aa r X i v : . [ m a t h . QA ] N ov FREE QUANTUM ANALOGUE OF COXETER GROUP D DANIEL GROMADA
Abstract.
We define the quantum group D +4 – a free quantum version ofthe demihyperoctahedral group D (the smallest representative of the Coxeterseries D ). In order to do so, we construct a free analogue of the propertythat a 4 × N = 4. The free D +4 is then defined by imposingthis generalized determinant condition on the free hyperoctahedral group H +4 .Moreover, we give a detailed combinatorial description of the representationcategory of D +4 . Introduction
Quantum groups constitute a generalization of the concept of a group in non-commutative geometry. In this work, we deal with compact quantum groups asdefined by Woronowicz in [Wor87]. Many examples of quantum groups arise bydeforming the commutativity relation in the associated Hopf ∗ -algebra O ( G ) (inparticular, we have the q -deformations such as U qN , SU qN , O qN , SO qN and so on),other examples of quantum groups are defined by liberating the commutativity re-lations. Here the canonical examples are the so-called free quantum groups such asthe free orthogonal group O + N , the free symmetric group S + N or the free hyperocta-hedral group H + N .Although we have a free analogue of O N , we have no free analogue of SO N ;although we have the free symmetric group, we have no free alternating group; andalthough we have the free analogues for the Coxeter series A (the symmetric group), B and C (the hyperoctahedral group), we have so far no free analogue of the Cox-eter groups of type D . All the mentioned examples, where the definition of a freecounterpart is missing, have one thing in common: Classically, as matrix groups,they can be obtained as normal subgroups imposing some kind of a determinantcondition det M = 1 (which indeed always defines a subgroup since determinantis multiplicative; in other words, it is a one dimensional representation). So, thereason what makes it hard to find a suitable free analogues for these groups is thatthere is no free determinant. More precisely, there is no non-trivial one-dimensionalrepresentation of the free quantum groups such as O + N , H + N or S + N (see e.g. [Fre17, Date : November 30, 2020.2010
Mathematics Subject Classification. D . I thank him also for reading my work and providing valuable com-ments. I would also like to thank Simon Schmidt for discussions regarding quantum symmetriesof demihypercubes and folded hypercubes. Thm. 5.1.1]). Although it is not possible to find some one-dimensional representa-tion of the free quantum groups being the free analogue of the determinant, it stillcan make sense just to look for a free analogue of the relations det = 1.Let G be a group and G + a quantum group. We say that G + is a liberated version of G and that G is a classical version of G + if G is a quantum subgroup of G + givenby imposing commutativity in the associated Hopf algebra. Now we would like tosay that G + is a free quantum group if O ( G ) contains no commutativity relationsat all. It is not entirely clear, how precisely the formal definition of this statementshould look like. In case of homogeneous groups (i.e. S N ⊆ G ), a natural conditionis to require that the representation category of G + can be described by some linearcategory of non-crossing partitions. In this article, we define a quantum group D +4 ,which forms a free analogue of the Coxeter group of type D of size N = 4 in thisabove mentioned sense. This is the first main result of this work formulated inSection 2: Theorem A (Theorem 2.12) . We have D ( D +4 ( H +4 . The subgroup D is obtained from D +4 imposing commutativity or ( − -commutativityon the entries of the fundamental representation. In Sections 3 and 4, we are looking for a combinatorial description of the rep-resentation category of this new quantum group D +4 . In the following theorem,we summarize the main results. The strategy and goals for those two sections aresummarized more in detail in the beginning of Section 3. We use the framework ofbilabelled graphs recently introduced in [MR19], see Sect. 3.1 for a summary. Theorem B (Propositions 3.20, 4.1, Theorem 4.3) . For every k, l ∈ N , let C ( k, l ) be the set of all bilabelled graphs K = ( K, a , b ) that satisfy the following conditions (i) K is planar, (ii) ˜ d v is even for all v ∈ V ( K ) , ( ˜ d v is the degree counting also the outputstrings, see Def. 3.15) (iii) K is bipartite and all the vertices a , . . . , a k , b , . . . , b l are elements of oneof the parts. (iv) There is no vertex v with d v = ˜ d v = 2 . (v) There are no multiple edges. (vi)
Every component of K has at least one vertex among ( a , b ) .Then the following holds: (1) The sets C ( k, l ) are finite for every k, l . (2) They model the representation theory of D +4 through the functor T A , where A is a matrix given by A ij = δ ij − / . That is, if we denote by u thefundamental representation of D +4 , then Mor( u ⊗ k , u ⊗ l ) = span { T A K | K ∈ C ( k, l ) } . The set C is not closed under the graph category operations introduced in[MR19]. Nevertheless, we can make it into a category by slightly modifying thoseoperations (taking certain quotient essentially). In this way, we obtain a very in-teresting diagram category, which is worth studying on its own. In Section 5, wesuggest some directions for further research and state some open problems regardingthis category. REE QUANTUM ANALOGUE OF COXETER GROUP D Preliminaries
In this section we recall the basic notions of compact matrix quantum groupsand Tannaka–Krein duality. For a more detailed introduction, we refer to themonographs [Tim08, NT13].1.1.
Compact matrix quantum groups. A compact matrix quantum group is apair G = ( A, u ), where A is a ∗ -algebra and u = ( u ij ) ∈ M N ( A ) is a matrix withvalues in A such that(1) the elements u ij i, j = 1 , . . . , N generate A ,(2) the matrices u and u t = ( u ji ) are similar to unitary matrices,(3) the map ∆ : A → A ⊗ A defined as ∆( u ij ) := P Nk =1 u ik ⊗ u kj extends toa ∗ -homomorphism.Compact matrix quantum groups introduced by Woronowicz [Wor87] are gen-eralizations of compact matrix groups in the following sense. For a matrix group G ⊆ M N ( C ), we define u ij : G → C to be the coordinate functions u ij ( g ) := g ij .Then we define the coordinate algebra A := O ( G ) to be the algebra generated by u ij . The pair ( A, u ) then forms a compact matrix quantum group. The so-called comultiplication ∆ : O ( G ) → O ( G ) ⊗ O ( G ) dualizes matrix multiplication on G :∆( f )( g, h ) = f ( gh ) for f ∈ O ( G ) and g, h ∈ G .Therefore, for a general compact matrix quantum group G = ( A, u ), the al-gebra A should be seen as the algebra of non-commutative functions defined onsome non-commutative compact underlying space. For this reason, we often denote A = O ( G ) even if A is not commutative. We can actually define the structureof a Hopf ∗ -algebra on A . The matrix u is called the fundamental representation of G . Let us note that compact matrix quantum groups are special cases of compactquantum groups, see [NT13, Tim08] for details.A compact matrix quantum group H = ( O ( H ) , v ) is a quantum subgroup of G = ( O ( G ) , u ), denoted as H ⊆ G , if u and v have the same size and there isa surjective ∗ -homomorphism ϕ : O ( G ) → O ( H ) sending u ij v ij . We say that G and H are identical if there exists such a ∗ -isomorphism (i.e. if G ⊆ H and H ⊆ G ). If there is a ∗ -isomorphism mapping u ij [ T vT − ] ij for some invertiblematrix T ∈ GL N , we say that G is similar to H and write G = T HT − .One of the most important examples is the quantum generalization of the or-thogonal group – the free orthogonal quantum group defined by Wang in [Wan95]through the universal ∗ -algebra O ( O + N ) := ∗ -alg( u ij , i, j = 1 , . . . , N | u ij = u ∗ ij , uu t = u t u = 1 N ) . In this article, we are going to study only orthogonal quantum groups G ⊆ O + N .In other words, we will always assume that the fundamental representation u of G is orthogonal (satisfies u ij = u ∗ ij and uu t = u t u = 1).For a compact matrix quantum group G = ( O ( G ) , u ), we say that v ∈ M n ( O ( G ))is a representation of G if ∆( v ij ) = P k v ik ⊗ v kj , where ∆ is the comultipli-cation. The representation v is called unitary if it is unitary as a matrix, i.e. P k v ik v ∗ jk = P k v ∗ ki v kj = δ ij . In particular, an element a ∈ O ( G ) is a one-dimensional representation if ∆( a ) = a ⊗ a . Another example of a quantum grouprepersentation is the fundamental representation u . DANIEL GROMADA
For two representations v ∈ M n ( O ( G )), w ∈ M m ( O ( G )) of G we define the spaceof intertwineres Mor( v, w ) = { T : C n → C m | T v = wT } . Deformed commutativity.
Although the goal of this article is to constructsome free quantum group, that is, to liberate the commutativity relations, wealso need to say something about deformations of commutativity – namely the q -commutativity at q = −
1. Here, the situation is much closer to the commutativesetting than in the free case. For example, we have the quantum determinant here.(See [KS97, Section 9.2] for more details.)We say that a matrix u has ( − -commutative entries if the following relationshold u ik u jk = − u jk u ik , u ki u kj = − u kj u ki , u ik u jl = u jl u ik assuming i = j and k = l .As an example, let us mention the q = − O − N , which is defined by O ( O − N ) = ∗ -alg( u ij | u = ¯ u , u orthogonal, u ( − . For q -commutative matrices, we define the q -determinant . The definition for q = − − u = X σ ∈ S N u σ (1) · · · u Nσ ( N ) = X σ ∈ S N u π (1) σ (1) · · · u π ( N ) σ ( N ) = X π ∈ S N u π (1)1 · · · u π ( N ) N = X π ∈ S N u π (1) σ (1) · · · u π ( N ) σ ( N ) Since we assume the ( − permanent . Proposition 1.1.
Let G = ( O ( G ) , u ) be a compact matrix quantum group such that u has ( − -commutative entries. Then det − u ∈ O ( G ) forms a one-dimensionalrepresentation of G .Proof. Indeed,∆(det − u ) = X σ ∈ S N N X k ,...,k N =1 ( u k · · · u Nk N ) ⊗ ( u k σ (1) . . . u k N σ ( N ) )= X σ ∈ S N X π ∈ S N ( u π (1) · · · u Nπ ( N ) ) ⊗ ( u π (1) σ (1) . . . u π ( N ) σ ( N ) ) = det − u ⊗ det − u, where the second equality follows from the fact that only tuples ( k , . . . , k N ) thatare permutations contribute to the sum since if k i = k j then u k σ (1) · · · u k N σ ( N ) = − u k ( σ ◦ τ )(1) · · · u k N ( σ ◦ τ )( N ) , where τ is the transposition of i and j . (cid:3) REE QUANTUM ANALOGUE OF COXETER GROUP D Monoidal involutive categories and Tannaka–Krein duality.
All cate-gories appearing in this article will be rigid monoidal involutive categories with theset of natural numbers including zero N as the set of self-dual objects. That is,by a category C we mean a collection of sets C ( k, l ) with k, l ∈ N together witha colection of operations • tensor product ⊗ : C ( k , l ) × C ( k , l ) → C ( k + k , l + l ), • composition · : C ( l, m ) × C ( k, l ) → C ( k, m ), • involution ∗ : C ( k, l ) → C ( l, k )such that • ⊗ and · are associative, ∗ is involutive • we have ( R ⊗ T )( S ⊗ U ) = ( RS ⊗ T U ) for every R ∈ C ( l , m ), S ∈ C ( k , l ), T ∈ C ( l , m ), U ∈ C ( k , l ), • there is the scalar identity id ∈ C (0 ,
0) such that T ⊗ id = id ⊗ T = T for every T ∈ C ( k, l ), • there is the identity id ∈ C (1 ,
1) such that T · id k = id l · T = T for every T ∈ C ( k, l ), where id k := id ⊗ k for k ≥ • there is the duality morphism T ∈ C (0 ,
2) such that(id ⊗ T ∗ )( T ⊗ id) = id and ( T ∗ ⊗ id)(id ⊗ T ) = id . By a linear category we mean a category in the above sense such that the sets C ( k, l ) are vector spaces, the operations ⊗ and · are bilinear and the involution ∗ is antilinear.The role of homomorphisms between two categories is played by monoidal uni-tary functors. A monoidal unitary functor F : C → D is a collection of maps F : C ( k, l ) → D ( k, l ) such that F ( ST ) = F ( S ) F ( T ) and F ( T ∗ ) = F ( T ) ∗ for every T ∈ C ( k, l ) and S ∈ C ( l, m ). We call F a category isomorphism if all the maps arebijections. Remark 1.2 (Frobenius reciprocity) . The duality morphism defines a linear iso-morphism between the spaces C ( k, l ) with fixed k + l . For a morphism T ∈ C ( k, l ),we define its right rotation and left rotation asRrot T := (id l − ⊗ T ∗ )( T ⊗ id) ∈ C ( k + 1 , l − , Lrot T := (id ⊗ T )( T ⊗ id k − ) ∈ C ( k − , l + 1) . In particular, any category is determined by the spaces C (0 , l ) since then we cancompute C ( k, l ) = Rrot k C (0 , k + l ).Let G = ( O ( G ) , u ) be a compact matrix quantum group. We can associate to G the linear category C G with C G ( k, l ) := Mor( u ⊗ k , u ⊗ l ) = { T : ( C N ) ⊗ k → ( C N ) ⊗ l | T u ⊗ k = u ⊗ l T } . Another examples of monoidal categories will be presented in sections 1.5, 3.1with the aim to model somehow such representation categories. Now the key ingre-dient for our work is the so-called Tannaka–Krein duality formulated by Woronow-icz in [Wor88]. We give here a simplified version of this statement (see also[NT13, Mal18, Fre19]).
Theorem 1.3 (Tannaka–Krein duality) . Let C be a linear category with C ( k, l ) ⊆ L (( C N ) ⊗ k , ( C N ) ⊗ l ) for some N ∈ N . Suppose that the duality morphism T : DANIEL GROMADA C → ( C N ) ⊗ is of the form P Ni =1 e i ⊗ e i . Then there exists a compact matrixquantum group G = ( O ( G ) , u ) ⊆ O + N such that C = C G . Finally, let us mention a well known correspondence between quantum subgroupsand subcategories.
Proposition 1.4.
Consider
G, H ⊆ O + N . Then H ⊆ G if and only if C H ( k, l ) ⊇ C G ( k, l ) for every k, l ∈ N . Considering two quantum groups G and G , we may define their intersection G := G ∩ G to be the largest quantum subgroup of both G and G . Theassociated representation category C G would then consequently be the smallestcategory containing both C G and C G . We may say that C G is generated by C G and C G .1.4. Partitions.
Let k, l ∈ N , by a partition of k upper and l lower points we meana partition of the set { , . . . , k }⊔{ , . . . , l } ≈ { , . . . , k + l } , that is, a decompositionof the set of k + l points into non-empty disjoint subsets, called blocks . The first k points are called upper and the last l points are called lower . The set of allpartitions on k upper and l lower points is denoted P ( k, l ). We define the union P := S k,l ∈ N P ( k, l ). The number | p | := k + l for p ∈ P ( k, l ) is called the length of p .We illustrate partitions graphically by putting k points in one row and l pointson another row below and connecting by lines those points that are grouped in oneblock. All lines are drawn between those two rows.Below, we give an example of two partitions p ∈ P (3 ,
4) and q ∈ P (4 ,
4) definedby their graphical representation. The first set of points is decomposed into threeblocks, whereas the second one is into five blocks. In addition, the first one is anexample of a non-crossing partition, i.e. a partition that can be drawn in a waythat lines connecting different blocks do not intersect (following the rule that alllines are between the two rows of points). On the other hand, the second partitionhas one crossing.(1.1) p = q =A block containing a single point is called a singleton . In particular, the parti-tions containing only one point are called singletons and for clarity denoted by anarrow ∈ P (0 ,
1) and ∈ P (1 , Linear categories of partitions.
Below, we define operations on partitionsthat give the collection of sets P ( k, l ) the structure of a (monoidal involutive)category. These operations probably first appeared in [Mar96]. In the context ofquantum groups, the partition categories were first considered in [BS09]. Note thatin contrast with [BS09], we consider linear categories of partitions here, that is, wealso allow linear combinations of partitions; see also [GW20a, GW19].Fix a natural number N ∈ N . Let us denote P N -lin ( k, l ) the vector space offormal linear combination of partitions p ∈ P ( k, l ). That is, P N -lin ( k, l ) is a vectorspace, whose basis is P ( k, l ). Let us denote P N -lin := S k,l P N -lin ( k, l ).Now, we are going to define some operations on P N -lin . First, let us define thoseoperations just on partitions. REE QUANTUM ANALOGUE OF COXETER GROUP D • The tensor product of two partitions p ∈ P ( k, l ) and q ∈ P ( k ′ , l ′ ) isthe partition p ⊗ q ∈ P ( k + k ′ , l + l ′ ) obtained by writing the graphicalrepresentations of p and q “side by side”. ⊗ = • For p ∈ P ( k, l ), q ∈ P ( l, m ) we define their composition qp ∈ P N -lin ( k, m )by putting the graphical representation of q below p identifying the lowerrow of p with the upper row of q . The upper row of p now represents theupper row of the composition and the lower row of q represents the lowerrow of the composition. Each extra connected component of the diagramthat appears in the middle and is not connected to any of the upper or thelower points, transforms to a multiplicative factor N . · = = N • For p ∈ P ( k, l ) we define its involution p ∗ ∈ P ( l, k ) by reversing itsgraphical representation with respect to the horizontal axis. (cid:18) (cid:19) ∗ =Now we can extend the definition of tensor product and composition linearly.We extend the definition of the involution antilinearly. Then the operations definethe structure of a linear category on the collection P N -lin ( k, l ).Any collection of subspaces K = S k,l ∈ N K ( k, l ), K ( k, l ) ⊆ P N -lin ( k, l ) con-taining the identity partition ∈ K (1 ,
1) and the pair partition ∈ K (0 , linear category of partitions .Note that for any linear category of partitions, the space C (0 ,
0) is one-dimensional,formed by scalar multiples of the empty partition playing the role of the scalar iden-tity and hence can be identified with C .For given p , . . . , p n ∈ P N -lin , we denote by h p , . . . , p n i N -lin the smallest lin-ear category of partitions containing p , . . . , p n . We say that p , . . . , p n generate h p , . . . , p n i N -lin . Note that the pair partitions are contained in the category bydefinition and hence will not be explicitly listed as generators.For any partition, its right and left rotations are given by simply rotating thepicture. That is, right rotation is given by taking the rightmost point of the lowerrow and putting it to the top row. For the left rotation, take the leftmost point ofthe upper row and put it to the bottom row. For example,Rrot (cid:18) (cid:19) = , Lrot (cid:18) (cid:19) = . Remark 1.5.
As we already mentioned, Banica and Speicher in [BS09] and manyothers following their work do not use the linear structure for partitions. Anycategory of partitions C in the Banica–Speicher sense defines a linear category ofpartitions K by taking K ( k, l ) := span C ( k, l ). Such categories are much easierto study by combinatorial means and therefore we call them easy . Many examples DANIEL GROMADA of categories that are not of this form (and hence called non-easy ) were recentlydiscovered in [GW20a, GW19].1.6.
Linear maps associated to partitions.
In this section, we review the con-nection between partitions and quantum groups [BS09]. More concretely, we showhow partition categories can be used to model representation categories of quantumgroups.Consider again a fixed natural number N ∈ N . Given a partition p ∈ P ( k, l ),we can define a linear map T p : ( C N ) ⊗ k → ( C N ) ⊗ l via(1.2) T p ( e i ⊗ · · · ⊗ e i k ) = N X j ,...,j l =1 δ p ( i , j )( e j ⊗ · · · ⊗ e j l ) , where i = ( i , . . . , i k ), j = ( j , . . . , j l ) and the symbol δ p ( i , j ) is defined as follows.Let us assign the k points in the upper row of p by numbers i , . . . , i k (from left toright) and the l points in the lower row j , . . . , j l (again from left to right). Then δ ( i , j ) = 1 if the points belonging to the same block are assigned the same numbers.Otherwise δ ( i , j ) = 0.As an example, we can express δ p and δ q , where p and q come from Equation(1.1), using multivariate δ function as follows δ p ( i , j ) = δ i i i j j , δ q ( i , j ) = δ i j j δ i j . We extend this definition for linear combinations of partitions linearly, that is, δ αp + q = αδ p + δ q and hence T αp + q = αT p + T q .Given a linear combination of partitions p ∈ P N -lin ( k, l ), we can interpret themap T p as an intertwiner T p u ⊗ k = u ⊗ l T p for some compact matrix quantumgroup G . Substituting the definition of T p , this implies the following relations N X t ,...,t k =1 δ p ( t , s ) u t i · · · u t k i k = N X j ,...,j l =1 δ p ( i , j ) u s j · · · u s l j l for every i , . . . , i k , s , . . . , s l ∈ { , . . . , N } .For example, considering p = ∈ P (0 , δ s s = N X j =1 u s j u s j . Thus, for any quantum group G ⊆ O + N , we have that T ∈ Mor(1 , u ⊗ u ). Similarly,we also have T ∈ Mor( u ⊗ u,
1) for any G ⊆ O + N . Proposition 1.6.
The map T • : p T p is a monoidal unitary functor. That is,we have the following (1) T p ⊗ q = T p ⊗ T q , (2) T qp = T q T p whenever one of the sides makes sense, (3) T p ∗ = T ∗ p . Corollary 1.7.
For every linear category of partitions K there exists a compactmatrix quantum group G = ( O ( G ) , u ) ⊆ O + N such that C G ( k, l ) = Mor( u ⊗ k , u ⊗ l ) = { T p | p ∈ K ( k, l ) } . Proof.
Follows by applying the Tannaka–Krein duality to the image of K by thefunctor p T p . (cid:3) REE QUANTUM ANALOGUE OF COXETER GROUP D In addition, we can express the associated quantum group G very concretelyby considering the universal ∗ -algebra given by the intertwiner relations T p u ⊗ k = u ⊗ l T p . That is, O ( G ) = ∗ -alg( u ij , i, j = 1 , . . . , N | u = ¯ u, T p u ⊗ k = u ⊗ l T p ∀ p ∈ K ) Remark 1.8.
The functor p T p is not injective. Consequently, given a linearcategory of partitions K and the associated quantum group G , there may exista different partition category K ′ that corresponds to the same quantum group G .In particular, having p K does not imply that T p C G . Lemma 1.9.
Consider linear categories of partitions K , K and the associatedquantum groups G and G . If K ( k, l ) = K ( k, l ) for some k, l ∈ N with k + l ≤ N ,then G = G .Proof. It is obvious that two identical quantum groups must have identical repre-sentation categories. However, as was mentioned in Remark 1.8, there may existtwo distinct categories of partitions that model the same quantum group since thefunctor p T p is not injective.Nevertheless, in [GW20a, Corollary 3.4], it was proven that p T p acts in-jectively on all the morphism spaces P N -lin ( k, l ) with k + l ≤ N . Therefore, if K ( k, l ) = K ( k, l ) for k + l ≤ N , their images under T • must be distinct and hencealso the quantum groups are distinct. (cid:3) Important examples of partition quantum groups.
In this subsection,we summarize some important partition categories and the corresponding quantumgroups.First of all, note that the relation T p u ⊗ k = u ⊗ l T p corresponding to the crossingpartition p = reads u ij u kl = u kl u ij , i.e. stands for commutativity. This meansthat the quantum group G corresponding to a given partition category K is actuallya group if ∈ K . Such a category is also called a group category . Let us mentiona few important examples [BS09]. • The category P N -lin = h , , i N -lin of all partitions corresponds tothe symmetric group S N represented by permutation matrices. • The category h , i N -lin spanned by all partitions with blocks of evensize corresponds to the hyperoctahedral group H N represented by signedpermutation matrices. • The category h i N -lin spanned by all pair partitions (all blocks have sizetwo) corresponds to the orthogonal group O N .In general, partitions containing some crossing always correspond to relationscontaining some commutation. Following this idea, free quantum groups are thosethat correspond to categories spanned by non-crossing partitions only. The set ofall non-crossing partitions itself forms a category N C N -lin . As a consequence, forany category K , we can define a new non-crossing category K + := K ∩ N C N -lin (taking K + ( k, l ) = K ( k, l ) ∩ N C N -lin ( k, l )). Using this construction, we can definefree counterparts of the group examples above [BS09]. • The category
N C N -lin = h , i N -lin of all non-crossing partitions corre-sponds to the free symmetric quantum group S + N . • The category h i N -lin spanned by all non-crossing partitions with blocksof even size corresponds to the free hyperoctahedral quantum group H + N . • The smallest possible category hi N -lin spanned by all non-crossing pair par-titions corresponds to the free orthogonal quantum group O + N .We did not define H + N and S + N in this article. Thanks to Tannaka–Krein duality,one can consider the above given characterization as the definition. Nevertheless,similarly as in the case of O + N , the quantum groups were actually defined earlier.The definition of S + N goes back to [Wan98], whereas H + N comes from [Bic04]. How-ever, many other quantum groups were discovered subsequently by studying othercategories of partitions. Definition 1.10.
Let K ⊆ P N -lin be a linear category of partitions containing thecrossing partition . A category K + ⊆ N C N -lin is called a non-crossing version of K if h K + , i N -lin = K . Remark 1.11.
One category can have more than one non-crossing version. As anexample, take the category K = h , ⊗ i N -lin . One of its non-crossing versionsis obviously h ⊗ i N -lin . Another is given by intersection with N C N -lin , which canbe written as K ∩ N C N -lin = h i N -lin and it is different from the previous one[Web13]. Infinitely many new categories interpolating these two were discoveredrecently [GW20b]. The non-crossing version constructed as an intersection with N C N -lin is always the largest one and could be somehow considered as the canonicalone. Nevertheless, smaller non-crossing versions may be actually more interestingsince the smaller the category, the “more free” the quantum group. Remark 1.12.
If we are in the “easy case”, i.e. if we work only with partitions andnot their linear combinations, then every group category K has its non-crossingversion – namely the intersection K ∩ N C N -lin . This might not hold when workingwith linear combinations of partitions.Similarly as commutativity is represented by the partition , the ( − − + 2 . In particular the quan-tum group O − N is represented by the linear category of partitions h − i N -lin [GW19]. Remark 1.13.
As a consequence, we have that H N = O N ∩ O − N . So, O − N can beseen not only as a deformation of O N , but also as a liberation of H N .Actually, we have the following. Proposition 1.14 ([GW19, Propositions 4.5, 6.9]) . For any N ∈ N , there is anisomorphism of monoidal ∗ -categories ϕ : h , i N -lin → h , i N -lin map-ping
7→ − + 2 , . In this sense, we can extend Definition 1.10 as follows.
Definition 1.15.
Let K ⊆ P N -lin be a linear category of partitions containing − + 2 . A category K + ⊆ N C N -lin is called a non-crossing version of K if h K + , − + 2 i N -lin = K .Note that the definition may not be compatible with Def. 1.10. That is, consid-ering a category K that contains both and − + 2 , then its non-crossingversion according to Def. 1.10 may not satisfy Def. 1.15 and vice versa.Finally, we mention certain category isomorphism from [GW20a]. We denote τ ( N ) := − N ∈ P N -lin . It holds that τ ( N ) · τ ( N ) = (hence also T τ ( N ) = I ). REE QUANTUM ANALOGUE OF COXETER GROUP D Consequently, the collection of maps T ( N ) : P N -lin ( k, l ) → P N -lin ( k, l ) defined as p τ ⊗ l ( N ) pτ ⊗ k ( N ) for p ∈ P N -lin ( k, l ) is a category isomorphism. Given a category K ⊆ P N -lin corresponding to some quantum group G , its image under T ( N ) cor-responds to a similar quantum group G ′ = T τ ( N ) GT τ ( N ) .1.8. Alternative maps associated to partitions and M¨obius inversion.
In[Maa20] alternative maps associated to partition were defined. Take N ∈ N . Fora partition p ∈ P ( k, l ), we define ˆ T p : ( C N ) ⊗ k → ( C N ) ⊗ l asˆ T p ( e ⊗ · · · ⊗ e k ) = N X j ,...,j l =1 ˆ δ p ( i , j )( e j ⊗ · · · ⊗ e j l ) , where ˆ δ p ( i , j ) equals to one if and only if the indices are equal precisely when thecorresponding points are in the same block, otherwise it equals to zero. (Recallthat the condition for δ p ( i , j ) = 1 was: if the points are in the same block, theindices must coincide. In contrast, for the ˆ δ p we require equivalence: points are inthe same block if and only if indices coincide.) Then the definition can be linearlyextended to the whole space P N -lin ( k, l ). Example 1.16.
Take p = ⊗ l . Then δ ⊗ l ( ∅ , j ) = 1 for every tuple ( j , . . . , j l ). Onthe other hand ˆ δ ⊗ l ( ∅ , j ) = 1 if and only if all the indices j , . . . , j l are mutuallydistinct. If we choose l = N , then ˆ δ ⊗ N ( ∅ , j ) = 1 if and only if ( j , . . . , j N ) isa permutation of the set { , . . . , N } . If we choose l > N , then ˆ δ ⊗ l ( ∅ , j ) = 0 forany j .It was shown in [Maa20, Lemma 4.24] that for any partition p ∈ P ( k, l ), wehave T p = X q ≥ p ˆ T q , where we denote q ≥ p for q ∈ P ( k, l ) if the partition q was made from p bymerging some blocks.It should be pointed out that the map p ˆ T p is not a functor. Nevertheless,it still might be quite useful. The above mentioned formula shows us, that we candescribe the linear map ˆ T p by applying the functor p T p on a certain linearcombination of partitions. More precisely, we can do the following.For every partition p ∈ P ( k, l ), we define a linear combination ˆ p ∈ P N -lin ( k, l )such that we have p = P q ≥ p ˆ p . It can be easily seen that such an operation isindeed well defined. Actually, one can express explicitly the image ˆ p using theM¨obius inversion formula as ˆ p = P q ≥ p µ ( p, q ) q , where µ is the M¨obius function onthe lattice of partitions, see [NS06, Lecture 10]. We extend this definition linearlyto the whole space P N -lin ( k, l ). Then we have T ˆ p = ˆ T p . Example 1.17.
Let us compute d from the relation p = P q ≥ p ˆ q . In order todo that, we actually have to compute ˆ p for every p ∈ P (0 , ≤ is the block partition . Here we directlyhave = d . Now, we have = d + d , so d = − . Similarly,we can compute d and d . Finally, we have= d + d + d + d + d = d + + + − , so d = − − − + 2 . Coxeter groups of type D . The Coxeter group of type D with N generators,sometimes refered to as the demihyperoctahedral group , is defined by the CoxeterdiagramStrictly speaking, Coxeter groups of type D are defined only for N ≥ N = 1 , , A . Hence, Coxeter group withfour generators corresponding to the diagram can be defined as h g , g , g , g | g i g j = g j g i , g i g g i = g g i g ∀ i, j = 1 , , } . Coxeter group of type D with N generators can be also realized as a matrixgroup using matrices of size N as follows D N := { X ∈ H N | perm X = 1 } , where H N is the hyperoctahedral group represented by signed permutations andperm X is the permanent of a matrix X defined asperm X = X σ ∈ S N X σ (1) · · · X Nσ ( N ) . In this case, the permanent just multiplies the signs of the signed permutation.
Remark 1.18.
The permanent as a map perm : GL( n, C ) → C is not a homomor-phism. Nevertheless it acts as a homomorphism on the subgroup H N ⊆ GL( n, C ) ofsigned permutation matrices. There is a quantum group explanation for that. Thesigned permutation matrices actually have ( − − − D .Note that we already know one quantum group, whose classical version is D N ,namely the ( − SO − N . Indeed, SO − N is obtained from O − N by the relation perm u = 1. Since the classical version of O − N is H N , adding the permanent relation pushes us to D N . Nevertheless, SO − N should not be considered as a good candidate for free counterpart of D N – itscommutativity relations are deformed, not liberated.2. Free permanent and free D +4 As indicated in Sect. 1.7, a good approach to define a free version of a group isto study its representation category using partitions. In the following, we find theintertwiner corresponding to the ( − Intertwiner corresponding to the permanent.
Let us fix a natural num-ber N ∈ N . We define the following vector in ( C N ) ⊗ N P ( N ) := X σ ∈ S N e σ (1) ⊗ · · · ⊗ e σ ( N ) , REE QUANTUM ANALOGUE OF COXETER GROUP D That is, in coordinates[ P ( N ) ] i ,...,i N = ( i , . . . , i N ) is a permutation,0 otherwise. Proposition 2.1.
Let G be a compact matrix quantum group with N × N fundamen-tal representation u with ( − -commutative entries. Then det − u is a subrepre-sentation of u ⊗ N corresponding to the one-dimensional invariant subspace spannedby the vector P ( N ) .Proof. Let us write[ u ⊗ N P ( N ) ] i ,...,i N = N X j ,...,j N =1 u i j · · · u i N j N [ P ( N ) ] j ,...,j N = X σ ∈ S N u i σ (1) · · · u i N σ ( N ) . Now, using the deformed commutativity, we can see that actually[ u ⊗ N P ( N ) ] i ,...,i N = ( det − u if ( i , . . . , i N ) is a permutation , i , . . . , i N ) is a permutation, then we have exactly the defining formulafor the ( − i j = i k for some j, k , we have that u i σ (1) · · · u i N σ ( N ) = − u i ( σ ◦ τ )(1) · · · u i N ( σ ◦ τ )( N ) , where τ is the transposition of j and k , so all the termswill cancel out.So, we have u ⊗ N P ( N ) = det − u · P ( N ) ∈ O ( G ) ⊗ span { P ( N ) } , which is what wewanted. (cid:3) Lemma 2.2.
It holds that P ( N ) = ˆ T ⊗ N = T d ⊗ N .Proof. Follows from Example 1.16 (cid:3)
In terms of intertwiners, Proposition 2.1 tells us that if u has ( − P ( N ) P ∗ ( N ) ∈ Mor( u ⊗ N , u ⊗ N ) (since P ( N ) P ∗ ( N ) is the projectiononto the subspace spanned by P ( N ) ). In addition, the equality det − u = 1 isequivalent to saying P ( N ) ∈ Mor(1 , u ⊗ N ). Lemma 2.2 then allows us to formulateeverything in terms of partitions since the vector P ( N ) can be represented by thepartition d ⊗ N . Remark 2.3.
Although H N is represented by a matrix with ( − h , i N -lin , which is usually used to describe this group,does not contain the element d ⊗ N d ⊗ N ∗ . This serves as an illustration of what wasmentioned in Remark 1.8The above formulated proposition is is by no means a new result. The intertwiner P ( N ) was used already in [Wor88] to define the quantum group SU qN . The work ofWoronowicz actually provides also kind of a converse to Proposition 2.1 Proposition 2.4.
Let G be a compact matrix quantum group with N × N fundamen-tal representation u such that P ( N ) ∈ Mor(1 , u ⊗ N ) . Then u has ( − -commutativeentries. Proof.
We follow the computation from [Wor88, page 66]. It is enough to showthat P ( N ) ∈ C G implies T − +2 ∈ C G . Let us compute the following[(id ⊗ id ⊗ P ∗ ( N ) )( P ( N ) ⊗ id ⊗ id)] ( i i ) , ( j j ) = X k ,...,k N − [ P ( N ) ] i i k ··· k N − [ P ( N ) ] k ··· k N − j j = ( ( N − i = j, { i, j } = { k, l } N − δ ( i i , j j ) + δ ( i i , j j ) − δ ( i i , j j )) . Consequently, C G ∋ (1 N ⊗ N ⊗ P ∗ ( N ) )( P ( N ) ⊗ N ⊗ N ) = ( N − T + T − T ) (cid:3) Example 2.5.
We can define the following categories. • The category h d ⊗ N i N -lin = h− + 2 , d ⊗ N i N -lin corresponds to the quan-tum group SO − N . • The category h d ⊗ N , i N -lin = h− + 2 , d ⊗ N , i N -lin correspondsto the Coxeter group D .2.2. Freeing the permanent relation, free SO − . For this subsection, we set N = 4. The key observation here is the fact that(2.1) \ = − T (4) − (2 − ) + + . Hence, it might be a good idea to substitute \ with T (4) . Proposition 2.6.
The category hT (4) i is a non-crossing version of thecategory h \ i .Proof. Obviously is hT (4) i noncrossing. From Eq. (2.1), we see that(2.2) hT (4) , − + 2 i = h \ , − + 2 i = h \ i . (cid:3) Definition 2.7.
We denote S − O +4 := T ⊗ τ (4) H +4 T ⊗ τ (4) the quantum group corre-sponding to the category hT (4) i and call it the free ( − -special orthogonalgroup . Theorem 2.8.
We have SO − ( S − O +4 ( O +4 . The subgroup SO − is obtained from S − O +4 imposing ( − -commutativity on theentries of the fundamental representation.Proof. The statement follows from the partition picture. For the associated cate-gories of partitions, we surely have h \ , − + 2 i ) hT (4) i ) hi . The right inclusion including the strictness is obvious. The left one follows fromEq. (2.2). It is strict since the second category is non-crossing while the firstone is not. The inclusions for the categories imply corresponding inclusions forthe quantum groups in the sense of Proposition 1.4. The inclusions indeed carry
REE QUANTUM ANALOGUE OF COXETER GROUP D over including the strictness thanks to Lemma 1.9 – we work with N = 4 and allthe partitions are defined on k + l = 4 points. The last assertion follows fromProposition 2.6. (cid:3) Remark 2.9.
Let us state two obvious remarks following directly from the defi-nition of S − O +4 . First, the quantum group is similar (and hence isomorphic) tothe free hyperoctahedral group H +4 . Secondly, the associated category C S − O +4 isisomorphic to the category C H +4 .2.3. Free version of Coxeter group D . In this section, we are going to presentthe first main result of this article – the definition of the free quantum analogue forthe demihyperoctahedral group of rank four.
Proposition 2.10.
The category hT (4) , i is a non-crossing versionof the category h \ , i in the sense of both Def. 1.10 and Def 1.15.Proof. Again, using Eq. (2.1), we can easily see that adding either the partitionor the linear combination − + 2 to the category hT (4) , i weobtain h \ , i in both cases. (cid:3) Definition 2.11.
We denote D +4 := H +4 ∩ S − O +4 the quantum group correspond-ing to the category hT (4) , i and call it the free demihyperoctehedralquantum group . Theorem 2.12.
We have D ( D +4 ( H +4 . The subgroup D is obtained from D +4 imposing commutativity or ( − -commutativityon the entries of the fundamental representation.Proof. The proof is the same as in case of Theorem 2.8. This time, we use thecategory inclusions h \ , i ) hT (4) , i ) h i . The right inclusion is obvious, its strictness follows from the fact that h i contains only partitions with blocks of even length, whereas T (4) contains alsoodd blocks as summands. The left inclusion again follows from Eq. (2.2). It is strictsince the second category is non-crossing while the first one is not. Again, thoseinclusions carry over to the corresponding quantum group including the strictnessthanks to Proposition 1.4 and Lemma 1.9. Finally, the last assertion follows fromProposition 2.10. (cid:3) Remarks on N = 4 . As a first remark, let us stress that our considerationsare very specific for N = 4. It is completely unclear, how to define the free analoguesfor other N . See also Question 5.11.Secondly, we can still ask whether the categories we just defined are interestingalso for N = 4. (Although we should definitely not interpret them as free analoguesof D N or SO − N .) Unfortunately, we have the following. Lemma 2.13.
Assume N = 2 , . Then hT ( N ) , i N -lin = h , ⊗ i N -lin = { all NC partitions of even length } . Proof.
It is straightforward to compute · ( T ( N ) ⊗ )= (cid:18) − N + 12 N (cid:19) − N (cid:18) − N (cid:19) (cid:18) − N (cid:19) ⊗ . It is known that h , ⊗ i N -lin is the category spanned by all non-crossingpartitions of even length. In particular, T ( N ) ∈ h , ⊗ i N -lin . (cid:3) On the other hand, the category hT ( N ) i N -lin makes perfect sense and, aswe already mentioned, corresponds to the quantum group T ⊗ Nτ ( N ) H + N T ⊗ Nτ ( N ) , which issimilar (and hence isomorphic) to H + N .Finally, one might want to study the category K := h d ⊗ k i N -lin for general k and N . But we obtain nothing new except for the above discussed case k = N .For k > N , we have d ⊗ k = 0, so K = hi N -lin . For k = 1, we have K = h i ,for k = 2, we have K = h ⊗ i N -lin . One can show that for 2 < k < N , we have K = h , i N -lin = { all NC partitions } if k is odd and K = h , ⊗ i N -lin = { all NC partitions of even length } if k is even.3. The representation category of D +4 as a graph category Let us now informally summarize the goals and strategy for the next sections.Our aim is to study the representation category associated to the new free demi-hyperoctahedral quantum group D +4 . Namely the most basic question reads: Question 3.1.
Find an explicit description of the representation category C D +4 .First, notice that we indeed do not have such a description yet. Although wedefined D +4 through its associated linear category of partitions, we know only thegenerators of the category, not all its elements explicitly. In contrast with the“easy” quantum groups, where we know, for instance, that h i contains all non-crossing partitions with even blocks and hence we can explicitly write down howthe associated representation category of H + N looks like. In case of D +4 , due to thefact that we are using linear combinations of partitions, it is not so straightforwardto see, what are the explicit elements of our category.A naive idea to describe all elements of the category K := hT (4) , i is to notice that the category is linearly spanned by elements that were made fromthose two generators by finite amount of the category operations – tensor product,composition, and involution. We can then model these operations by some graphsand these graphs would then correspond to elements of this category. To be moreconcrete, let us denote the four-block by a black point (more precisely, bythe diagram ) and the linear combination T (4) by a white point (moreprecisely, by the diagram ). Let us now ignore the distinction between thinand thick lines for a moment. Now, we can construct more complicated diagramsthat stand for some other elements of the category such as= · ( ⊗ T (4) ) . In the end, every element of K can be expressed as a linear combination ofsuch diagrams. Before discussing how much sensible and useful this approach is,let us note that there is also another way how to construct and interpret suchdiagrams with the same result: We forget about the distinction between black and REE QUANTUM ANALOGUE OF COXETER GROUP D white points – both will now stand for the fourblock – but we are going todistinguish between thick and thin lines – thin lines will be simply identities, butthe thick ones will stand for the partition τ (4) .This latter approach fits perfectly to the framework of graph categories intro-duced recently in [MR19]. We formalize this approach in Section 3.Nevertheless, one might object that such naive descriptions brings us basicallynothing – we are still not able to explicitly say what are the elements of the mor-phism spaces C D +4 ( k, l ) or K ( k, l ) since there are infinite amount of graphs/diagramswe would have to go through to determine this for every given k, l . In other words,there is an infinite amount of ways how to combine the generators together. Ofcourse that if we iterate the category operations ad infinitum, we eventually obtainall elements of K ( k, l ) for some given k, l , but this is not practically possible. Wehave to find, for every k, l ∈ N , a finite amount of graphs (of manipulations withthe generators) that already span the morphism space K ( k, l ). We solve this taskin Section 4.3.1. Graph categories.
As another example of monoidal involutive categories, wemention graph categories defined in [MR19]. Actually, we generalize the approachof [MR19] slightly.
Definition 3.2. A bilabelled graph K is a triple ( K, a , b ), where K is a graph, a = ( a , . . . , a k ), b = ( b , . . . , b l ) are tuples of vertices of K .The definition works for any type of graph. However, in this article, by a graphwe will always mean an undirected graph with the possibility of loops and multipleedges (unlike [MR19]). We will call a the tuple of input vertices while b are outputvertices (the role of a and b is switched in comparison with [MR19] to be consistentwith the notation for partitions). For any k, l ∈ N we denote by G ( k, l ) the set ofall bilabelled graphs K = ( K, a , b ) with | a | = k , | b | = l . The set of all bilabelledgraphs is denoted simply by G .We define a structure of a (monoidal involutive) category on the set of all bi-labelled graphs by introducing some operations. Consider K = ( K, a , b ), H =( H, c , d ). We define • tensor product K ⊗ H = ( K ⊔ H, ac , bd ), • composition (only defined if | b | = | c | ) H · K = ( H · K, a , d ), where H · K is a graph that is created from H ⊔ K by contracting the vertices b i and c i for every i (in contrast with [MR19], we keep the multiple edges), • involution K ∗ = ( K, b , a ).We denote M k,l := ( M, v k , v l ), where M is a graph with a single vertex v . Wedenote by the bilabelled graph corresponding to the null graph with no vertices.Any collection of bilabelled graphs C containing M , (playing the role of iden-tity), M , (playing the role of the duality morphism), (playing the role of thescalar identity) and closed under the above defined operations forms a categorycalled a graph category . Definition 3.3.
Let A ∈ M N ( R ) be a symmetric matrix, let K = ( K, a , b ) bea bilabelled graph and denote by k and l the length of a and b . Then we define a linear map T A K : ( C N ) ⊗ l → ( C N ) ⊗ k as[ T A K ] ij := X ϕ : V ( K ) → [ N ] ϕ ( a )= j , ϕ ( b )= i h ϕ i A , where h ϕ i A := Y { i,j }∈ E ( K ) A ϕ ( i ) ϕ ( j ) . Proposition 3.4.
The map T A is a monoidal unitary functor. That is, T A K ⊗ H = T A K ⊗ T A H , T A H · K = T A H T A K , T A K ∗ = T A ∗ K . Proof.
The proof in all cases is straightforward. As an example, let us do thecomputation for the most complicated part, which is the composition.[ T A H T A K ] ij = X k [ T A H ] ik [ T A K ] kj = X k X ϕ : V ( H ) → [ N ] ϕ ( c )= k , ϕ ( d )= i X ψ : V ( K ) → [ N ] ψ ( a )= j , ψ ( b )= k h ϕ i A h ψ i A == X ω : V ( H ⊔ K ) → [ N ] ω ( a )= j , ω ( d )= i ω ( b )= ω ( c ) h ω | H i A h ω | K i A = X ω : V ( H · K ) → [ N ] ω ( a )= i , ω ( d )= j h ω i A = [ T A HK ] ij (cid:3) Remark 3.5.
The category of all partitions essentially embeds into the categoryof all graphs as p K p := ( K p , i , j ), where K p is an edgeless graph with verticescorresponding to blocks of p , the tuples i and j describe the blocks of upper resp.lower points of p [MR19, Def. 6.13]. See the examples below.By essentially embeds , we mean in particular that T p = T A K p for all partitions p and for any matrix A with Tr A = N . In order to make the mapping p K p a true embedding, we would have to slightly modify either the definition of graphcategories (by introducing a linear structure and imposing the relation M , = N ,cf. Sect. 5.1; alternatively, by not considering the linear structure on partitions andallowing empty parts, see [MR19, pp. 11–12] or [Gro20]).We will denote bilabelled graphs pictorially. Unlike [MR19], we will draw them“top to bottom” instead of “right to left”. Vertices will be denoted as black pointsand edges as thick lines. As in case of partitions, the whole graph should be drawnin a strip between two horizontal lines. Input vertices are marked by drawing a thinstring from the top line to the corresponding vertex. Output vertices are connectedby strings to the bottom line. The left-right order in which the strings are attachedto the horizontal lines should correspond to the order of the tuples a and b .Let us mention a couple of examples. First, the edgeless graphs associated topartitions. M , = K = , M , = K = , M , = K = , M , = K = , K =Now, let us make some more complicated graphs and also illustrate how theoperations work. Similarly to partitions, we can describe the operations as manip-ulations with the pictures. Tensor product is simply putting “side by side” whilecomposition is putting “one above the other” and then contracting all the thin REE QUANTUM ANALOGUE OF COXETER GROUP D lines. Any bilabelled graph can be constructed using those operations starting withthe edgeless partition graphs and the graph containing two points connected by anedge . ( ⊗ ) · · ( ⊗ ) = = · ( ⊗ ) · = =In contrast with the thin strings, we draw the graph edges by thick lines.As we already mentioned, we have T A K p = T p for any partition p . In general,the idea of the T A K map is similar as with partitions. The vertices correspond toKronecker deltas, the thin strings correspond to identity maps and the thick edgescorrespond to the map A . In particular, we have T A = A, T A = A • A, where • denotes the entrywise Schur product , so [ A • A ] ij = A ij . Remark 3.6.
In the original work [MR19], the matrix A is considered to be anadjacency matrix of some graph Γ. In such a case, it is indeed possible to ignorethe multiplicity of the edges since we have A ij = A ij . Remark 3.7.
Also here we can apply the Frobenius reciprocity. Categories ofgraphs are closed under rotations (and their inverses) defined asLrot K = ( K, ( a , . . . , a k ) , ( a , b , . . . , b l )) , Rrot K = ( K, ( a , . . . , a k , b l ) , ( b , . . . , b l − ))for any K = ( K, a , b ) ∈ G ( k, l ). Hence, it is not that important which of thevertices are inputs and which are outputs. However, their cyclic order a k , . . . , a , b , . . . , b l is important. This means that when drawing the bilabelled graphs, it isnot that important to draw all the input string pointing up from the vertex andthe output vertices pointing down. But it is important that all the strings pointsomehow outside the region, where the graph is drawn, so it is clear in which (saycounter-clockwise) direction they appear. Remark 3.8.
Let us mention here the biggest disadvantage of describing a repre-sentation category as a graph category in comparison with categories of partitions.Let C be a graph category. Then the morphism spaces C ( k, l ) are typically infinitefor all k, l ∈ N . So, even if we know all the elements of C more or less explicitly, itmight still be very hard to determine, how exactly the associated intertwiner spacesMor( u ⊗ k , u ⊗ l ) = span { T A K | K ∈ C ( k, l ) } look like.Therefore, given a graph category C and a symmetric matrix A interpreting C ,we may be interested in finding a collection of finite subsets C ( k, l ) ⊆ C ( k, l ) suchthat T A C ( k, l ) = T A C ( k, l ). We may even ask that C ( k, l ) is the smallest possiblecollection. That is, to ask for T A C ( k, l ) being linearly independent.3.2. Replacing edges by τ . Now our goal is to express the representation cat-egory C D +4 as an image of a graph category by the functor T A . To do that, weneed to make a convenient choice for the matrix A . As we already mentioned, webasically need to be able to replace the thick edges by the element τ (4) . Proposition 3.9.
Consider any π ∈ P N -lin (1 , . Then there is a monoidal unitaryfunctor F π : G → P N -lin mapping K p p for every p ∈ P and π . Moreover,we have T T π K = T F π K for every K ∈ G .Proof. Denote π = α + β . Given a bilabelled graph K = ( K, a , b ), we denote by p K the partition of ( a , b ) according to the connected components of K . We denoteby rl( K ) the number of connected components C of K such that a i C and b j C for all i, j .We can write explicitly the action of F π as F π K = X S ⊆ E ( K ) α | E ( K ) \ S | β | S | N rl( K \ S ) p K \ S . Now we only need to check that this map indeed defines a functor. (cid:3)
The idea of the functor F π is that given a bilabelled graph K , we replace everyedge of K by the partition π . Remark 3.10.
Note that the construction K p K from the proof is a specialcase of the functor F taking π := . More precisely, F p = N rl( K ) p K .Another canonical way of assigning partitions to bilabelled graphs is to map K = ( K, a , b ) ker( a , b ) [MR19, Def. 6.2]. Here, ker( a , b ) denotes a partition of k upper and l lowere points, where the upper points are labelled by the vertices a , . . . , a k , the lower points are labelled by b , . . . , b l , and two points are in thesame part if the associated vertices coincide [RW15, Sect. 2.4]. In this case, wehave F K = N ker( a , b ), where by inner vertices we mean all verticesthat are not input/output.Now, we can put N = 4 and π := τ := τ (4) to obtain the following. Proposition 3.11.
The quantum group D +4 is described by the graph category h , i through the functor T A with A = T τ .Proof. Using the preceding proposition. We see that F τ h , i = h , T (4) i = h , T (4) i (cid:3) Explicit description of the associated graph category.Definition 3.12 ([MR19, Def. 5.1]) . Given a bilabelled graph K = ( K, a , b ) ∈ G ( k, l ), we define the graph K ◦ as the graph obtained from K by adding the cycle α k , . . . , α , β , . . . , β l of new vertices and edges a i α i , b j β j for every, i, j . We referto the cycle as the enveloping cycle . We further define K ⊙ by adding an additionalvertex adjacent to every vertex of the enveloping cycle. Definition 3.13 ([MR19, Def. 5.3, 5.4]) . A bilabelled graph K is called planar if K ⊙ is a planar graph. Equivalently, K ◦ must be planar and the enveloping cyclemust be facial.Basically, this means that we can draw the bilabelled graph K according to therules described in Sect. 3.1 in such a way that all the lines (both thick and thin,that is, edges of K together with the input/output strings) do not cross each other.The face of K ◦ corresponding to the enveloping cycle or the face of K neighbouringwith the input/output vertices is then drawn as the unbounded face and we willalso refer to this face as the unbounded face in the subsequent text. REE QUANTUM ANALOGUE OF COXETER GROUP D Remark 3.14 ([MR19, Thm. 5.15]) . The planar bilabelled graphs are closed underthe category operations. Hence, they form a graph category.
Definition 3.15.
Let K = ( K, a , b ) be a bilabelled graph and let v be a vertexof K . The degree of v in K ◦ will be called the extended degree of v and denotedby ˜ d v . Definition 3.16.
Let K be a bilabelled graph containing an inner vertex v (i.e. v is not an input/output vertex) of degree two. Let e, f be the two edges incidentwith v and let K ′ be a bilabelled graph that was made from K by contracting thosetwo edges. We will call K ′ a two-path contraction of K . Definition 3.17.
For any set S of bilabelled graphs, we denote by h S i c the smallestgraph category containing S that is closed also under two-path contraction. Proposition 3.18.
Consider a bilablled graph K and its two-path contraction K ′ . (1) If A = 1 , then T A K = T A K ′ . (2) If π · π = , then F π K = F π K ′ .In particular, the image of a given graph category under such a functor T A , resp. F π coincides with the image of its two-path-contraction closure.Proof. The idea is the same in both cases: The relation A = 1, resp. π · π =means that two consecutive edges may be replaced by a single (thin) string andhence contracted without changing the image. (cid:3) So, instead of studying the category h , i , we can study its two-path-contrac-tion closure h , i c . Definition 3.19.
We define X k,l := ( X k + l , ( v , . . . , v k ) , ( v k +1 , . . . , v k + l )), where X n denotes the star graph with central vertex v and surrounding vertices v , . . . , v n . Proposition 3.20.
We have h , i c = h M k,l , X k,l | k + l even i = C , where C consists of all bilabelled graphs K = ( K, a , b ) ∈ G ( k, l ) such that (i) K is planar, (ii) ˜ d v is even for all v ∈ V ( K ) , (iii) K is bipartite and all the vertices a , . . . , a k , b , . . . , b l are elements of oneof the parts. (We will call this part even and the other will be odd .) Note in particular that C does not contain graphs with loops since these wouldnot be bipartite. We are going to prove this proposition by showing a series ofinclusions. Lemma 3.21.
We have C ⊇ h , i c .Proof. Obviously we have , ∈ C . Now it is easy to check that C is closedunder the category operations and two-path contractions. (cid:3) Lemma 3.22.
We have h , i c ⊇ h M k,l , X k,l | k + l even i .Proof. The fact that all M k,l for k + l even are generated by is basically knownfrom the theory of partition categories (see [Web13, Prop. 2.7(2)]). To prove it, itis enough to show that M ,l ∈ h i ⊆ h , i c for every l even. For l = 2 , have it by definition. For l = 0, we have M , = M , M , . For larger l we proveit by induction. We have M , l +2 = ( M , ⊗ ( l − ⊗ M , ⊗ M , ⊗ )( M , l ⊗ M , ).The proof for X k,l ∈ h i c ⊆ h , i c goes the same way except for the factthat we have to use a two-path contraction at the end. (cid:3) Lemma 3.23.
We have h M k,l , X k,l | k + l even i ⊇ C .Proof. The idea of the proof is that given any K ∈ C , we take M k + l for every evenvertex of K and X k + l for every odd vertex of K and compose them together asprescribed by the graph K . We illustrate the idea on the following example, wherewe for clarity denote the even vertices by black circles and the odd vertices bywhite circles . == ( M , ⊗ M , ) X , M , ( X , ⊗ X , )( M , ⊗ M , ⊗ M , ) . A proper proof should be formulated using the induction. This requires somepreparation and we devote to it a special subsection 3.4. (cid:3)
Proof of Proposition 3.20.
Follows from the lemmata above. (cid:3)
Inductive description of C . Given a bilabelled graph K = ( K, a , b ) ∈ C ,we define K • to be the subgraph of K obtained by removing all vertices v with d v = 1, ˜ d v = 2 in K . We also define a • and b • as follows: if a i ∈ K • , then a • i := a i ;otherwise, a • i is the unique neighbor of a i . The same definition is used for b • . Forany vertex v of K • we will denote d • v its degree inside K • (while d v denotes itsdegree inside K ). Note that when representing a bilabelled graph by a picture, itis perfectly enough to draw K • instead of K if we keep the distinction betweenthin and thick lines or, equivalently, between even and odd vertices. Here followsan example. ↔ We say that an element v appears consecutively in the sequence c , . . . , c m if itsoccurence forms an interval up to rotation. That is, there is an index i ∈ { , . . . , m } and a number r ∈ { , . . . , m − } such that c i + j = v if and only if j ∈ { , . . . , r } ,where the indices are taken modulo m .Now, we present the inductive description of C , where the induction is on thenumber of vertices of K • . So, consider K = ( K, a , b ) ∈ C ( k, l ). Take any vertex v bordering with the unbounded face in K • that appears consecutively among a • k , . . . , a • , b • , . . . , b • l (or does not appear there at all). For simplicity, we can assumethat b • = ( v, v, . . . , v ) and that a • does not contain v (otherwise rotate K ). Inparticular l = ˜ d v − d • v . Denote also m := d • v and c , . . . , c m the neighbors of v in K • .We distinguish two cases – (a) v is even or (b) v is odd. It holds that REE QUANTUM ANALOGUE OF COXETER GROUP D (a) K = M m,l H , resp.(b) K = X m,l H ,where, in both cases, H was made from K by removing the vertex v from K • . To bemore precise, we have (a) H = ( H, a ′ , d ), where H was made from K by removingthe vertex v , adding vertices d , . . . , d m and edges { c i , d i } , and (b) H = ( H, a ′ , c ),where H was made from K by removing the vertices v, b , . . . , b l .Let us formalize this procedure to an algorithm. Algorithm 3.24.
Constructing the sets C ( k, l ), k, l ∈ N .(1) Add the empty graph to C (0 , k, l, m ∈ N such that k + m and l + m are even and for every H = ( H, a , b ) ∈ C ( k, m ), add (b) X m,l H to C ( k, l ) and (a), if all b • , . . . , b • m are odd, add also M m,l H to C ( k, l ).(3) Add also all rotations of the graphs constructed in (2) into C .(4) Repeat (2) and (3). Remark 3.25.
The algorithm itself is very banal. Basically it says nothing morethan “graphs are made from smaller graphs by adding vertices”. Nevertheless, weare going to improve this algorithm in the following section and hence it will beuseful to have this naive version formulated as well.Actually, it is not even an algorithm since it obviously never terminates. Tomake it terminate, we would have to introduce some condition such as restrictingthe number of vertices of K • . In this way, we would end up with a collection ofsome subsets C ( k, l ) ⊆ C ( k, l ). As we already mentioned in Remark 3.8, it is ingeneral not clear how to formulate such a terminating condition in order to obtainsufficiently large sets C ( k, l ) in order to describe the whole image by the associatedfunctor T A . We attack this problem in the following section.4. Finite generating set for the intertwiner spaces
Denote C := h , i c as in Proposition 3.20. In this section, we are going toattack the problem indicated in Remark 3.8. We are going to define a collection offinite subsets C ( k, l ) ⊆ C ( k, l ) such that span T T τ C ( k,l ) = T T τ C ( k,l ) = C D +4 ( k, l ). Onlythis allows us to explicitly compute the intertwiner spaces C D +4 ( k, l ).4.1. Defining the subsets C ( k, l ) . Denote C ( k, l ) ⊆ C ( k, l ) a subclass of bil-abelled graphs K = ( K, a , b ) that satisfies in addition the following(iv) there is no vertex v with d v = ˜ d v = 2,(v) there are no multiple edges,(vi) every component of K has at least one vertex among ( a , b ).The condition (iv) can be equivalently formulated as ˜ d v ≥ v of K • unless v is isolated. The properties are not preserved under the categoryoperations, so C is not a graph category. Nevertheless, the following holds. Proposition 4.1.
We have F τ C ( k, l ) = F τ C ( k, l ) . Hence, T T τ C ( k,l ) = T T τ C ( k,l ) .Proof. It is enough to show that F τ K is invariant (up to scalar factors) under(iv) two-path contractions, (v) erasing pairs of edges between two vertices in K ,(vi) erasing components of K with no input/output vertex.Invariance of F τ C ( k, l ) under (iv) was shown already in Proposition 3.18. For (v), let K be a bilabelled graph containing two vertices v, w that are con-nected by multiple edges. Let K ′ be made from K by erasing two of those edges.Then it holds that F τ K = 1 / F τ K ′ . This follows from the fact that F τ = 1 / K has a component with no input/output vertex, then it is of the form K = K ′ ⊗ H , where H has no input output vertex and hence F τ H = α ∈ C . Butthis means that F τ K = α F τ K ′ . (cid:3) Lemma 4.2.
Consider K ∈ C ( k, l ) . Then every bilabelled graph H induced bya connected component H of K is in C .Proof. Straightforward observation. (cid:3)
Theorem 4.3.
For every k, l ∈ N , C ( k, l ) is a finite set. We prove this theorem in Section 4.3.4.2.
Algorithm for generating C ( k, l ) . In this subsection, we give an algorithmto compute explicitly the sets C ( k, l ) for any k, l ∈ N . In the following subsec-tion, we are going to prove Theorem 4.3 by showing that this algorithm alwaysterminates. Lemma 4.4.
Consider K ∈ C ( k, l ) such that K is connected and | V ( K • ) | ≥ .Then there is a vertex v of K • with d • v ≤ such that K • \ { v } is connected.Proof. Suppose the contrary. That is, there are m vertices v , . . . , v m in K • suchthat d • v i = 2 and K \{ v i } is not connected. Otherwise all vertices in K • have degreeat least three. Denote n := | V ( K • ) | the number of vertices in K • . Denote by c thenumber of vertices bordering with the unbounded face. From (ii), (iv) it followsthat those are the only vertices that can have degree two or three, all other verticeshave degree at least four. Consequently, we can make the following estimate(4.1) | E ( K • ) | = 12 X v ∈ V ( K • ) d • v ≥ n − c − m. Now, recall also the Euler’s formula for non-empty connected planar graphs − . From this formula, we can infer that the number of edges surrounding the un-bounded component is at least c −
1. Denote by Φ the set of all faces in K • . Fora face Ω ∈ Φ, denote by ν (Ω) the number of edges bordering Ω, where we count anedge twice if it borders with Ω from both sides. In particular ν (Ω ∞ ) ≥ c − m ,where Ω ∞ is the unbounded face. Since the graph is bipartite and has no multipleedges, we have ν (Ω) ≥ ∈ Φ. Now, we can estimate2 | E ( K • ) | = X Ω ∈ Φ ν (Ω) ≥ | Φ | −
1) + c − m ≥ | E ( K • ) | − n ) + c − m, so(4.2) | E ( K • ) | ≤ n − − c − m. This is, however, in contradiction with Inequality (4.1). (cid:3)
REE QUANTUM ANALOGUE OF COXETER GROUP D Remark 4.5.
Given K satisfying conditions (i)–(v), then the condition (vi) isequivalent to saying that there is no isolated vertex in K ◦ . Indeed, suppose K satisfies (i)–(v), but not (vi). Then the same must hold also for the bilabelledgraph H ∈ C (0 ,
0) induced by the connected component with no input/outputvertex. Since H has no input/output vertices, we have H • = H . In the proof ofLemma 4.4 we actually did not use the assumption (vi), so there is a vertex v of H = H • with d v = ˜ d v = d • v ≤
2. From (ii), (iv), we actually have d v = ˜ d v = 0. Lemma 4.6.
Consider K = ( K, a , b ) ∈ C ( k, l ) such that K is connected. Takeany v ∈ V ( K • ) such that K \ { v } is connected. Then v appears consecutively in a • k , . . . , a • , b • , . . . , b • l .Proof. Supposing v does not appear consecutively, and assuming that K \ { v } isconnected, we can find K as a minor of K ⊙ . Indeed, v not appearing consecutivelymeans that there exist vertices w , w ∈ V ( K ) • such that the sequence a • k , . . . , a • , b • , . . . , b • l contains . . . w . . . v . . . w . . . v . . . Denote by α , β , γ , and δ the corre-sponding vertices on the enveloping cycle. Since K \ { v } is connected, there mustbe a path between w and w not containing v . So, we can construct K as a minoras follows: Kvw w (cid:3) Algorithm 4.7.
Input: k ∈ N . Output: The sets C ( k, l ) for all k, l , k + l ≤ k .(1) Add into C (0 , k, l ∈ N , 4 ≤ k + l ≤ k add M k,l and X k,l into C ( k, l ).(2A) For every k, l ∈ N odd, k + l ≤ k , l ≥ H = ( H, a , b ) ∈ C ( k, M ,l H (if b = b • i.e. b • is odd) or X ,l H (if b = b • is even) to C ( k, l ).(2B) For every k, l ∈ N even, k + l ≤ k , l ≥ H = ( H, a , b ) ∈ C ( k, b • = b • and both are either (a) odd or (b) even add either (a) M ,l H or (b) X ,l H to C ( k, l ).(3) Add into C also all rotations of the bilabelled graphs obtained in (2A) and(2B).(4) Repeat (2A)–(2B)–(3) until no new graphs appear.(5) Add M , , M , , M , into C . Add all possible tensor products and theirrotations to obtain bilabelled graphs K with unconnected K . Proposition 4.8.
Suppose that Algorithm 4.7 terminates after a finite number ofsteps. Then it indeed constructs all the elements of C ( k, l ) , k + l ≤ k , that is,bilabelled graphs satisfying (i)–(vi). In particular, this means that the sets C ( k, l ) are finite. Proof.
The algorithm is a modification of Algorithm 3.24. We just need to justifythe changes.First change is that in the steps (1)–(4) we construct only graphs K ∈ C ( k, l )such that K is connected. This is possible thanks to Lemma 4.4 that says that toconstruct K with K connected, we only need to add a vertex to a graph H such that H is also connected. So, to obtain all connected graphs K ∈ C ( k, l ), we do not haveto consider the unconnected ones. Finally, as we mentioned in Lemma 4.2, takinga bilabelled graph K satisfying (i)–(vi), its connected components also satisfy (i)–(vi). So, supposing the steps (1)–(4) terminate in finite time, we can obtain allunconnected elements of C ( k, l ) just using the tensor product on the connectedelements.Second change is that, in the new algorithm, we perform the step (2) only for m = 1 (step (2A)) and for m = 2 (step (2B)). First, notice that we do not have toconsider the case m = 0 since this would add a new connected component to thegraph. This would result in obtaining an unconnected graph unless the original one H contains no vertex. This forces us, however, to change the starting point of theinduction (1), where it is not enough to consider just the empty graph, but we haveto consider all the possible graphs with one vertex – those are M k,l and X k,l (whilewe have to exclude the rotations of X , as they do not satisfy (iv); the rotationsof M , can be included at the end, because M , acts as identity with respect tocomposition). The fact that we can keep m ≤ C ( k, l ) of graph satisfying (i)–(iii), but only the subsets C ( k, l )satisfying in addition the conditions (iv)–(vi). In step (2A), we cannot consider l = 1 as X , H would violate (iv) (and M , H = H , so this also need not beconsidered), so l ≥ b • i to be all oddwhen adding M m,l H since otherwise we would violate (iv). In step (2B) we require b • = b • since otherwise we would obtain a double edge and hence violate (v). Wedo not consider l = 0 since this would violate (iv), so l ≥ l − m , which is non-negative since m ≤ l ≥
2. Actually the only way it can stay the same is whenwe choose l = 2 in step (2B), otherwise it strictly increases. This means that ifwe want to compute all C ( k, l ) for k + l ≤ k , we can restrict to those spaces with k + l ≤ k already in the beginning, which gives us hope to get the result in a finiteamount of steps. Restricting m ≤ C ( k, l )only a finite amount of new elements. Consequently, if the algorithm finishes aftera finite amount of steps, it means that the sets C ( k, l ) are finite. (cid:3) Example 4.9.
As an example, let us go through the algorithm for k = 8. Wewill not distinguish between input and output vertices, so everything will be here“up to rotation”. Also we will always draw the graph K • instead of the full graph K . For clarity, we keep the distinction between even vertices by black circles andodd vertices by white circles . REE QUANTUM ANALOGUE OF COXETER GROUP D We initialize the algorithm by adding all the M ’s and X ’es, that is, , , , , , . Now, step (2A) adds the following graphs , , .
The step (2B) adds nothing at this point. We can apply again step (2A), namelyto the first graph of the last three, to obtain , , , .
Finally, to the first and to the third graph of the last results, we can apply (2B)and in both cases we get the same result . Proof of Theorem 4.3.
In this subsection, we are going to show that Al-gorithm 4.7 terminates and hence the sets C ( k, l ) are finite. In order to do so, wecan actually ignore the structure of the graphs and focus only on the vectors ofinput/output vertices ( a , b ).Let Σ be a countable alphabet partitioned into even and odd part Σ = Σ ∪ Σ (both parts countable, mutually disjoint). We define a language (i.e. a set of words) L ⊆ Σ ∗ as follows. For any a ∈ Σ and k ∈ N \ { } , we put a k ∈ L . In addition,we define the following two families of production rules.(A) Substitute any letter of a word w by l copies, l ∈ { , , , . . . } , of a new onethat has opposite parity and does not occur in w .(B) Suppose there are two consecutive letters (alternatively the first and the lastletter) in w that are not equal, but have the same parity. Then substitutethem by l copies, l ∈ { , , , . . . } , of a new letter that has opposite parityand does not occur in w .The following example illustrates the derivation of some words in L . We considerlower case letters to be even and upper case letters to be odd. aaaa (A) −−→ l =3 BBBaaa (A) −−→ l =5 BBBCCCCCaa (B) −−→ l =2 BBddCCCCaa
Note that all the production rules make the given word longer except for (B)with l = 2, which preserves the length. We say that a word w ∈ Σ ∗ is infinitelyiterable if one can iterate rule (B) with l = 2 forever.There are surely words, where one can never apply rule (B). For example, aaaa since all the letters are the same or aBcD since the parity alternates. It is not hardto think of examples, where one can use it only once such as abbb . On the otherhand, there are examples of words w ∈ Σ ∗ , where one can do the iteration foreversuch as aabb → aCCb → DCCD → aaCD → aabb → . . . We will say that a word w is strongly infinitely iterable (s.i.i.) if one can iteraterule (B) for l = 2 forever always taking two neighbouring letters and never the firstand last one. If a word is not infinitely iterable (i.i.), then it is not s.i.i. If a wordis not s.i.i., then none of its subwords is s.i.i. Lemma 4.10.
There is no infinitely iterable word in L . Proof.
We will prove it by induction. For the sake of contradiction, suppose that w ∈ L is i.i., but any shorter w ′ ∈ L is not. Note that the words a k , k ∈ N \ { } arenot i.i.; moreover, one cannot apply the rule (B) on them at all. Hence, without lossof generality, we can assume that w was made by some production rule from someother word w ′ that is not i.i. Equivalently, this means that w ′ is strictly shorterand w was not made by applying (B), l = 2 (otherwise w ′ would necessarily be i.i.).Consequently, w contains at least three identical consecutive letters. Withoutlost of generality, assume w = aaav , a ∈ Σ , v ∈ Σ ∗ , where those letters a weremade by the last operation. Undoing the last operation (and maybe doing somedifferent one), we see that we have also w ′ = bv ∈ L for some letter b ∈ Σ (possiblydifferent from a ). Since w ′ is shorter, it cannot be i.i. and hence v is not s.i.i.Since we assume that w is i.i., we can denote by ( w i ) i ∈ N the correspondingsequence of iterations. Since v is not s.i.i., there must be an index i ∈ N such that w i − = aaav i − but w i = aaBBv i or w i = Baav i B , where B ∈ Σ and v i wasmade from v i − by deleting the first resp. last letter. Since v i − was made bysome strong iterations from v and v i is its subword, it also cannot be s.i.i. Hence,we can repeat the argument.Choosing for example the first possibility, we must then reach some i ∈ N suchthat w i − = aaBBv i − , but w i = aaBccv i or w i = CaBBv i C , where c ∈ Σ , C ∈ Σ . Again, v i cannot be s.s.i., so we can iterate this procedure.Eventually, we end up with some i n ∈ N such that v i n is empty and, up torotation, we have w i n = a a a a · · · a n − a n a n such that the parity of the a i ’salternates. Obviously all words of L are of even length, so also w i n is. Consequently,the parity of a and a n is also different. But this means that one can no more applythe rule (B) on w i n . This is a contradiction. (cid:3) Proof of Theorem 4.3.
We are going to show that Algorithm 4.7 terminates. Be-cause of the restriction k + l ≤ k , the only way how the algorithm may notterminate is that there is some bilabelled graph K ∈ C ( k, l ) such that one can iter-ate the step (2B) with l = 2 on an appropriate rotation of the corresponding graphinfinitely many times as this is the only possibility how to preserve the number ofinput/output vertices.The rules (A) and (B) for the language L exactly correspond to steps (2A) and(2B) of Algorithm 4.7. Hence (if we choose the alphabet Σ appropriately), for anyconnected graph K = ( K, a , b ) ∈ C ( k, l ) (that is any graph that is obtained insteps (1)–(4) of the algorithm), we have w := a k · · · a b · · · b l ∈ L . If now onecould iterate the step (2B) with l = 2 on K ad infinitum, it would mean that wecan iterate step (B) with l = 2 on the word w , but this is impossible as we justproved in Lemma 4.10 (cid:3) Concluding remarks and open problems
Let us mention some additional observations and concluding remarks here. Inparticular, we believe that this work opens wide possibilities for further research,so we mention a number of open problems in this section.5.1.
Defining a category structure on C . We defined the collection of sets C ( k, l ) by restricting to special graphs in C ( k, l ) in order to obtain sets that arefinite while keeping the image under the functor T T τ . Much more natural, however, REE QUANTUM ANALOGUE OF COXETER GROUP D is not to look for subsets of C ( k, l ), but for quotient sets. This allows us to keepthe category structure. Definition 5.1.
We define an equivalence relation on the sets C ( k, l ) by taking thesymmetric and transitive closure of the following. Graph K is equivalent to K ′ if K ′ was made by (iv) a two-path contraction, (v) by erasing exactly two edges betweengiven two vertices or (vi) by erasing an isolated vertex that is not an input/outputvertex. The quotient sets are denoted by ¯ C ( k, l ) and form again a category. Remark 5.2.
As follows from Proposition 4.1, the functors F τ and T T τ do notpass to ¯ C ( k, l ). If K ′ was made from K by erasing a pair of edges between twovertices, then we have F τ K = 1 / F τ K ′ . If K ′ was made from K by erasing anisolated vertex of K ◦ , then F τ K = 4 F τ K ′ . There are two ways how to deal withthis.First possibility is to define T T τ on ¯ C ( k, l ) by fixing some representative – namelywe can take the smallest one, which is a graph from C ( k, l ). Then T T τ is well de-fined, but it is not a functor. The functorial property holds only up to a multiplica-tive constant (similarly as in the case of Banica–Speicher easy quantum groups,cf. [BS09, Prop. 1.9]). Second possibility is to introduce a linear structure on C asin the following definition. Definition 5.3.
Let C N -lin ( k, l ) be the vector space of formal linear combinationsof elements in C ( k, l ). We extend the category operations to C N -lin ( k, l ) to definea linear category. We define ¯ C N -lin ( k, l ) to be a quotient vector space of C N -lin ( k, l )with respect to the relations(iv) K = K ′ if K ′ was made from K by a two-path contraction,(v) K = 1 /N K ′ if K ′ was made from K by erasing a pair of edges betweentwo vertices(vi) K = N K ′ if K ′ was made from K by erasing a vertex v with ˜ d v = 0.This defines an interesting diagrammatic category, which is maybe worth study-ing further. Let us suggest some questions in the following sections.5.2. Looking for a fibre functor for ¯ C N -lin . The mapping T T τ defines a fibrefunctor (i.e. a functor to the category of matrices) on ¯ C N -lin for N = 4. But thedefinition of ¯ C N -lin works essentially for any complex number N . Therefore, wehave the following natural question. Question 5.4.
Are there some other fibre functors for ¯ C N -lin ? (Considering maybe N ∈ { , , , . . . } .)We are essentially looking for some pair of tensors interpreting the even and theodd vertices. Note an important fact that thanks to Proposition A.1 these tensorsdo not have to be permutation invariant in their indices.5.3. Functor injectivity, semisimplicity.
We were able to find finite subsets C ( k, l ) ⊆ C ( k, l ) describing the quantum group D +4 . The natural question is now,whether those are the minimal subsets. Question 5.5.
Is the set of intertwiners { T T τ K | K ∈ C (0 , k ) } linearly independentfor every k ∈ N ? Equivalently, is the functor T T τ injective on ¯ C ? That is, isthe category ¯ C isomorphic to C D +4 . It is known that such questions can actually be characterized within the diagramcategory itself without knowing the particular form of the corresponding functor.Hence, we can ask the same question for ¯ C N -lin for arbitrary N even though we donot know any fibre functor here yet: Question 5.6.
Fix any N ∈ C . Is the bilinear form ( K , H ) K ∗ H , where K , H ∈ ¯ C N -lin (0 , k ), non-degenerate for every k ∈ N ?Note also that this is equivalent to the Karoubi envelope of the correspondingcategory being semisimple. See e.g. [Jun19, FM20] for more detailed discussion ofsuch questions in case of partition categories.5.4. Law of characters and category isomorphisms.
We were able to provethat the sets C ( k, l ) are finite and we are able to generate them. However, we arenot able to count them. This is also an important question: Question 5.7.
Compute the number of elements C ( k, l ) = dim ¯ C N -lin ( k, l ).Note that if the functor T T τ is injective, answering this question gives us thedimensions of the fixed point spaces C D +4 (0 , k ) = Mor(1 , u ⊗ k ), which is a veryimportant quantity. A related question then is to determine the law of the character χ = P i u ii , see [BS09, Wor87].Computing these numbers for small k actually leads to a very exciting conjecture: Conjecture 5.8.
We have C (0 , k ) = C k , where C k are the Catalan numbers.Note that Catalan numbers are counting the non-crossing partitions as well asthe non-crossing pair partitions. More precisely, C k = N C (0 , k ) = N C (0 , k ) , where N C is the category of all non-crossing partitions and
N C is the category ofall non-crossing pair partitions (in the Banica–Speicher sense, without consideringthe linear structure). This leads to another even more exciting conjecture: Conjecture 5.9.
The category C is isomorphic to the product category N C × N C .Here, we consider the graph categories and partition categories without the linearstructure. Generalizing this to the linear case should be straightforward, one justneeds to tune the loop parameters N appropriately.Note that if such an isomorphism exists, then it has to map
7→ × ,
7→ × or the other way around (up to scaling in the linear case). We checked on someexamples that such a mapping indeed behaves well for small k , but we were notable to prove this in full generality.Cartesian products of partition categories are studied in [CW16]. An interest-ing fact is that the category C is by definition generated by and ; incontrast, it is an open problem whether N C × N C is generated by × and × .Solving this conjecture would then help solving the problems that we presentedbefore – it would provide us a new fibre functor for ¯ C N -lin and it would solvethe problem about semisiplicity since the bilinear form associated to N C is wellunderstood. REE QUANTUM ANALOGUE OF COXETER GROUP D Questions concerning (free) Coxeter D N . There are also many open ques-tions regarding our original motivation – finding a free analogue for Coxeter groupsof type D . An obvious question is whether we can do the same also for different N . Question 5.10.
Is there a free analogue of Coxeter groups D N for N > D N , that is, C := h d ⊗ N i N -lin and compute C + := C ∩ N C .Then we can ask whether C + is a non-crossing version of C . That is, whether h C + , i N -lin = C . This is not clear now as we are not able to compute thiscategory explicitly. Question 5.11.
Find an explicit description for the category h d ⊗ N i N -lin for arbi-trary N ∈ N .We may also study the quantum group D +4 itself. Another question might behow meaningful this quantum group is as a free analogue of D . Does it naturallyappear in some applications? For instance, we can ask the following. Question 5.12.
Is there a finite graph Γ such that its quantum automorphismgroup is isomorphic to D +4 ? Remark 5.13.
The Coxeter groups of type D of rank N are symmetry groups ofthe so-called N -dimensional demihypercube (created from the ordinary hypercubeby “taking only the odd vertices”). A natural candidate for the graph Γ fromthe question above should be the 4-demihypercube then. However, there is anexception exactly for N = 4. The 4-demihypercube is actually the complementof a graph consisting of four isolated segments. Hence, its automorphism group isthe hyperoctahedral group H = Z ≀ S and the quantum automorphism group isthe free hyperoctahedral quantum group H +4 = Z ≀ ∗ S +4 . (Here ≀ ∗ denotes the freewreath product , which describes the quantum automorphim group of n copies of agiven graph [Bic04].)In addition, note that D N is also the automorphism group of the so-called N -dimensional folded hypercube , but again only for N > N -dimensionalfolded hypercube is created from the ordinary N -hypercube by identifying oppositevertices or, alternatively, from ( N − SO − N for N > N -demihypercube. For N = 4, the folded hypercube graph coincideswith the full bipartite graph K , , which is the complement of K ⊔ K . Hence, itsautomorphism group is S ≀ Z and its quantum automorphism group is S +4 ≀ ∗ Z . Appendix A. The way of drawing the elements of C is unique As a side remark, we would like to mention a result that does not tell us muchabout the category C D +4 , but might be useful when studying some of the openproblems sketched in Section 5.We prove namely that the way of drawing the bilabelled graphs in C is unique.This may be good to know when studying ¯ C N -lin as an abstract category and, inparticular, when looking for its fibre functors. Proposition A.1.
For any K = ( K, a , b ) ∈ C ( k, l ) , there is a unique planar draw-ing of K . More precisely, there is (up to orientation and choice of the unbounded face) unique planar embedding of the graph K ⊙ and the orientation is fixed by fixingthe orientation of the enveloping cycle.Proof. First of all, we need to assume that K is connected. For the unconnectedcase, we can then use induction. If K is not connected, then, up to rotation, wehave K = H ⊗ H . From induction, H and H have unique drawing and finallyone can see that the only possible drawing of K is to put H and H side by side.We also need to assume that | V ( K • ) | ≥
2. But obviously all the graphs , M k,l , X k,l have a unique planar drawing being the only instances of K with | V ( K • ) | ≤ K ⊙ directlysince it is rarely 3-connected. Instead, we construct the graph K •⊙ from K • the same way as K ⊙ is constructed from K . That is, add the enveloping cycle α • k , . . . , α • , β • , . . . , β • k and edges { α • i , a • i } and { β • i , b • i } and finally one last vertexconnected to all the vertices of the enveloping cycle. Now, one can see that K •⊙ differs from K ⊙ just by the fact that some edges in K •⊙ are two-paths in K ⊙ , butanyway K •⊙ has a unique planar drawing if and only if K ⊙ has.So, take any K ∈ C ( k, l ) such that K (equivalently K • ) is connected and | V ( K • ) | ≥
2. We are going to prove that K •⊙ is 3-connected. To illustrate the idea,let us first prove that it is 2-connected. For the sake of contradiction, suppose that v is a separating vertex in K •⊙ . Since K • is connected, we must have v ∈ K • . Takesome connected component H • of K •⊙ \ { v } that does not contain the envelopingcycle and denote by m the number of edges between v and H ′ . This componentinduces a bilabelled graph H = ( H, ∅ , c ) ∈ C with c • , . . . , c • m being the neighboursof v in H • . (If v is odd then H = H • and c i = c • i are the neighbours of v . If v iseven, then we have to introduce new vertices c , . . . , c m that split the edges between v and H • .) Now, H • must contain at least two vertices since if there was a singlevertex u , then we would have ˜ d u = 1 in K , which violates (ii). Consequently, wecan use Lemma 4.4, to find a vertex u in H • with d • u ≤ H . This means that u appears at least twice in ( c • , . . . , c • m ) and hence there is a double edge between u and v in K , which is a contradiction.The proof of 3-connectedness is similar just slightly more complicated. Let { v, w } be the separating pair. Suppose first that both are actually elements of K • . Again,we find a connected component H • of K •⊙ \ { v, w } not containing the envelopingcycle and define the corresponding bilabelled graph H = ( H, ∅ , c ). Some of thevertices c • i are connected to v in K • and some are connected to w . Now, onecan modify the proof of Lemma 4.4 to show that there actually must be at leasttwo vertices u , u in H such that d • u i ≤ H and H • \ { u i } is connected. So,according to Lemma 4.6, both u , u must appear at least twice and consecutivelyin ( c • , . . . , c • m ). From this, it follows that at least of those two vertices must havea double edge with either v or w .Secondly, assume { v, α } is a separating pair, where v ∈ K • and α is an element ofthe enveloping cycle adjacent to some vertex a ∈ K • . Let H • be the component of K •⊙ \{ v, α } not containing the enveloping cycle but containing a (such a componentexists, otherwise v is a separating vertex and K •⊙ is not even 2-connected, whichwas already excluded above). Now, again there are at least two vertices u in H • such that d • u ≤ H , so one of them is not a and must be connected witha multiple edge to v in K . REE QUANTUM ANALOGUE OF COXETER GROUP D A pair of vertices of the enveloping cycle can be separating only if k + l = 2,which happens only for M , and this case was already discussed above as it doesnot satisfy | V ( K • ) | ≥ K = ( K, a , b ) ∈ C ( k, l ) and let us show that the orientation of thedrawing of K ⊙ is uniquely determined by the orientation of the enveloping cycle.So, suppose that reflecting a drawing of K ⊙ preserves the order of the envelopingcycle. In other words, define the word w := a k · · · a b · · · b l and suppose that thereflection of w (reading the word backwards) is a rotation of w . For now, supposealso that K is connected. We prove that K consists of a single path and hence itsdrawing is equivalent to the reflection.Without loss of generality, we can assume w = x m a i · · · a i k k y n a i k k · · · a i , where a i = a i +1 . Recall the algorithm generating all elements of C . The word w musthave been constructed by applying rules (A) and (B) on some initial word w = a k , k ∈ N \ { } . It is enough to show that we only used rule (A) as this leads tothe graph consisting of a single path. Note that the last operation must havecreated either x m or y n , without loss of generality, assume the second possibil-ity. Let us undo the last operation that created w . We claim that it must havebeen the operation (A). If it was (B), then the predecessor of w was of the form w ′ = x m a i · · · a i k k uva i k k · · · a i . If neither of u and v equals a k , then it is a contradic-tion with the generalization of Lemma 4.4 since there is only one letter appearingconsecutively at least twice in w ′ . If, say u = a k , then it is a contradiction with the3-connectivity of the associated graph we proved above since removing the vertex a k and the element of the enveloping cycle corresponding to v isolates the vertex v .So, w was created from some w ′ by applying rule (A). But this means that w ′ wasalso a “palindrom up to rotation”, so by induction w was created only by repeatedapplication of rule (A).If K is not connected, we again use induction. We have K = H ⊗ H (up torotation) and by induction hypothesis both H and H have planar drawings thatare equivalent to their reflections. Then this must hold also for K . (cid:3) References [Bic04] Julien Bichon. Free wreath product by the quantum permutation group.
Algebras andRepresentation Theory , 7:343–362, 2004. doi:10.1023/B:ALGE.0000042148.97035.ca.[BS09] Teodor Banica and Roland Speicher. Liberation of orthogonal Lie groups.
Advances inMathematics , 222(4):1461–1501, 2009. doi:10.1016/j.aim.2009.06.009.[CW16] Guillaume C´ebron and Moritz Weber. Quantum groups based on spatial partitions.2016, arXiv:1609.02321v1.[FM20] Johannes Flake and Laura Maassen. Semisimplicity and indecomposable objects in in-terpolating partition categories. 2020, arXiv:2003.13798v3.[Fre17] Amaury Freslon. On the partition approach to Schur–Weyl duality and free quantumgroups.
Transformation Groups , 22(3):707–751, 2017. doi:10.1007/s00031-016-9410-9.[Fre19] Amaury Freslon. Applications of non-crossing partitions to quantum groups, 2019. URL .[Gro20] Daniel Gromada. Group-theoretical graph categories. 2020, arXiv:2009.06998v1.[GW19] Daniel Gromada and Moritz Weber. Generating linear categories of partitions. 2019,arXiv:1904.00166v2.[GW20a] Daniel Gromada and Moritz Weber. Intertwiner spaces of quantum group sub-representations.
Communications in Mathematical Physics , 376:81–115, 2020.doi:10.1007/s00220-019-03463-y.[GW20b] Daniel Gromada and Moritz Weber. New products and Z -extensions of compact matrixquantum groups. To appear in Annales de l’Institut Fourier , 2020, arXiv:1907.08462v3. [Jun19] Stefan Jung. Linear independences of maps associated to partitions. 2019,arXiv:1906.10533v1.[KS97] Anatoli Klimyk and Konrad Schm¨udgen.
Quantum Groups and Their Representations .Springer-Verlag, Berlin, 1997.[Maa20] Laura Maaßen. The intertwiner spaces of non-easy group-theoretical quantum groups.
Journal of Noncommutative Geometry , 14(3), 2020. doi:10.4171/JNCG/384.[Mal18] Sara Malacarne. Woronowicz Tannaka–Krein duality and free orthogo-nal quantum groups.
Mathematica Scandinavica , 122(1):151–160, 2018.doi:10.7146/math.scand.a-97320.[Mar96] Paul Martin. The structure of the partition algebras.
Journal of Algebra , 183(2):319–358, 1996. doi:10.1006/jabr.1996.0223.[Mir16] Seyed Morteza Mirafzal. Some other algebraic properties offolded hypercubes.
Ars Combinatoria , 124:154–159, 2016. URL .[MR19] Laura Manˇcinska and David E. Roberson. Quantum isomorphism is equivalent to equal-ity of homomorphism counts from planar graphs. 2019, arXiv:1910.06958v2.[NS06] Alexandru Nica and Roland Speicher.
Lectures on the Combinatorics of Free Probability .Cambridge University Press, Cambridge, 2006.[NT13] Sergey Neshveyev and Lars Tuset.
Compact Quantum Groups and Their RepresentationCategories . Soci´et´e Math´ematique de France, Paris, 2013.[RW15] Sven Raum and Moritz Weber. Easy quantum groups and quantum subgroups of a semi-direct product quantum group.
Journal of Noncommutative Geometry , 9(4):1261–1293,2015. doi:10.4171/JNCG/223.[Sch20] Simon Schmidt. Quantum automorphisms of folded cube graphs.
Annales de l’InstitutFourier , 70(3):949–970, 2020. doi:10.5802/aif.3328.[Tim08] Thomas Timmermann.
An Invitation to Quantum Groups and Duality . European Math-ematical Society, Z¨urich, 2008.[Wan95] Shuzhou Wang. Free products of compact quantum groups.
Communications in Math-ematical Physics , 167(3):671–692, 1995. doi:10.1007/BF02101540.[Wan98] Shuzhou Wang. Quantum symmetry groups of finite spaces.
Communications in Math-ematical Physics , 195(1):195–211, 1998. doi:10.1007/s002200050385.[Web13] Moritz Weber. On the classification of easy quantum groups.
Advances in Mathematics ,245:500–533, 2013. doi:10.1016/j.aim.2013.06.019.[Wor87] Stanis law L. Woronowicz. Compact matrix pseudogroups.
Communications in Mathe-matical Physics , 111(4):613–665, 1987. doi:10.1007/BF01219077.[Wor88] Stanis law L. Woronowicz. Tannaka–Krein duality for compact matrix pseu-dogroups. Twisted SU ( N ) groups. Inventiones mathematicae , 93(1):35–76, 1988.doi:10.1007/BF01393687.
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