation (1). The same equation also shows Σ(p r ) = Σ(p) by choosing for S the total set of points.
Not only does ρ map blocks to blocks but also subsequent legs to subsequent legs because rotation respects the cyclic order. And this is how all pairs of subsequent legs in p r arise. And Equation (1) shows that two subsequent legs α and β in p r satisfy σ p r ({α, β}) = 0 if and only if their preimages, subsequent legs in p, satisfy Since ρ maps crossing blocks to crossing blocks and since all crossings of p r come from crossings in p, Equation (2)  Proof. Same procedure as in Lemma 6.3. Let κ be the map that reflects the points of p.
Step 1: Relating σp to σ p . Reflection only moves points and does not change their native colors. As it sends upper points to upper points and lower points to lower points, the normalized colors are unaffected too. Verticolor reflection however involves not only reflection but also an inversion of (native and thus also normalized) color. We hence find for all sets of S of points ofp: (3) Step 2: Relating δp to δ p . The reflection operation reverses the cyclic order. Let α and β be arbitrary points ofp.
Note that the left-open interval has become a right-open one. Hence, by definition of δ p and by Equation (3), In short, δp is is given by flipping the arguments in δ p and inverting the sign.
Step Lemma 6.5. Let p, p ′ ∈ C be arbitrary.
Proof. One last time we proceed as in Lemmata 6.3 and 6.4. Let S p and S ′ p be the sets of points in p ⊗ p ′ stemming from p and p ′ respectively, and let τ denote the map translating the points of p ′ to their locations in S p ′ .
Step 1: Relating σ p⊗p ′ to σ p and σ p ′ . The tensor product leaves S p untouched, it only involves a shift of τ −1 (S p ′ ) to S p ′ . The native colors are not affected. Since τ maps upper points to upper points and lower points to lower points, the normalized colors do not change either. We will be able to confine our considerations to sets S of points with S ⊆ S p or S ⊆ S p ′ , for which then holds in conclusion: Step 2: Relating δ p⊗p ′ to δ p and δ p ′ . Again, we are only interested in the color distances of points α and β which both belong to S p or both to S p ′ . Then, Hence, Σ(p) = σ p (S p ) and Σ(p ′ ) = σ p ′ (τ −1 (S p ′ )), Equation (5) and the definition of δ p⊗p ′ yield in these cases if α, β ∈ S p ′ and ]α, β[ p⊗p ′ ∩S p = ∅.
In particular, it follows Step 3: Implications for Z. The blocks of p ⊗ p ′ are by definition precisely the blocks of p plus the sets τ (B ′ ) for blocks B ′ of p ′ . This proves Part (a). Especially, B ⊆ S p or B ⊆ S p ′ for all blocks B of p⊗p ′ . Hence, Equation (5) shows Part (b). The same equation shows Σ(p⊗p ′ ) = σ p⊗p ′ (S p ∪S p ′ ) = σ p (S p )+σ p ′ (τ −1 (S p ′ )) = Σ(p)+Σ(p ′ ) and thus Claim (c).
Since every block of B of p ′ satisfies B ⊆ S p or B ⊆ S p ′ any two subsequent legs of a block B of p ′ are also subsequent legs in the block of p or p ′ where they originate. Moreover, Equation (5) shows that their combined color sum in p ⊗ p ′ is identical to the one in p respectively p ′ . Thus, Equation (6) proves Part (d) for Y ∈ {L, K}.
Lastly, if B 1 and B 2 are crossing blocks of p ⊗ p ′ , then necessarily B 1 , B 2 ⊆ S p or B 1 , B 2 ⊆ S p ′ by definition of the tensor product: Indeed, if we suppose, e.g., B 1 ⊆ S p and B 2 ⊆ S p ′ , then a crossing of B 1 and B 2 would mean a crossing between S p and S p ′ , which is impossible since S p and S p ′ are disjoint consecutive sets. Hence, Equation (6) also proves Part (d) for Y = X, which completes the proof. 6.1.4. Behavior of Z under Erasing Turns. We have not understood yet how Z behaves under erasing turns. However, the effect of the erasing operation on Z is too complicated to treat well abstractly beyond the following simple (but important) observations. Lemma 6.6. Let p ∈ P ○• be a partition with no upper points and let T be a turn comprising exactly the two rightmost lower points of p.
6.2. Total Color Sum. The first family of elements Q ∈ Q is chosen such that all components except the one for Σ are trivial. That means we restrict the allowed total color sums of partitions. Proof. We use the Alternative Characterization (Lemma 4.3) to show that R Q is a category. By Remark 6.7 it suffices to show that Σ({ ⋅ }) ⊆ mZ is stable under the alternative category operations.
Erasing: Lastly, it remains to consider p ∈ R Q and a turn T in p and to show Σ(E(p, T )) ∈ mZ. Because we already know Σ({ ⋅ }) ⊆ mZ is respected by rotations, we can assume that p has no upper points and that T comprises exactly the two rightmost lower points. Let P p denote the set of all points of p. Since T is a turn, σ p (T ) = 0. Hence, by Lemma 6.6 (b), That concludes the proof.
6.3. Block Size and Color Sum. A second subset of Q gathers elements where only the components for F and V are non-trivial. So, we only impose (interdependent) conditions on the sizes and color sums of blocks. In contrast to the two previous examples, the constraint on block color sums is not merely a consequence of the range of allowed block sizes. The F -constraint restricts R Q to a subset of P ○• 2 and the V -constraint prohibits non-neutral blocks.
And also the non-trivial L-component ∅ of Q provides no additional constraints beyond the ones induced by the F -and V -components: A pair block which is neutral, i.e. has normalized coloring ○• or •○, cannot have subsequent legs of the same normalized color. Thus, for every p ∈ P ○• 2 with V ({p}) = {0} automatically holds L({p}) = ∅.
(d) Lastly, the set R Q where Q = ({1, 2}, ±{0, 1}, Z, ∅, Z, Z) (Part (d)) is given exactly by the set of all partitions all of whose blocks have at most two legs and all of whose pair blocks are neutral.
As said before, restricting F to {1, 2} reduces the allowed set of block color sums to ±{0, 1, 2} and the values −2 and 2 can then only stem from pair blocks. Excluding these two values then forces all pair blocks to be neutral. So, as in Part (c), the F -and V -components induce constraints in their own right.
However, the non-trivial value ∅ of the L-component of Q is merely a reflection of the F -and V -constraints because neutral pair blocks cannot have subsequent legs of the same normalized color.
We use the Alternative Characterization (Lemma 4.3) to show that R Q is a category. As ∈ R Q is clearly true, that means it suffices to prove that in Cases (a) Verticolor reflection: Given arbitrary p ∈ R Q we are guaranteed by Lemma 6.4 Tensor product: According to Lemma 6.5 (a) and (b), for all partitions p, Erasing: Lastly, let p ∈ R Q be arbitrary and let T be a turn in p. We have to show F ({E(p, T )}) ⊆ f Q and, but just in Cases (c) and (d), also V ({E(p, T )}) ⊆ v Q . Since we already know that we can assume that p has no upper points and that T comprises exactly the two rightmost lower points of p.
Let B be an arbitrary block of E(p, T ). What we need to prove is B ∈ f Q as well as σ E(p,T ) (B) ∈ v Q , the latter however just in Cases (c) and (d). If B is in fact a block of p, then B ∈ f Q and, by Lemma 6.6 (b), also σ E(p,T ) (B) = σ p (B) ∈ v Q since we have assumed p ∈ R Q . Thus, suppose that B is not a block of p. Then, there are (not necessarily distinct) blocks B 1 and B 2 of p with Step 1: Block size. We prove B ∈ f Q . From T = 2 and from T ⊆ B 1 ∪ B 2 follows By definition, f Q = {1, 2} or f Q = {2}. We treat the two cases individually. Step 2: Block color sum. We show σ p (B) ∈ v Q in Cases (c) and (d). Since σ p (T ) = 0 and T ⊆ B 1 ∪ B 2 , we infer by Lemma 6.6 (b):  1, 2}, ±{0, 1}). Then, B 1 and B 2 are singletons or pair blocks and σ p (B 1 ), σ p (B 2 ) ∈ ±{0, 1}. If both B 1 and B 2 were singletons, then B 1 ∩ T, B 2 ∩ T ≠ ∅ and T ⊆ B 1 ∪ B 2 would imply B 1 ∪ B 2 = T and thus the contradiction B = (B 1 ∪ B 2 ) T = ∅. Hence, we know that at least one of the blocks B 1 and B 2 must be a pair. The assumptions p ∈ R Q and v Q = ±{0, 1} force the pair blocks of p to be neutral (since non-neutral pair blocks have color sums 2 or −2). It follows σ p (B 1 ) = 0 or σ p (B 2 ) = 0. Now, no matter whether B 1 = B 2 or B 1 ≠ B 2 , Equation (9) proves σ E(p,T ) (B) ∈ ±{0, 1} = v Q , which concludes the proof. 6.4. Color Distances between Legs of the Same Block. For our third family of elements of Q the L-and K-components are non-trivial, implying constraints on the color distances between subsequent legs of the same block. In part, this translates to a condition on the color distances between arbitrary -not just subsequent -legs. Proof. Suppose δ p (α, β) ∈ mZ for all blocks B of p and all α, β ∈ B. Since we can especially choose α ≠ β and ]α, β[ p ∩B = ∅, it follows (L, K)({p}) ≤ (mZ, mZ). On the other hand, picking α = β shows Σ(p) = δ p (α, α) ∈ mZ, which lets us conclude Σ({p}) ⊆ mZ. As (F, V, X)({p}) ≤ (N, Z, Z) is trivially true, that proves one implication.
(a) For every m ∈ {0}∪N, if Q = (N, Z, mZ, mZ, mZ, Z) (Part (a) of Lemma 6.13 below), then R Q is given by the set of all partitions such that the color distance between any two legs of the same block is a multiple of m, as seen in Lemma 6.11. Mostly, it is the L-and K-components of Q inducing the characteristic constraints via Z({ ⋅ }) ≤ Q. The F -, V -and X-conditions are redundant. However, the Σ-condition is not. And it is not implied by the L-and K-constraints, either.
then the set R Q is a subset of the set from Part (a): Still, a partition p ∈ P ○• is required to have color distances in mZ between any two legs of the same block if it is to be an element of R Q . However, now, additionally, all blocks of p must have size one or two, the total color sum Σ(p) must be a multiple of 2m (and not just m) and, most importantly, the color distances between subsequent legs of the same block must satisfy two different conditions, depending on their normalized colors. Since blocks can have size two at most, two legs of the same block are subsequent if and only if they are distinct. Hence, R Q is the set of all partitions such that ◾ every block has at most two legs, ◾ the total color sum is an even multiple of m, ◾ the color distance between any two distinct legs of the same block is -an odd multiple of m if they have identical normalized colors and -an even multiple of m if they have different normalized colors. Also, note that, in contrast to Part (a), the parameter m = 0 is not allowed.
For points α and β in p ∈ P ○• , saying is a helpful technical way of expressing that the distance of the points is an odd multiple of m if they have the same normalized color and an even multiple otherwise. The constraints induced via Z({ ⋅}) ≤ Q by the F -, V -and Σ-components are non-trivial and not implied by the L-and K-restrictions. Of course, the X-condition is still redundant. Tensor product: Let p, p ′ ∈ R Q be arbitrary. Then, in particular Σ(p), Σ(p ′ ) ∈ k Q . Since k Q is a subgroup of Z we conclude gcd(Σ(p), Σ(p ′ ))Z ⊆ k Q . Therefore, In conclusion, (L, K)({ ⋅ }) ≤ (l Q , k Q ) is respected by tensor products.
Erasing: Lastly, we let p ∈ R Q have a turn T and show (L, K)({E(p, T )}) ≤ (l Q , k Q ). Because we have already shown invariance under rotation, we can suppose that p has no upper points and that T comprises exactly the rightmost two lower points of p. Let α and β be legs of the same block B of E(p, T ) such that α ≠ β and such that ]α, β[ E(p,T ) ∩B = ∅. What we then have to prove is that δ E(p,T ) (α, β) ∈ l Q if σ E(p,T ) ({α, β}) ≠ 0 and that δ E(p,T ) (α, β) ∈ k Q if σ E(p,T ) ({α, β}) = 0. By Lemma 6.6 (b) and (c) it suffices to show To prove (10) we now distinguish the two Cases (a) and (b). Case 1: Part (a). Since l Q = k Q = mZ, Assertion (10) simplifies to the claim Because Q is as required by Lemma 6.11, we can immediately deduce (11) from p ∈ R Q if α and β belong to the same block in p. Thus, it remains to assume that the blocks B 1 of α and B 2 of β in p are distinct and prove (11) under this premise.
Step 1.1: Decomposing δ p (α, β). Because B 1 ≠ B 2 , the definition of E(p, T ) requires the existence of α ′ ∈ B 1 ∩ T and β ′ ∈ B 2 ∩ T . Lemma 3.1 (c) yields where we have used the consequence Σ(p) ∈ mZ of p ∈ R Q . By this congruence, Assertion (11) can be seen by proving that all summands on the right hand side are multiples of m. Hence, that is what we now show.
Case 2: Part (b). In this case, l Q = m+2mZ and k Q = 2mZ mean that Assertion (10) can be expressed equivalently as If B is a block of p, then p ∈ R Q implies (12) immediately. Hence, we only need to prove (12) for the case that B is not a block of p. We again want to use Lemma 3.1 (c). Hence, we must find a suitable choice of points α ′ and β ′ to decompose δ p (α, β) with.
Step 2.1: Finding points α ′ and β ′ . As B is not a block of p, there exist blocks B 1 and B 2 of p with B 1 ∩ T, B 2 ∩ T ≠ ∅, with T ⊆ B 1 ∪ B 2 and with B = (B 1 ∪ B 2 ) T . In particular, α, β ∈ B 1 ∪ B 2 .
Step 2.3: Color distances of α and α ′ and of β and β ′ . The assumption p ∈ R Q furthermore guarantees because α, α ′ ∈ B α and α ≠ α ′ on the one hand and β ′ , β ∈ B β and β ′ ≠ β on the other hand.
With the proof of (12) thus complete, so is the proof overall.
6.5. Color Distances between Legs of Crossing Blocks. The last family of elements of Q we treat exhibits non-trivial X-components.
Remark 6.14. If Q = (N, Z, mZ, mZ, mZ, Z E) for some m ∈ {0} ∪ N and some E ⊆ Z with E = −E = E +mZ (as in Lemma 6.15 below), we can employ Lemma 6.11 to understand the set R Q . It is given by the set of all partitions such that the color distance between any two legs of the same block is a multiple of m and such that two points belong to non-crossing blocks whenever the color distance between them is an element of E.
While the F -and V -components of Q, of course, effectively induce no conditions at all via Z({ ⋅ }) ≤ Q, the Σ-, L-and K-conditions are non-trivial and they are not implied by the X-constraint. Proof. Let Q be of the kind described in the claim. Once more, we use the Alternative Characterization (Lemma 4.3) to show that R Q is a category. By Lemma 6.13 (a) it suffices to consider the X-component of Z, i.e. to prove that the constraint X({ ⋅ }) ⊆ Z E is invariant under rotation, tensor products, verticolor reflection and erasing turns.
Rotation: With the help of Lemma 6.3 we can infer X({p r }) = X({p}) ⊆ Z E for all p ∈ R Q and all r ∈ {⤹, , , ⤸}. Thus, X({ ⋅ }) ⊆ Z E is preserved by rotations.
Verticolor reflection: For all p ∈ R Q holds X({p}) = −X({p}) ⊆ −Z E by Lemma 6.4. Since our assumption E = −E implies −Z E = Z E, this proves the condition X({ ⋅ }) ⊆ Z E stable under verticolor reflection.
Erasing: Let p ∈ R Q have a turn T . We show X({E(p, T )}) ⊆ Z E. Since X({ ⋅ }) ⊆ Z E is preserved by rotations, no generality is lost assuming that p has no upper points and that the two rightmost lower points of p make up T . Let α 1 and α 2 be points in E(p, T ) whose blocks B 1 and B 2 cross in E(p, T ). By Lemma 6.6 (c) all we have to show is If B 1 and B 2 are both blocks of p as well, then (16) is true by the assumption p ∈ R Q . Thus, we only need to show (16) in the opposite case. If so, then, by nature of the erasing operation, exactly one of the blocks B 1 and B 2 is also a block of p.
Step 3: Showing that B 1,i and B 2 cross in p. If we can establish that B 1,1 and B 2 or B 1,2 and B 2 cross each other in p, then Equation (17) proves (16) as Z E = (Z E) + mZ. So, this is all we have to show.
Theorem 6.16. For every Q ∈ Q the set R Q is a non-hyperoctahedral category: Proof. For every family Q ′ ⊆ Q holds by Lemma 6.2 We show that for every Q ∈ Q there exists a set Q ′ ⊆ Q such that Q = ⋂ × Q ′ and such that R Q ′ is a category for every Q ′ ∈ Q ′ . As PCat ○• is a complete lattice, that then proves that R Q is a category of two-colored partitions for every Q ∈ Q. It is straightforward to check that indeed ⊗ ∈ R Q or ∉ R Q for every Q ∈ Q. For this proof only, we use names for specific elements of the set Q: The below table applies for all m ∈ {0} ∪ N and E ⊆ Z with E = −E = E + mZ.
The corresponding sets R Q for all these elements Q ∈ Q are categories of partitions, as shown by the respective lemma cited in the last column. The ensuing table lists how every element of Q can be written as a meet in L of the specific elements defined above. Here, u ∈ {0} ∪ N, m ∈ N, D ⊆ {0} ∪ ⟦⌊ m 2 ⌋⟧, E ⊆ {0} ∪ N and N is a subsemigroup of (N, +).
That concludes the proof.

Further Categories
Although we are only interested in non-hyperoctahedral categories in this article, it deserves pointing out that the proof Lemma 6.10 also shows the existence of certain hyperoctahedral categories.
Proof. We use the Alternative Characterization (Lemma 4.3) of categories. Lemmata 6.3, 6.4 and 6.5 (b) imply that V ({ ⋅ }) ⊆ gZ is stable under rotations, verticolor reflection and tensor products. While verifying Lemma 6.10 we showed that for every p ∈ P ○• , every turn T in p and every block B of E(p, T ) the following is true: Either B is a block of p and then σ E(p,T ) (B) = σ p (B) or B is not a block of p and then there are blocks B 1 and B 2 of p such that σ E(p,T ) (B) = σ p (B 1 ) + σ p (B 2 ) or σ E(p,T ) (B) = σ p (B 1 ). Thus, V ({E(p, T )}) ⊆ V ({p}) ∪ (V ({p}) + V ({p})), which proves the claim.