arXiv:2106.08787v1 [math.QA] 16 Jun 2021 u loteqatmoe.Ti a rtntcdb ag[ Wang by noticed first was c This only gra not ones. of possess sym quantum property often of the remarkable they also theory A objects, but classical classical groups. the are quantum of they of though terms terms stud in in to only also allowed but not This objects geometry. different non-commutative of the in groups pact bet partitions. – objects yWrnwc n[ in Woronowicz by [ group Banica later and group symmetric quantum free rniiegah pt 1vrie n eetetniewr of work [ extensive thesis recent PhD a his and in vertices summarized 11 is to up graphs transitive ult:H rvdta opc unu ru suiul determ uniquely th with is came group [ then result Speicher quantum useful Ta compact very A so-called a category. the that representation proved of He generalization a duality: formulated Woronowicz twiners. ftemnindtoatos[ authors two j the mentioned is the mentioning Worth of graphs. different of groups automorphism emty n18,Wrnwc nrdcdtento of notion the introduced [ Woronowicz 1987, In geometry. pc of space if n avdhpruegraphs. hypercube halved and UNU YMTISO ALYGAH FABELIAN OF GRAPHS CAYLEY OF SYMMETRIES QUANTUM Wor87 N nipratto o tdigcmatqatmgop a intro was groups quantum compact studying for tool important An adjective The 2020 Date tdigqatmsmere fgah tre ihtewr fB of work the with started graphs of symmetries quantum Studying ol iet hn oSmnShitfrdsusosaott about discussions for Schmidt Simon to thank to like would I hswr a upre ytepoetOVVCA CZ.02.1.01 CAAS OPVVV project the by supported was work This e od n phrases. and words Key ≥ fagah ic hn ayatoswre ndtriigtequ the determining on worked authors many then, Since graph. a of ue1,2021. 17, June : floigerirwr fDifl n ib)a h eeaiainof generalization the as Jimbo) and Drinfeld of work earlier (following ] 4. ahmtc ujc Classification. Subject Mathematics n h unu ymtiso h avdcb rp,tefold the graph, graphs. cube p Hamming halved this the the det Applying of and for symmetries strategy graphs. quantum such the general find of a groups develop automorphism We quantum symmetries. quantum of Abstract. N BS09 ons hsqatmgopi uhlre hntecasclgroup classical the than larger much is group quantum This points. ,wosoe o omdltoeitrwnr sn combinatorial using intertwiners those model to how showed who ], quantum Ban05 esuyCye rpso bla rusfo h perspectiv the from groups abelian of graphs Cayley study We Wor88 alygah yecb rp,Hmiggah unu symm quantum graph, Hamming graph, hypercube graph, Cayley ,wogv h ento fthe of definition the gave who ], h monoidal the – ] ntetteo hsatcerfr onon-commutative to refers article this of title the in S BB07 AILGROMADA DANIEL N + Introduction Sch20b steqatmgopo ymtiso finite a of symmetries of group quantum the as GROUPS eemnn unu ymtiso vertex of symmetries quantum determining ] 04 Piay;0C5 82 (Secondary). 18M25 05C25, (Primary); 20G42 ]. 1 ∗ ctgr frpeettosadinter- and representations of -category eqatmsmere ffolded of symmetries quantum he opc unu groups quantum compact unu automorphism quantum /0.0/0.0/16 Wan98 oko aiaand Banica of work e dcb graph cube ed asclsymmetries, lassical riigthe ermining oeue we rocedure, itpublication oint cmd,which Schmidt, h sta al- that is phs groups, metry ,wodefined who ], symmetries y co [ ichon nnaka–Krein ue again duced 019/0000778. ndb its by ined e antum Bic03 com- etry. S N ] 2 DANIEL GROMADA

This formalism can also be used when working with quantum automorphisms of graphs. Given a graph X, its quantum automorphism group can be defined as the unique compact matrix quantum group G, whose representation category (N) (N) is generated by the intertwiners T , T and AX , where AX is the adjacency matrix of X. In this paper, we study Cayley graphs of abelian groups. We use the intertwiner formalism to formulate a general algorithm for determining the quantum automor- phism groups of such graphs. The result is presented in Section 3 as Algorithm 3.2. It is based on the idea, which was already used in [BBC07] to determine the quan- tum symmetries of the hypercube graph, namely that the Fourier transform on the underlying group Γ diagonalizes the adjacency matrix of the Cayley graph of this group. The case of the hypercube graph is presented in Section 2 as a motivating example. As a side remark, let us mention the work of Chassaniol [Cha19], who uses the intertwiner approach to determine quantum symmetries of some circulant graphs, i.e. Cayley graphs of the cyclic groups. But apart from using the intertwiners, his techniques are different from ours. Subsequently, we use our algorithm to determine the quantum automorphism groups of certain Cayley graphs, which were not known before. This constitutes the main result of this article, which can be summarized as follows: Theorem. We determine the quantum automorphism groups of the following graphs. (a) [Theorem 4.1] For n 6=1, 3, the quantum symmetries of the halved hypercube 1 graph 2 Qn+1 are described by the anticommutative special orthogonal group −1 SOn+1. (b) [Theorem 5.1] For n 6=1, 3, the quantum symmetries of the folded hypercube graph FQn+1 are described by the projective anticommutative orthogonal −1 group P On+1. (c) [Theorem 6.1] For m 6= 1, 2, the quantum symmetries of the Hamming + graph H(n,m) are described by the wreath product Sm ≀ Sn. Note that quantum symmetries of the folded hypercube and the Hamming graphs were already studied before [Sch20a, Sch20c], but determining the quantum auto- morphism group for a general value of the parameters was left open.

1. Preliminaries In this section we recall the basic notions of compact matrix quantum groups and Tannaka–Krein duality. For a more detailed introduction, we refer to the monographs [Tim08, NT13]. 1.1. Notation. In this work, we will often work with operators between some tensor powers of some vector spaces. Therefore, we adopt the “physics notation” with upper and lower indices for entries of these “tensors”. That is, given T : V ⊗k → V ⊗l for some V = CN , we denote N j1···jl T (ei1 ⊗···⊗ eik )= Ti1···ik (ej1 ⊗···⊗ ejl ) j ,...,j =1 1 Xl j We will sometimes shorten the notation and write Ti using multiindices i = (i1,...,ik), j = (j1,...,jl). QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 3

1.2. Compact matrix quantum groups. A compact matrix quantum group is a i pair G = (A, u), where A is a ∗-algebra and u = (uj ) ∈ MN (A) is a matrix with values in A such that i (1) the elements uj, i, j =1,...,N generate A, (2) the matrices u and ut (u transposed) are similar to unitary matrices, i N i k (3) the map ∆: A → A ⊗ A defined as ∆(uj ) := k=1 uk ⊗ uj extends to a ∗-homomorphism. P Compact matrix quantum groups introduced by Woronowicz [Wor87] are gen- eralizations of compact matrix groups in the following sense. For a matrix group i i i G ⊆ MN (C), we define uj : G → C to be the coordinate functions uj(g) := gj. Then we define the coordinate algebra A := O(G) to be the algebra generated by i uj. 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 on some non-commutative compact underlying space. For this reason, we often denote A = O(G) even if A is not commutative. Actually, A also has the structure of a Hopf ∗-algebra. In addition, we can also define the C*-algebra C(G) as the uni- versal C*-completion of A, which can be interpreted as the algebra of continuous functions of G. The matrix u is called the fundamental representation of G. 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 is O O i i a surjective ∗-homomorphism ϕ: (G) → (H) sending uj 7→ vj . We say that G and H are equal if there exists such a ∗-isomorphism (i.e. if G ⊆ H and H ⊆ G). We say that G and H are isomorphic if there exists a ∗-isomorphism ϕ: O(G) → O(H) such that (ϕ ⊗ ϕ) ◦ ∆G = ∆H ◦ ϕ. We will often use the notation H ⊆ G also if H is isomorphic to a quantum subgroup of G. 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 + i i i∗ t t (ON ) := ∗-alg(uj , i,j =1,...,N | uj = uj ,uu = u u = idCN ). Note that this example was then further generalized by Banica [Ban96] into the + universal free orthogonal quantum group O (F ), where F ∈ MN (C) such that F F¯ = ± idCN and O + i −1 (O (F )) := ∗-alg(uj , i,j =1,...,N | u is unitary, u = F uF¯ ), i i∗ where [¯u]j = uj . 1.3. Representation categories and Tannaka–Krein reconstruction. For a compact matrix quantum group G = (O(G),u), we say that v ∈ Mn(O(G)) i i k is a representation of G if ∆(vj ) = k vk ⊗ vj , where ∆ is the comultiplica- tion. The representation v is called unitary if it is unitary as a matrix, i.e. i j∗ k∗ k P O k vkvk = k vi vj = δij . In particular, an element a ∈ (G) is a one- dimensional representation if ∆(a)= a ⊗ a. Another example of a quantum group PrepresentationP is the fundamental representation u. We say that a representation v of G is non-degenerate if v is invertible as a matrix (in the classical group theory, we typically consider only non-degenerate 4 DANIEL GROMADA representations). It is faithful if O(G) is generated by the entries of v. The meaning of this notion is the same as with classical groups: Given a non-degenerate faithful representation v, the pair G′ = (O(G), v) is also a compact matrix quantum group, which is isomorphic to the original G. A subspace W ⊆ Cn is called an invariant subspace of v if the projection P : Cn → Cn onto W commutes with v, that is, P v = vP . This then defines the subrepresentation w := vP = P v. However, w as a representation is degenerate. If we need to express the subrepresentation as a non-degenerate representation, we had better consider a coisometry U : Cn → Cm with P = U ∗U and define w′ := UvU ∗. A representation v is called irreducible if it has no non-trivial subrepresentations. For two representations v ∈ Mn(O(G)), w ∈ Mm(O(G)) of G we define the space of intertwiners Mor(v, w)= {T : Cn → Cm | T v = wT }. The set of all representations of a given quantum group together with those inter- twiner spaces form a rigid monoidal ∗-category, which will be denoted by Rep G. Nevertheless, since we are working with compact matrix quantum groups, it is more convenient to restrict our attention only to certain representations related to the fundamental representation. If we work with orthogonal quantum groups + + G ⊆ On (or G ⊆ O (F ) in general), then it is enough to focus on the tensor powers u⊗k since the entries of those representations already linearly span the whole O(G). + Considering G = (O(G),u) ⊆ O (F ), F ∈ MN (C), we define ⊗k ⊗l N ⊗k N ⊗l ⊗k ⊗l CG(k,l) := Mor(u ,u )= {T : (C ) → (C ) | Tu = u T }. The collection of such linear spaces forms a rigid monoidal ∗-category with the monoid of objects being the natural numbers with zero N0. Remark 1.1. The term rigidity means that there exists a duality morphism R ∈ ∗ + C(0, 2) such that (R ⊗ idCN )(idCN ⊗R) = idCN . For quantum groups G ⊆ O (F ), ij j the duality morphism is given by R = Fi . An important feature of rigidity is the so-called Frobenius reciprocity, which basically means that the whole category C is determined by the spaces C(0, k), k ∈ N0 since C is closed under certain rotations. Conversely, we can reconstruct any compact matrix quantum group from its representation category [Wor88, Mal18]. Theorem 1.2 (Woronowicz–Tannaka–Krein). Let C be a rigid monoidal ∗-category N ⊗k N ⊗l with N0 being the set of objects and C(k,l) ⊆ L ((C ) , (C ) ). Then there exists a unique orthogonal compact matrix quantum group G such that C = CG. We have + j ij C G ⊆ O (F ) with Fi = R , where R is the duality morphism of . We can write down the associated quantum group very concretely. The relations satisfied in the algebra O(G) will be exactly the intertwining relations: O i −1 ⊗k ⊗l C (G)= ∗-alg(uj , i,j =1,...,N | u = F uF¯ , Tu = u T ∀T ∈ (k,l)). We say that S is a generating set for a representation category C if C is the small- est monoidal ∗-category satisfying the assumptions of Theorem 1.2 that contains S. We use the notation C = hSiN (it is important to specify the dimension N of the vector space V = CN associated to the object 1). If we know such a generating set, it is enough to use the generators for our relations: O i −1 ⊗k ⊗l (G)= ∗-alg(uj , i,j =1,...,N | u = F uF¯ , Tu = u T ∀T ∈ S(k,l)). QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 5

1.4. Partitions. Representation categories of homogeneous orthogonal quantum + groups, that is, those G such that SN ⊆ G ⊆ ON are conveniently described using partitions. A partition p ∈ P(k,l) is a decomposition of k upper and l lower points into non-empty disjoint subsets called blocks. For instance,

p = q =

(N) N ⊗k Given any partition p ∈ P(k,l), we define a linear map Tp : (C ) → (CN )⊗l, whose entries are given as “blockwise Kronecker delta” – we label the k upper points by indices i1,...,ik and the l lower points by indices j1,...,jl and (N) j1,...,jl define [Tp ]i1,...,ik to be one if and only if for any given block of p all the corre- sponding indices are equal. For instance, working with the example above, we may write (N) j1j2j3j4 (N) j1j2j3j4 [Tp ]i1i2i3 = δi1i2i3j2j3 , [Tq ]i1i2i3i4 = δi2j3j4 δi3j2 . C (N) P Theorem 1.3 ([Jon94]). It holds that SN (k,l) = span{Tp | p ∈ (k,l)} for every N ∈ N.

We define PartN (k,l) := span P(k,l) the space of formal linear combinations of partitions. On this collection of linear spaces, we may define the structure of a monoidal ∗-category in terms of simple pictorial manipulations (see e.g. [GW20] for C details) such that they respect the category structure in Sn . In other words, the (N) Part C mapping T : n → Sn is a monoidal unitary functor. (Note that passing to the linear spaces is often omitted, but in this article, we need them.) + As a consequence, any homogeneous quantum group ON ⊇ G ⊇ SN can be described using some diagrammatic category of partitions. Thanks to the Tannaka– Krein duality, we also have the converse – any category of partitions defines a some homogeneous compact matrix quantum group [BS09]. For more information, see the survey [Web17] or the author’s PhD thesis [Gro20a]. Finally, let us mention that when working with anticommutative deformations of groups (see the next section), then it might (although sometimes might not) be convenient to use a deformed functor. Let p ∈ P(k,l) be a partition that does not (N) contain any block of odd size. Then we define a linear operator T˘p by ˘(N) j (N) j [Tp ]i = σiσj[Tp ]i, where σi is a certain sign function: Given a multiindex i = (i1,...,ik), we count the number of pairs (k,l) such that kil. If this number is odd, then σi = −1; otherwise σi = 1. See [GW21, Section 7].

1.5. Anticommutative deformations. In this work, we will often work with certain anticommutative deformations of classical groups. We say that a matrix u has anticommutative entries if the following relations hold

i j j i k k k k i j j i ukuk = −ukuk, ui uj = −uj ui , ukul = ul uk assuming i 6= j and k 6= l. As an example, let us mention the anticommutative orthogonal quantum group O −1 i (ON )= ∗-alg(uj | u =u ¯, u orthogonal, u anticommutative). 6 DANIEL GROMADA

There is a whole theory about q-deformations of classical groups, where taking q = 1 gives the classical case and q = −1 gives usually the anticommutative one, see [KS97] for more details. 1.6. Exterior products – classical case. In this section, we would like to recall the definition of exterior products in connection with the representation theory of classical groups. Let V be a vector space. We define the exterior product V ∧ V and, more ∧k ∧k generally, the exterior powers Λk(V )= V as follows. V is the vector subspace of V ⊗k generated by the elements 1 v ∧···∧ v = sgn(σ)v ⊗···⊗ v 1 k k! σ(1) σ(k) σ∈S Xk ⊗k ∧k We denote by Ak : V → V the coisometry mapping v1 ⊗···⊗vk 7→ v1 ∧···∧vk ∗ ⊗k ⊗k and call it the antisymmetrizer. (That is, AkAk : V → V is the projection onto V ∧k taken as a subspace of V ⊗k.) The motivation for such a definition is the following. Proposition 1.4. Let G be any group and V some G-module. Then V ∧k is always a submodule of V ⊗k. n In particular, we can consider G = On acting on V = C by standard matrix multiplication. Let us prove this statement from a quantum group point of view. Proof. Let u ∈ C(G) ⊗ GL(V ) be the representation of G on V . We need to prove ⊗k ∗ ∗ ⊗k that u AkAk = AkAku . Let us express both sides in coordinates.

⊗k ∗ i 1 i1 ik [u AkAk]j = sgn(σ)u −1 ··· u −1 k! σ (j1) σ (jk ) σ∈S Xk ∗ ⊗k i 1 σ(i1 ) σ(ik ) [A Aku ] = sgn(σ)u ··· u k j k! j1 jk σ∈S Xk i1 ik σ(i1) σ(ik ) The terms u −1 ··· u −1 and u ··· u differ just by reordering of the σ (j1) σ (jk) j1 jk factors. Since we assume that G is a classical group, the entries of u are commuta- tive, so the terms must be equal.  We denote by ∧k ⊗k ∗ u := Aku Ak the corresponding subrepresentation. ∧k n The dimension of the k-th exterior power V equals k , where n = dim V . The highest nonzero power is therefore the n-th, which is one-dimensional. Given
a representation u of some group G, the n-th exterior power of u equals to the determinant u∧n = det u. 1.7. Exterior products – anticommutative case. The concept of exterior prod- uct does not work in general for quantum groups. Let us revise it here for the case of anticommutative deformations. For this purpose we need to introduce some sort of “anticommutative antisym- metrization”. This should be basically the same thing as the usual symmetrization, but, in addition, we have to “throw out the diagonal” again. (Recall that classically we have v1 ∧···∧ vk = 0 whenever va = vb for some a,b.) QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 7

We define V ∧˘ k to be the vector subspace of V ⊗k generated by the elements

0 if va = vb for some a 6= b v1 ∧˘ · · · ∧˘ vk = 1 v ⊗···⊗ v otherwise ( k! σ∈Sk σ(1) σ(k) ⊗k ∧˘ k We denote by A˘: V →PV the coisometry mapping v1 ⊗···⊗vk 7→ v1 ∧˘ · · · ∧˘ vk. Remark 1.5. If we view anticommutative deformations as 2-cocycle twists of usual ∗ groups, then this procedure amounts to twisting the intertwiner AkAk of u. In this sense, the rest of this subsection might be considered as obvious, but it does not harm to recall the facts explicitly.

−1 Proposition 1.6. Consider G ⊆ On and denote by u its fundamental represen- ⊗k ˘∗ ˘ ˘∗ ˘ ⊗k tation. Then u AkAk = AkAku . In other words, this means that (Cn)∧˘ k is an invariant space of the representation u⊗k. We denote the corresponding subrepresentation by ∧˘ k ˘ ⊗k ˘∗ u := Aku Ak. Proof. Let us write both sides of the equation entrywise.

⊗k ˘∗ ˘ i 0 if ja = jb for some a 6= b (1.1) [u AkAk]j = 1 i1 ik u −1 ··· u −1 otherwise ( k! σ∈Sk σ (j1) σ (jk )

∗ ⊗k i 0P if ia = ib for some a 6= b (1.2) [A˘ A˘ku ] = k j 1 uσ(i1) ··· uσ(ik ) otherwise ( k! σ∈Sk j1 jk

i1 ik σ(i1) σ(ik ) Notice again that u −1 P··· u −1 and u ··· u coincide up to ordering σ (j1) σ (jk) j1 jk of the factors. If both i and j consist of mutually distinct indices, then the factors commute and hence the terms are equal. If ja = jb for some a 6= b, then the factors uσ(ia) and uσ(ib) mutually anticommute. Consequently, the symmetrization ja jb σ(i1) σ(ik ) σ uj1 ··· ujk equals to zero. The same applies in the case when ia = ib for some a 6= b.  P The dimension of the anticommutative exterior powers are again given by the binomial coefficients. So, taking the n-th power, we can define the anticommutative determinant as follows

( ∧˘ n 1 n π(1) π(n) det u := u = uσ(1) ··· uσ(n) = uσ(1) ··· uσ(n) σ∈S σ∈S Xn Xn π(1) π(n) π(1) π(n) = u1 ··· un = uσ(1) ··· uσ(n). π∈S π∈S Xn Xn Since we assume the anticommutativity, all the factors of the terms always mutually commute. From this, the different ways to write down the determinant follow. The definition is very similar to the one of classical determinant – we are only missing the sign of the permutation in the sum. Such an object is sometimes considered ( also in the classical theory of matrices, where it is called the permanent. Since det u −1 −1 is a one-dimensional representation of On , it defines a quantum subgroup SOn called the anticommutative special orthogonal quantum group

O −1 i ( (SOn )= ∗-alg(uj | u =u ¯, u orthogonal, u anticommutative, det u = 1). 8 DANIEL GROMADA ( Note that the relation det u = 1 can be seen also as an intertwiner relation ˘ ⊗n ˘ −1 −1 Anu = An. In other words SOn is a quantum subgroup of On that was created ⊗n by adding the intertwiner A˘n ∈ Mor(u , 1) to its representation category. Simi- ⊗n larly, SOn can be created from On by imposing the intertwiner An ∈ Mor(u , 1).

+ 1.8. Projective versions. Let G ⊆ ON be an orthogonal compact matrix quan- tum group and denote by u its fundamental representation. Then u ⊗ u is surely its representation, but its matrix entries may not generate the whole algebra O(G). Denote by O(P G) the ∗-subalgebra of O(G) generated by entries of u ⊗ u, i.e. ele- i k O ments of the form ujul . Then P G := ( (P G),u⊗u) is a compact matrix quantum group called the projective version of G.

q q Proposition 1.7. Consider N ∈ N odd. Then P ON ≃ SON . For our work, we need only q = +1 (classical case) and q = −1 (anticommutative case). Nevertheless, the statement and its proof actually does not depend on q and we could take any deformation of the orthogonal group here.

q Proof. Denote by u the fundamental representation of ON , so that u ⊗ u is the q fundamental representation of P ON . Denote by v the fundamental representation + of SON . O q O q We claim that there is a ∗-homomorphism α: (SON ) → (P ON ) mapping i i vj 7→ uj detq u

i First, note that uj detq u is a polynomial of even degree in the entries of u (since N i is odd, so the determinant is of odd degree) and hence uj detq u is indeed an element O q q of (P ON ). Secondly, it is easy to check that all relations of SON are satisfied by i the image. In particular the determinant equals to one since detq(uj detq u)i,j = 2 (detq u) = 1. O q O q On the other hand, there is surely a ∗-homomorphism β : (P ON ) → (SON ) i k i k mapping uj ul 7→ vj vl since this is nothing but the restriction of the quotient map O q O q (ON ) → (SON ). Finally, it is easy to check that both β ◦ α and α ◦ β equal to the identity, so the maps must actually be isomorphisms. 

2. Warm-up: Quantum symmetries of the classical hypercube In this section, we would like to revisit the result [BBC07, Theorem 4.2] saying that the quantum automorphism group of the n-dimensional hypercube graph is −1 the anticommutative orthogonal group On . Our aim is to explain the proof in slightly more detail and provide more explicit computations to make everything clear. Throughout the whole paper, we are going to rely heavily on the theory of representation categories and we will express everything in terms of intertwiners. This may seem a bit clumsy in this particular case (in comparison with the approach of [BBC07] for instance), but it will become very handy in the following sections, where we are going to study quantum symmetries of some other graphs. Using intertwiners, we will be able to formulate our approach in a very general way for arbitrary Cayley graphs of abelian groups. QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 9

2.1. Quantum automorphism group of a graph. We define the free symmetric + + quantum group [Wan98] SN = (C(SN ),u), where

O + i i 2 i i∗ i (SN )= ∗-alg(uj , i,j =1,...,N | (uj) = uj = uj , uk = 1). k X + It holds that SN describes all quantum symmetries of the space of N discrete + points. What we mean by this is that SN is the largest quantum group that faithfully acts on the space of N points. Let us look on this property in even more detail. Denote by XN = {1,...,N} the set of N points. We can associate to XN the algebra of all functions C(XN ), which has a basis δ1,...,δN of the canonical projections, that is, functions δi(j)= δij . An action of a quantum group G on XN is described by a coaction of the associated Hopf ∗-algebra, that is, a ∗-homomorphism C(XN ) → C(XN ) ⊗ O(G) satisfying some axioms. Now, note that since (δi) is a linear basis of C(XN ), the action of any compact N i quantum group on XN must be of the form δj 7→ i=1 δi ⊗ vj . The axioms of a coaction are now equivalent to the fact that v is a representation of the acting P quantum group g. The algebra C(XN ) can be defined as the universal C*-algebra 2 ∗ generated by δi satisfying the relations δi = δi = δi and i δi = 1. Now, it is easy to check that the requirement of the coaction being a ∗-homomorphism exactly + P corresponds to the defining relations of C(SN ). Alternatively, one can see the homomorphism condition also as some kind of an + + intertwiner relation. SN can be seen as a quantum subgroup of ON with respect to the relation uT (N) = T (N)(u⊗u), that is, requiring T (N) ∈ Mor(u⊗u,u). Here T (N) N N N (N) k is a tensor C → C ⊗ C defined by [T ]ij = δijk. See also [Ban99, Ban02]. + Actually, it is easy to check that the partition defining ON together with + + defining SN ⊆ ON generate all non-crossing partitions, that is, partitions where the strings do not cross. Let us denote by NC(k,l) the set of all partitions p ∈ P(k,l), which are non-crossing. This provides a “free quantum analogue” to Theorem 1.3 of Jones.

(N) Proposition 2.1 ([BS09]). It holds that C + (k,l) = span{Tp | p ∈ NC(k,l)} SN for every N ∈ N.

Now let X be a finite graph, so it has a finite set of vertices V := V (X). Let us number those vertices, so we can write V = {1,...,N}. The adjacency matrix of X is then an N × N matrix AX with entries consisting of zeros and ones such that j [AX ]i = 1 if and only if (i, j) is an edge in X. If X is undirected, then AX should i be symmetric, otherwise it need not. If we have [AX ]i = 1, we say that X has a loop at the vertex i. All concrete examples of graphs mentioned in this paper will be simple, i.e. undirected and without loops, but the general considerations will hold also for the directed case with loops. We say that a quantum group G acts on the graph X through the coaction j α: δi 7→ j δj ⊗ ui if the coaction α commutes with the adjacency matrix, that is, α ◦ A = (A ⊗ id) ◦ α. Equivalently, this means the uA = Au. The quantum P automorphism group of X is defined to be the universal quantum group acting on X. 10 DANIEL GROMADA

Definition 2.2 ([Ban05]). Let X be a graph on N vertices. We define the quantum automorphism group of X to be the compact matrix quantum group Aut+ X = (O(Aut+ X),u) with

O + i i 2 i i∗ i (Aut X)= ∗-alg(uj , i,j =1,...,N | (uj ) = uj = uj , uk =1, Au = uA). k X Equivalently, it is determined by its representation category

C (N) (N) (N) (N) Aut+ X = hT ,T , AX iN = hT ,T , AX iN , where ∈ P(0, 1) is the singleton partition. The equivalence of the two generating sets can be easily seen by noticing that T (N) = T (N) ∗T (N) on one hand and T (N) = T (N)T (N) on the other hand. See also [Cha19].

2.2. The n-dimensional hypercube. Consider a natural number n ∈ N. The n-dimensional hypercube graph Qn is determined by the vertices and edges of an n-dimensional hypercube. It can be parametrized as follows. n n The set of N := 2 vertices can be identified with the elements of the group Z2 . n We are going to denote these group elements by Greek letters α = (α1,...,αn) ∈ Z2 with αi ∈ {0, 1} ≃ Z2. We will denote the group operation as addition. We also denote the canonical generators by εi = (0,..., 0, 1, 0,..., 0). Two vertices α, β ∈ V (Qn) are connected by an edge if and only if they differ in exactly one of the n indices, that is, if β = α + εi for some i ∈ {1,...,n}. n Equivalently, we can say that Qn is the Cayley graph of Z2 with respect to the generating set ε1,...,εn. We can also express the edges via the adjacency matrix

α 1 if β = α + εi for some i, [AQn ]β = (0 otherwise.

Example 2.3 (n = 3). We mention the example of the ordinary three-dimensional cube and its parametrization using triples of zero/one indices. Note that shifting the vertex by εi, that is, flipping the i-th index, we move in the direction of the i-th dimension.

(0,1,1) (1,1,1)

(0,1,0) (1,1,0) ε2

(0,0,1) (1,0,1) ε3

(0,0,0) (1,0,0) ε1

Below, we show the corresponding adjacency matrix. Since it is not clear, how the n elements of Z2 should be ordered, we also labelled the rows with the corresponding tuples (the columns are ordered the same way). We also indicated the division of the matrix into blocks with respect to the number of ones in the tuple (essentially QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 11 the distance from the (0, 0, 0)-vertex).

0 1 1 1 0 0 0 0 (0,0,0)  1 0 0 0 0 1 1 0  (1,0,0)  1 0 0 0 1 0 1 0  (0,1,0)    1 0 0 0 1 1 0 0  (0,0,1) AQ =   3    0 0 1 1 0 0 0 1  (0,1,1)    0 1 0 1 0 0 0 1  (1,0,1)      0 1 1 0 0 0 0 1  (1,1,0)    0 0 0 0 1 1 1 0  (1,1,1)     2.3. Functions on the hypercube and Fourier transform. We denote by n C(Qn) the algebra of functions on the vertex set V (Qn)= Z2 . It has the canonical basis of δ-functions δα defined by δα(β)= δαβ. 2 We can also define the structure of a Hilbert space l (Qn) on the vector space of functions simply by putting hf,gi = α f(α)g(α). The basis (δα) is then or- thonormal with respect to this inner product. 2P There is another important basis on l (Qn) given by functions of the form

µ1 µn n τµ = τ1 ··· τn , where µ = (µ1,...,µn) ∈ Z2 and αi τi(α) = (−1) , so α1µ1+···+αnµn α·µ τµ(α) = (−1) = (−1) . n First, note that the elements τµ form a presentation of the group Z2 itself or, n alternatively, of the group algebra CZ2 . That is, we have τµτν = τµ+ν ; indeed, α·µ α·ν α·(µ+ν) (τµτν )(α) = (−1) (−1) = (−1) = τµ+ν .

Secondly, note that the basis (τµ) is orthogonal: n α·(µ+ν) 2 if µ + ν = 0, i.e. µ = ν, hτµ, τν i = (−1) = α (0 otherwise. X Let us explain more in detail the last equality, which will actually be useful also in subsequent computations. n Lemma 2.4. For β ∈ Z2 , it holds that 2n if β =0, (−1)α·β = α∈Zn (0 otherwise. X2 Proof. If β = 0, then (−1)α·β = (−1)0 = 1, so in the sum we are summing over 2n ones. Hence the result. If β has some non-zero entry, say βi = 1, then for every α, we can flip the i-th entry of α in order to flip the sign of (−1)α·β. Thus, there is an equal amount of +1 as −1 in the sum, so they cancel out.  2 2 α α·µ We define the transformation matrix F : l (Qn) → l (Qn) by Fµ = (−1) called the Fourier transform. It provides a transformation between the two bases: α τµ = α δαFµ . Thanks to the orthogonality property above, we have that F is, up to scaling, orthogonal. More precisely F −1 = 1 F ∗. P 2n 12 DANIEL GROMADA

Example 2.5 (n = 3). Let us again look on the case n = 3. The matrix F consists only of ±1 elements. For easier reading, we write only + or − in the matrix instead of +1 and −1. The order for the bases (δα) and (τµ) or, better to say, the order of 3 the tuples α ∈ Z2 are again indicated behind the matrix.

+ + + + + + + + (0,0,0)  + − + + + − − −  (1,0,0)  + + − + − + − −  (0,1,0)    + + + − − − + −  (0,0,1) F =      + + − − + − − +  (0,1,1)    + − + − − + − +  (1,0,1)      + − − + − − + +  (1,1,0)    + − − − + + + −  (1,1,1)     2.4. Applying Fourier transform to the intertwiners. Recall that the quan- + tum automorphism group of the hypercube Aut Qn should be the quantum sub- + group of SN with respect to the intertwiner relation uA = Au, where we denote for + short A := AQn . Equivalently, it is the quantum subgroup of ON with respect to relations uT (N) = T (N)(u ⊗ u) and uA = Au. On the first sight, it is not clear, which quantum group these relations define. In order to see this, we first apply the Fourier transform to the intertwiners. (Recall + that F is up to scaling orthogonal, so ON is invariant under F.) Let us look on an example first. Example 2.6 (n = 3). The most important intertwiner seems to be the adjacency matrix of the graph as only this carries the data of the graph itself. A straightfor- ward computation gives us a diagonal matrix

3 (0,0,0)  1  (1,0,0)  1  (0,1,0)    1  (0,0,1) Aˆ := F −1AF =      −1  (0,1,1)    −1  (1,0,1)      −1  (1,1,0)    −3  (1,1,1)     Writing some explicit matrices for T (N) would be slightly complicated, so we will get back to this tensor later. So, let us now do the computations for general n. For the adjacency matrix, we have n 1 1 [F −1AF]µ = (−1)α·µAα(−1)β·ν = (−1)α·µ(−1)(α+εi)·ν ν 2n β 2n α,β α i=1 X X X n 1 = (−1)α·(µ+ν) (−1)νi = δ (n − 2deg ν), 2n µν α ! i=1 ! X X QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 13 where deg ν is the number of ones in ν, which we will subsequently call the degree of ν. So, we can say that the Fourier image of the adjacency matrix is a direct sum of identities with some scalar factors n −1 (2.1) Aˆ := F AF = (n − 2k)I n (k) k M=0 Imposing Aˆ to be an intetwiner is equivalent to saying that every subspace 2 of l (Qn) consisting of elements with a given degree d (w.r.t. the basis (τµ)) is invariant. So, let us denote the invariant subspaces by 2 Vk := {f ∈ l (Qn) | deg f = k}. n Note that dim Vk = k . Note also that those spaces do not define a grading, but only a filtration on the algebra C(Qn).
So, denote byu ˆ := F −1uF the Fourier transform of the fundamental rep- resentation of the quantum automorphism group of the hypercube and byu ˆ = (0) (1) (n) uˆ ⊕ uˆ ⊕···⊕ uˆ the decomposition according to the invariant subspaces Vk. This was only how the adjacency matrix A transform under F. But we also need + to transform the intertwiners defining the free symmetric quantum group SN . Let (N) N N N n (N) αβ us consider the tensor T : C ⊗ C → C ⊗ C defined by [T ]γδ = δαβγδ. + (N) (N) ∗ (N) (This is indeed an intertwiner of SN since T = T T .) Thanks to the fact that u must be block diagonal (as we just derived), we can study such intertwiners restricted to such blocks. So, denote by Pk the (Fourier 2 transform composed with the) orthogonal projection l (Qn) → Vk and define the ˆ(N) −1 (N) (1) tensor T 1 1 := (P ⊗ P ) T (P ⊗ P ), which must be an intertwiner ofu ˆ . We 1 1 can compute its entries:

N N 1 ˆ( ) ij −1 ( ) εi,εj αi+βj +γk+δl [T 1 1 ] = [(F⊗F) T (F⊗F)] = (−1) δα,β,γ,δ kl εk ,εl 22n 1 1 α,β,γ,δ X 1/2n if in the tuple (i,j,k,l), 1 two and two indices are = (−1)αi+αj +αk +αl =  the same 22n α  X 0 otherwise  One can write down this result also using the partition notation as n ˆ(N) (n) (n) (n) (n) 2 T 1 1 = T + T + T − 2T 1 1 −1 This is well known to be the intertwiner defining the quantum group On . (See e.g. [GW21, Section 7]). Now, we may already see, where is this heading towards. As a side remark, not that one might also want to directly compute the projection (N) of T to the subspace V1. This is of course possible by doing a very similar computation. However, this intertwiner turns out to be equal to zero (since one can never “pair” a triple of indices).

2.5. Quantum automorphism group of the hypercube. In this section, we −1 prove that the quantum automorphism group of the hypercube graph is the On – a result originally obtained in [BBC07, Theorem 4.2]. 14 DANIEL GROMADA

Theorem 2.7. The quantum automorphism group of the n-dimensional hypercube −1 graph is the anticommutative orthogonal quantum group On with the representation F 1 ⊕ v ⊕ (v ∧˘ v) ⊕···⊕ v∧˘ n F −1, −1 where v is the standard fundamental representation of
On . Proof. Denote by u the fundamental representation of the quantum automorphism + −1 group of the hypercube Aut Qn. Denote byu ˆ := F uF the Fourier transform of u. We prove the theorem by a series of lemmata. We start with results derived in the previous section. Lemma 2.8. The representation uˆ decomposes as uˆ =u ˆ(0) ⊕ uˆ(1) ⊕···⊕ uˆ(n). Proof. Follows from the form of the Fourier transform of the adjacency matrix (2.1).  (1) −1 Lemma 2.9. The subrepresentation uˆ satisfies the defining relation for On . ˆ(N) (1)  Proof. Follows from T 1 1 being an intertwiner ofu ˆ . 1 1 (1) + Lemma 2.10. The subrepresentation uˆ is a faithful representation of Aut Qn. ν Proof. The representationu ˆ acts on Qn through the coaction τµ 7→ ν τν ⊗ uµ. n Since the algebra C(Qn) is generated by the invariant subspace V1 = span{τi} , P i=1 this coaction is uniquely determined by the coaction ofu ˆ(1) on this invariant sub- space. In particular, the entries ofu ˆ must be generated by the entries ofu ˆ(1).  + −1 Consequently, Aut Qn is a quantum subgroup of On . Now it remains to prove the opposite inclusion. n j Lemma 2.11. The mapping τi 7→ j=1 τj ⊗ vi extends to a ∗-homomorphism α: C(Q ) → C(Q ) ⊗ O(O−1). The subspaces V are invariant subspaces of this n n n P l action for every l =0,...,n.

Proof. We have to check that the images of the generators τi satisfy the generating relations. That is: ∗ 2 α(τi) = α(τi), α(τi) =1, α(τi)α(τj )= α(τj )α(τi). j The first one is obvious as both τi and vi are self-adjoint. For the second, we 2 j k j have α(τi) = j,k τj τk ⊗ ui ui . While τj commutes with τk, we have that ui k anticommutes with ui unless j = k. So, the entries for j 6= k subtract to zero and P j j we are left with j τj τj ⊗ ui ui = 1. Finally, the last one (assuming i 6= j) is again easy as we have α(τ )α(τ )= τ τ ⊗ ukul , where everything commutes (unless P i j k,l k l i j k = l, but for those summands we have τ τ ⊗ ukuk = 0). P k k k i j It remains to show that Vl are invariant subspaces. Take arbitrary element P τi1 ··· τil ∈ Vl (assuming i1 < ···

So, we are actually summing over elements τj1 ··· τjl ∈ Vl only, which is what we wanted to prove.  QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 15

−1 + This means that On is a quantum subgroup of Aut Qn, which finishes the + −1 proof that Aut Qn ≃ On . Finally, from the explicit expression (2.2), it is clear −1 that On acts on the invariant subspaces Vl indeed through the representations uˆ(l) = v∧˘ l. 

3. Cayley graphs of abelian groups: General picture The fact that the Fourier transform diagonalizes the adjacency matrix of the hypercube graph is not a coincidence. In the graph theory, it is a well known fact, which holds for any Cayley graph of an abelian group. (See [LZ19] for a nice survey on the spectral theory of Cayley graphs.) Let Γ be a finite abelian group. We denote by IrrΓ ⊆ C(Γ) the set of all irreducible characters (that is, one-dimensional representations; since Γ is abelian, all irreducible representations are in fact one-dimensional). Note that IrrΓ forms a basis of C(Γ) and expressing a function in this basis is exactly the Fourier transform on Γ. Let S ⊆ Γ be a set of generators of Γ. As in the last section, we are going to denote the elements of Γ by Greek letters and the operation on Γ as addition. The Cayley graph of the group Γ with respect to the generating set S denoted by Cay(Γ,S) is a directed graph defined on the vertex set Γ with an edge (α, β) for every pair of elements such that β = α + ϑ for some ϑ ∈ S. If S is closed under the group inversion, then the Cayley graph is actually undirected (for every edge, one also has the opposite one). So, the adjacency matrix of Cay(Γ,S) is of the form

β 1 if β = α + ϑ for some ϑ ∈ S, [A]α = (0 otherwise. Proposition 3.1 ([Lov75, Bab79]). Let Γ be a finite abelian group and S its gen- erating set. Denote by A the adjacency matrix associated to the Cayley graph Cay(Γ,S). Then IrrΓ forms the eigenbasis of A. Given χ ∈ IrrΓ, its eigenvalue is given by

(3.1) λχ = χ(−ϑ). ϑ∈S X Proof. This is just a straightforward check. Take any χ ∈ Irr Γ then we have α α  [Aχ] = Aβ χ(β)= χ(α − ϑ)= χ(−ϑ)χ(α)= λχχ(α). β∈ ϑ∈S ϑ∈S XΓ X X This result suggests the strategy for determining the quantum automorphism group for any Cayley graph corresponding to an abelian group: express everything in the basis of irreducible characters. Since this diagonalizes the adjacency matrix, the meaning of it as an intertwiner becomes obvious. On the other hand, it might be slightly more complicated to discover the meaning of the intertwiner T (N). Algorithm 3.2 (Determining Aut+ Cay(Γ,S)). Let Γ be an abelian group and S its generating set. We are trying to determine the quantum automorphism group Aut+ Cay(Γ,S) with fundamental representation u. In order to do so, we perform the following steps:

(1) Determine the irreducible characters of Γ. Suppose that Γ = Zm1 ×···× n αiµi Zmn . Then IrrΓ = {τµ}µ∈Γ with τµ(α) = i=1 γi , where γi is some primiteve mi-th root of unity for every i. Q 16 DANIEL GROMADA

(2) Determine the spectrum of AΓ using Equation (3.1). (3) Denoting by λ0 > λ1 > ··· > λd the mutually distinct eigenvalues of AΓ, determine the corresponding eigenspaces V0, V1,...,Vd. (Note that we will always have V0 = span{τ0}, where 0 is the group identity.) (4) The eigenspaces are invariant subspaces of u. To formulate it slightly differ- ently: We may define the Fourier transform F as the matrix corresponding α to the change of basis (δα) 7→ (τµ), that is Fµ = τµ(α). Then we can define uˆ = F −1uF, which decomposes as a direct sumu ˆ =u ˆ(0) ⊕ uˆ(1) ⊕···⊕ uˆ(d).

(5) Choose some of the spaces and define W := Vi1 ⊕Vi2 ⊕· · · in such a way that W generates C(Γ) as an algebra. (In our examples below, it will be enough to take W := V1, but it does not always have to be like this.) This means that v :=u ˆ(i1) ⊕ uˆ(i2) ⊕ · · · is a faithful representation of Aut+ Cay(Γ,S). (Since the coaction of u oru ˆ restricted to W must then again uniquely extend to the whole space C(X) and hence we can recover the whole u this way.) (N) (6) Any non-crossing partition p ∈ NC(k,l) defines an intertwiner Tp ∈ ⊗k ⊗l (N) −1 ⊗l (N) ⊗k Mor(u ,u ), where N = |Γ|. We define Tˆp := F Tp F , which (N) ⊗k ⊗l has to be an intertwiner Tˆp ∈ Mor(ˆu , uˆ ). It is actually enough to study the intertwiners corresponding to the block partitions bk,l ∈ P(k,l) – partitions, where all the k upper and l lower points are in a single block, (N) so [T ]β1,...,βl = δ . The entries of the Fourier transformed bk,l α1,...,αk α1,...,αk,β1,...,βl intertwiner are easily computed as

ˆ ν1,...,νl 1−k (3.2) [Tbk,l ]µ1,...,µk = N δµ1+···+µk ,ν1+···+νl . In particular, we can focus on the subspace W and study the relations (W ) ⊗k ⊗l (W ) (W ) ⊗l (N) ∗⊗k Tˆp v = v Tˆp , where Tˆp = U Tˆp U , where U is the coiso- metry V0 ⊕···⊕ Vd → W .

4. Halved hypercube The hypercube graph is bipartite and hence we can create a new graph of it by the procedure of halving – taking one of the two components of the associated distance- two-graph. Taking a natural number n ∈ N, we define the (n+1)-dimensional halved 1 hypercube graph 2 Qn+1 obtained by halving the ordinary hypercube Qn+1. That is, we take all the even vertices in Qn+1 (equivalently, all odd vertices) and connect by an edge every pair of vertices that were in the distance two in the original hypercube Qn+1. 1 There is also a simpler definition of 2 Qn+1. Take the n-dimensional hypercube Qn and add an additional edge for every pair of vertices in distance two. This is also 2 1 known as squaring the graph. It holds that Qn = 2 Qn+1. Using this description, we can write the adjacency matrix as follows

1 if β = α + εi for some i α [A]β = 1 β = α + εi + εj for some i 6= j 0 otherwise 1 2 Consequently, we see that 2 Qn+1 = Qn is a Cayley graph corresponding to the n n group Γ = Z2 with respect to the generating set S = {εi}i=1 ∪{εi + εj }i

Now we would like to determine the quantum automorphism group of the halved 1 hypercube graph 2 Qn+1. Let us first summarize some known results. For n ≤ 2, the graph is actually the full graph on N =2n vertices, so the quantum automorphism + group is the free symmetric quantum group SN (for n =0, 1, i.e. N =1, 2, this ac- tually coincides with the classical one SN ). For n = 3, the graph is the complement of the graph consisting of four isolated segments. Hence, its quantum automor- + + phism group is the free hyperoctahedral quantum group H4 = Z2 ≀∗ S4 . (Here ≀∗ denotes the free wreath product, which describes the quantum automorphism group of n copies of a given graph [Bic04].) 1 So, the question is what is the quantum automorphism group of 2 Qn+1 for n> 3. n The classical automorphism group is known to be Z2 ⋊Sn+1. More precisely, it is the n+1 index two subgroup of the hyperoctahedral group Hn+1 = Z2 ≀ Sn+1 = Z2 ⋊ Sn+1 (the symmetry group of the hypercube) imposing that the product of all the Z2- signs is equal to one. This group is also known as the Coxeter group of type D. −1 Since the quantum automorphism group of Qn+1 is On+1, we may expect that the 1 answer for the halved hypercube 2 Qn+1 should be the anticommutative special −1 orthogonal group SOn+1.

+ 1 4.1. Determining Aut 2 Qn+1. We follow Algorithm 3.2. We start by computing the spectrum using Equation (3.1): n 1 n λ = τ (ε )+ τ (ε + ε )= (−1)µi + (−1)µi+µj = l + l2 − , µ µ i µ i j µ 2 µ 2 i=1 i

−1 Proof. We follow the proof of Theorem 2.7. Let us first prove that SOn+1 ⊆ + 1 −1 Aut 2 Qn+1, that is, SOn+1 really acts on the halved hypercube. To do this, we are going to show that the mapping n+1 i τj 7→ τi ⊗ uj i=1 X extends to a ∗-homomorphism. Most of the work was already done in Lemma 2.11. It only remains to prove that the extra relation we have here τ1 ··· τn+1 = 1 is also preserved under this action. That is, we need to show that n+1

1= τi1 ··· τin+1 ⊗ ui11 ··· uin+1,n+1 i ,...,i =1 1 Xn+1 ( This is indeed true thanks to the anticommutative determinant relation det u = 1. Now for the converse direction, consider again the intertwiner Tˆ(N) and compute its restriction to V˜1. It is easy to check that we obtain the same formula 1/2n if in the tuple (i,j,k,l), two and two indices are ˆ(N) ij the same [T ]kl =   0 otherwise ˜ even on the “extended” spaceV1 = V1 ⊕Vn (this is the point, where the assumption −1 + 1 n 6=1, 3 is needed). Consequently, we have proven that On+1 ⊇ Aut 2 Qn+1. −1 Finally, it remains to find some intertwiner that would push us to the SOn+1. Consider the block partition bn+1 = ··· ∈ P(n +1, 0). The corresponding (N) intertwiner is then of the form [T ··· ]α1,...,αn+1 = δα1,...,αn+1 . We are interested ˜ ˜ ⊗(n+1) in restriction of its Fourier transform on V1 (more precisely, V1 ):

ˆ(N) 1 if (i1,...,in+1) is a permutation of (1,...,n + 1) [T ··· ]i1,...,in+1 = (0 otherwise (1) ⊗n This is exactly the antisymmetrizer A˘n ∈ Mor((ˆu ) , 1), which exactly corre- ( sponds to the relation detu ˆ(1) = 1.  4.2. Open problem. Let us finish this section with links to some open questions and related research. Note that there are free quantum analogues of the Coxeter groups of series A (that is, the symmetric groups Sn) and series B or C (that is, the hyperoctahedral groups Hn), but so far we do not have any really free analogue of the Coxeter series D (recall that series D consists exactly of the symmetry 1 groups of halved hypercubes 2 Qn). The series of the anticommutative special −1 orthogonal groups SOn is a liberation of the Coxeter groups of type D, but we should not call them free as they obey some sort of commutativity laws (namely the anticommutativity). Question 4.2. Is there a free analogue for the Coxeter groups of series D? One particular candidate was recently discovered in [Gro20b] for n = 4. For general n, the question is still open. If some candidates appear, then the natural follow-up question would be: Is there some graph, whose quantum symmetry is this? QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 19

5. Folded hypercube Folded hypercube is another graph that can be derived from the hypercube graph. Consider again n ∈ N. The (n + 1)-dimensional folded hypercube graph FQn+1 is a quotient of the hypercube Qn+1 obtained by identifying the opposite corners, so α ≡ α + ι, where ι = (1,..., 1) = ε1 + ··· + εn. By this, we end up with a graph having only half of the vertices, that is, N =2n. Also here, there is a more convenient description. The (n + 1)-dimensional folded hypercube can be obtained from the n-dimensional ordinary hypercube by connecting all the opposite corners by an additional edge. So, the adjacency matrix can be written as 1 if β = α + εi for some i, α [A]β = 1 if β = α + ι, 0 otherwise. In other words, it is the Cayley graph of Zn with respect to the generating set  2 {ε1,...,εn, ι}. Now, let us again review, what is known about its classical and quantum sym- metries. For n ≤ 2, the folded hypercube FQn+1 is just the complete graph on n + N = 2 vertices, so its quantum automorphism group is SN . For n = 3, it is the complete bipartite graph on 4 + 4 vertices, which is the complement of the disjoint + union K4 ⊔ K4, so its quantum automorphism group is S4 ≀∗ Z2 (again, we refer to [Bic04] for explanation of the free wreath product). So, the interesting area is n> 3. n The classical automorphism group of FQn+1 for n> 3 is of the form Z2 ⋊ Sn+1 [Mir16], but it is a different semidirect product than in the case of the halved cube. n+1 Here, we take the quotient group of Hn+1 = Z2 ⋊Sn+1 by identifying the (n+1)- tuple of signs with the opposite ones. In other words, it is the projective version −1 PHn+1. Therefore, we expect P On+1 to be the quantum automorphism group. In [Sch20c], it was proven for n even that the quantum automorphism group −1 −1 is actually SOn+1, which matches our guess since it is isomorphic with P On+1 according to Proposition 1.7.

+ 5.1. Determining Aut FQn+1. Let us now compute the eigenvalues of the ad- jacency matrix: n n µi µ1+···+µn λµ = τµ(εi)+ τµ(ι)= (−1) + (−1) i=1 i=1 X X = n − 2deg µ + (−1)deg µ = n − 4⌈1/2 deg µ⌉ Again, the eigenvalue depends only on the degree of µ, but again the values of λd := λµ with deg µ = d are not mutually distinct. In this case, we have λ2d−1 = λ2d. So, we have invariant subspaces V˜0 := V0, V˜2 := V1 ⊕ V2 and so on. That is, V˜2i = V2i−1 ⊕ V2i, i = 0, 1,..., ⌈n/2⌉ (with V˜n+1 = Vn if n is odd). + Denoting by u the fundamental representation of Aut FQn+1, we denote by uˆ =u ˆ(0) ⊕ uˆ(2) ⊕ u(4) ⊕ · · · the decomposition of its Fourier transform according to the invariant subspaces. Denote τn+1 := 1, so the elements τij := τji := τiτj with 1 ≤ i < j ≤ n + 1 form ˜ ˜ n+1 a basis of V2. In general, the basis of V2i is (τα) with α ∈ Z2 , deg α = 2i. We 20 DANIEL GROMADA

˜ n can use the basis of V2 as a generating set of C(Z2 )= C(FQn+1): τ 2 =1, τ ∗ = τ , C(FQ )= C∗ τ , 1 ≤ i < j ≤ n +1 ij ij ij . n+1 ij τ τ = τ τ , τ τ = τ  ij kl kl ij ij ik jk

Alternatively, one can view C(FQ ) as the subalgebra of C(Zn+1)= C(Q ) n+1 2 n+1 ∗ 2 ∗ C(Qn+1)= C (τ1,...,τn+1 | τi =1, τi = τi, τiτj = τj τi) generated by the elements τij = τji = τiτj . This exactly corresponds to the fact that FQn+1 is a quotient graph of Qn+1. Theorem 5.1. Consider n ∈ N \{1, 3}. The quantum automorphism group of the folded hypercube graph FQn+1 is isomorphic to the anticommutative projective −1 orthogonal group P On+1. It acts through the fundamental representation u with (i) ∧˘ 2i −1 uˆ = v , where v is the fundamental representation of On+1. Before proving this theorem, we need to do some preparatory work first. At this point, it is not even clear whether the prescribed representation v∧˘ 2i is a faithful −1 representation of P On+1. We are actually going to prove that v ∧˘ v is a faithful O L−1 representation. As a second step, we need to characterize (P On+1) by generators and relations in terms of this representation v ∧˘ v. Equivalently, we need to find suitable generators of the representation category C associated to v ∧˘ v. Only this allows us to use our standard machinery and prove Theorem 5.1. This preparatory work is done in Subsection 5.2. The proof of Theorem 5.1 itself is then formulated in Subsection 5.3.

5.2. Projective version represented by exterior product. In Section 1.8, we defined the projective version of a compact matrix quantum group again as a compact matrix quantum group with the fundamental representation of the form u ⊗ u. In the classical case, we have also another faithful representation at our disposal.

Proposition 5.2. Consider G ⊆ On, n > 1. Denote by u its fundamental repre- sentation. Then u ∧ u is a faithful representation of P G. Before formulating the proof, let us clarify the definition of projective groups. Our definition from Section 1.8 works for orthogonal groups only. For a general ma- trix group G ⊆ GLn, one typically defines its projective version to be the quotient P G = G/λI, λ ∈ C \{0} (thus, in particular, P GLn = GLn/Z(GLn)). Note that if G is orthogonal, then λI ∈ G only for λ = ±1. Therefore, assuming G ⊆ On, our definition P G = {A ⊗ A | A ∈ G} is compatible with the general one (since we can reconstruct A from A ⊗ A up to a global sign). Finally, let us mention that over C, we obviously have P GLn = PSLn, which is known to be a simple group, that is, it has no non-trivial normal subgroups.

1 n Proof. Denote V := C , so that GLn = GL(V ). Consider the homomorphism ϕ: GL(V ) → GL(V ∧ V ) mapping A 7→ A ∧ A. Since it maps multiples of identity to multiples of identity, it induces a homomorphismϕ ˜: P GL(V ) → P GL(V ∧ V ). Since the P GL(V ) is simple (and the mappingϕ ˜ is obviously non-trivial), we

1Credit for this proof goes to a math.StackExchange user Joshua Mundinger https://math.stackexchange.com/a/4049085/359512 QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 21 must have thatϕ ˜ is injective. Consequently, the kernel of ϕ is contained in scalar matrices. Now, we can restrict ϕ to any subgroup G ⊆ GLn. Note that we can factor ϕ ϕ ϕ as ϕ: G →1 G′ →2 G′′, where G′ = {A ⊗ A | A ∈ G} and G′′ = {A ∧ A | A ∈ G}. ′ ′′ We need to prove that ϕ2 : G → G is an isomorphism for G ⊆ On as we have ′ G ≃ P G in this case. Recall that we have the property ker ϕ ⊆{λI}. If G ⊆ On, we must actually have ker ϕ ⊆ {±I}. But we know that ker ϕ1 = {±I} and hence ϕ2 must indeed be an isomorphism. 

It is known that the anticommutative orthogonal group can be obtained from the ordinary one by a 2-cocycle twist. Consequently, the two quantum groups are monoidally equivalent. More precisely, there exists a monoidal isomorphism −1 of the corresponding representation categories Rep On → Rep On mapping the fundamental representations one to another (that is, there is also a monoidal iso- morphism C → C −1 ). See e.g. [BBC07, GW21]. This implies the following On On corollary.

−1 Corollary 5.3. Denote by u˘ the fundamental representation of On , n> 1. Then −1 u˘ ∧˘ u˘ is a faithful representation of P On .

Proof. Denote by u the fundamental representation of On. As mentioned above, −1 there is a monoidal equivalence Rep On → Rep On mapping u 7→ u˘. It is easy ∗ to check that this monoidal equivalence also maps A2A2 ∈ Mor(u ⊗ u,u ⊗ u) to ˘∗ ˘ A2A2 ∈ Mor(˘u⊗u,˘ u˘⊗u˘). Consequently, it must map u∧u tou ˘ ∧˘ u˘. Since the former is a faithful representation of P On, the latter must be a faithful representation of −1  P On .

In the following text, we will denote the projective orthogonal group represented by the matrix u ∧ u by Pˆ On = (O(P On),u ∧ u). (The hat should remind us about C the wedge product ∧.) Expressing the representation category PˆO in terms of the C n representation category On is easy: We only have to compose all the intertwiners with the antisymmetrizer A2 : x ⊗ y 7→ x ∧ y. That is, C (k,l)= {A⊗l T A∗⊗k | T ∈ C (2k, 2l)} Pˆ On (2) (2) On ⊗l ∗⊗k Pair = {A(2)TpA(2) | p ∈ n(2k, 2l)}, where Pairn = h , in = span{p | p is a pairing}. (Pairing is a partition, where all blocks have size two. By an old result of Brauer [Bra37], this is exactly the category corresponding to the orthogonal group. See also e.g. [BS09, Web17, Gro20a].) However, the question is, what is the generating set of this category. Finding some small set of generators is actually not so easy as it may seem on the first sight. In order to understand the following text, one needs to familiarize the category operations on linear categories of partitions (or at least the linear category of all pairings Pairn). The rough idea is that in order to perform the composition of two pairings, one should simply follow the lines and, if needed, replace all loops by the factor n. We refer to [GW20, Section 3] for more details. In the following computations, we are going to treat the antisymmetrizer as a ∗ projection rather than a coisometry, so A2 = A2. In this sense, it can be expressed 1 in terms of partitions as 2 ( − ). However, we are going to use a more con- venient notation: In the diagrams, we will denote the antisymmetrizer by . So, 22 DANIEL GROMADA for example, the antisymmetrization of will be denoted by , the antisym- 1 P metrization of the identity is = 2 ( − ). For any partition p ∈ (2k, 2l), we are going to denote its antisymmetrization by ˚p = ⊗l · p · ⊗k. Consequently, we can write (5.1) C (k,l)= {T | p ∈ Pair (2k, 2l)}, Pˆ On ˚p n so C is modelled by a diagrammatic category Pair◦ := {˚p | p ∈ Pair }. PˆOn n n Pair◦ Theorem 5.4. Consider n 6=2, 4, 6, 8. Then the category n is generated by the set { , }. Before proving the theorem, let us mention two important Corollaries: Corollary 5.5. Consider n 6=2, 4, 6, 8. Then the representation category C is Pˆ On generated by {T (n),T (n)}.

Proof. Follows directly from Theorem 5.4 and Equation (5.1). 

Corollary 5.6. Consider n 6=2, 4, 6, 8. Then the representation category C −1 is Pˆ On generated by {T˘(n), T˘(n)}.

−1 Proof. As we mentioned earlier, On is monoidally equivalent with On , the monoidal equivalence maps the fundamental representation u of On to the fundamental rep- −1 ⊗k ⊗l resentationu ˘ of On . As for the intertwiners, it maps Tp ∈ Mor(u ,u ) for ⊗k ⊗l p ∈ Pairn to T˘p ∈ Mor(˘u , u˘ ) defined at the end of Section 1.4. The Corol- lary then follows by restricting the monoidal equivalence to the full subcategory C C  P On ⊆ On and applying to Corollary 5.5. Now, we focus on the proof of Theorem 5.4. To make it easier to follow, we split it into several lemmata. Pair◦ First, let us do a small remark on rotations in the category n. This category C (as well as the category P On ) is rigid and the duality morphism looks like this: . This again allows to do rotations in the category, but those rotations look a bit different than in the original category Pairn. Consider some p ∈ P(2k, 2l). Let us call each pair of some (2i +1)-st and (2i +2)-nd point on either lower or upper row in p a two-point. When drawing ˚p, those two-points are highlighted by the ellipse . The element then allows to rotate those two-points in ˚p as a whole, not separately. For instance, rotating , we may obtain , but we cannot obtain . (The latter would actually equal to zero due to the antisymmetrization: 1 = 2 ( − ) = 0.) Pair◦ Lemma 5.7. Consider n 6= 2. Then the category n is generated by the set { , , , , }. Proof. Denote by C the category generated by the given generators. Notice that we have the duality morphism among the generators, so C is a rigid category and we can consider everything up to rotation now. For instance, we could equivalently consider instead of or instead of in the generating set. As the first step, we prove that ˚pk ∈ C for every k ∈ N \{1}, where pk = ⊗k ··· ∈ NC2(0, 2k). That is, pk is the rotation of . We do this by induction. The ˚pk for k = 2, 3, 4, 5 are among the generators, so we have the QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 23 initial step and a couple of others already by assumption. We construct any ˚pk by precomposing ˚pk−1 with ··· : 1 n − 2 1 = − − + = ˚p + ˚p ⊗ ˚p , 4 4 k 4 3 k−2   As the second step, we prove that ˚p ∈ C for every pairing p ∈ P2(0, 2k). We use the element , which allows to permute the two-points in ˚p. We claim that any p ∈ P2(0, 2k) can be obtained by such two-point permutations from some C C ˚pi1 ⊗···⊗˚pil . Since we already proved that ˚pi ∈ for every i> 1 and ∈ , this will finish the proof of the theorem. Note that thanks to the antisymmetrization, the order of the two points in a two-point is irrelevant (only affecting the ± sign). So consider any p ∈ P2(0, 2k). Take the first two-point and denote the corre- sponding points by pt1 and pt2. Take the point which is paired with pt2 and denote it by pt3. We denote by pt4 its neighbour that form a two-point with pt3. Perform a two-point permutation of p such that (pt3, pt4) is the second two-point. Call pt5 the point, which is paired with pt4 and continue in this manner until we find some pt2i1 , which is paired with pt1. At this point, we have that ˚p is a two-point per- P mutation of ˚pi1 ⊗ ˚q, where q ∈ 2(0, 2k − 2i1). If we use mathematical induction,  we may assume that q is already a two-point permutation of pi2 ⊗···⊗ pil . Lemma 5.8. Suppose n 6=4, 6, 8. Then both and are generated by { , }. Proof. Denote by C the category generated by { , }. Recall that is a rotation of . So, we can do the following computation in C : 1 1 = − = ( − ) 2 2   By rotation, this means that C contains the linear combinations − and − . Now, we can do the following (we leave out the detailed computation now): 4( − )2 = (n − 4) − 2 − 2 + + + . Subtracting 2( − ), 2( − ), , – those all being elements of C – we get that C must contain (n − 8) + . Finally, squaring this element, we get 2 2 2 (n − 8) (n − 2) (n − 8) (n − 8) + = + + 2(n − 8) + 4 4 Doing all the possible
subtractions and multiplying by 4, we get that C contains α , where α = (n − 8)2(n − 2)+8(n − 8)=(n − 4)(n − 6)(n − 8) Consequently, ∈ C unless n = 4, 6, 8. Since we also proved that (n − 8) + ∈ C , we now have that ∈ C .  Lemma 5.9. Suppose n 6=4. Then is generated by { , , , }. Proof. First, we generate the following two elements of C : (5.2) 2 = − ∈ C ,

(5.3) 2 = − ∈ C . 24 DANIEL GROMADA

Now, take the second element and permute the third and fourth two-point to obtain − . Adding the element (5.2), we get (5.4) − ∈ C . Now we go another way: Start with the element (5.2) and precompose it with 1 1 . We obtain 4 (n − 3) − 4 . Multiplying by four and adding (5.4), we finally get (n − 4) ∈ C .  Proof of Theorem 5.4. Follows directly from the lemmata above. Lemma 5.8 tells us that and generate and . Lemma 5.9 shows that those to- gether generate . Finally, Lemma 5.7 shows that all those together already Pair◦  generate the whole category n. In the formulation of Theorem 5.4, we needed to make the assumption n 6= 2, 4, 6, 8. We can easily repair the formulation to include the cases n = 6, 8 as well. Notice that the only place, where we needed the assumption n 6= 6, 8 was Lemma 5.8. So, we can modify the formulation of the theorem as follows: Pair◦ Proposition 5.10. Consider n 6=2, 4. Then the category n is generated by the set { , , , }. We will actually need a slightly more technical result in the sequel. Pair◦ Proposition 5.11. Consider n 6=2, 4. Then the category n is generated by the set { , ,p}, where (5.5) p = − − + + + .

Proof. We only need to show that the given intertwiners generate and . We do this by modifying the proof of Lemma 5.8. In Lemma 5.8, we showed that { , } actually generate the following el- ements (for which we did not yet need the assumption n 6= 6, 8): The element − . A rotation of this one is the element − . We also con- struct the element (n−8) + , which can be rotated into (n−8) + . Subtracting those together with , ∈ C from the element (5.5), we get that C contains (n − 7) . Consequently, it must contain both and . (For n = 7, we can use Lemma 5.8 directly.)  Finally, let us reveal the meaning of the misterious linear combination p from Equation (5.5). One can easily check that given a tuple of indices i1, j1, i2, j2, i3, j3, i4, j4 such that ik 6= jk, we have 1 if the indices can be paired ˘(n) (i1,j1),(i2,j2),(i3,j3),(i4,j4) (5.6) [Tp ] = (0 otherwise Indeed, the individual partitions just depict the ways of how the indiced can be paired. (The minus signs are there because of the crossings to compensate the minus sign given by the deformed functor T˘(n).) Thus, we have the following Corollary

Corollary 5.12. Consider n 6= 2, 4. Then the representation category C −1 is PˆOn (n) (n) (n) (n) generated by {T˘ , T˘ , T˘p }, where T˘p is given by Eq. (5.6). QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 25

5.3. Proof of Theorem 5.1. Denote by v the fundamental representation of −1 + −1 On+1. Theorem 5.1 says that Aut FQn is isomorphic with P On+1 acting through u withu ˆ = v∧˘ 2i. First step of the proof was provided by Corollary 5.3, where we showed that v ∧˘ v is a faithful representation of P O−1 and hence the whole L n+1 v∧˘ 2i must be a faithful representation. −1 −1 Secondly, we show that P On+1 acts on FQn+1, which then implies that P On+1 ⊆ L + Aut FQn+1. But there is no work to be done here as this is just a restriction of the −1 action of On+1 on Qn+1. Indeed, recall that we have the coaction α: C(Qn+1) → O −1 n i C(Qn+1) ⊗ (On+1) by τj 7→ i=1 τivj and that C(FQn+1) is the subalgebra of C(Qn+1) generated by even polynomials in τj . Restricting to this subalgebra, we P O −1 get exactly the desired coaction C(FQn+1) → C(FQn+1) ⊗ (P On+1). + −1 Finally, we need to prove the converse inclusion Aut FQn+1 ⊆ P On+1. So, we (2) −1 need to show thatu ˆ is a representation of P On+1. Assume for a moment that n +1 6= 2, 4, 6, 8. Thanks to Corollary 5.6, we only need to show that T˘(n+1) and T˘(n+1) are intertwiners ofu ˆ(1). The first one is just the orthogonality, which is automatic as Fourier transform preserves orthogonality. The second one is easy to (N) ⊗2 obtain by looking at the Fourier transform of T ∈ Mor(u,u ) projected to V˜2. Its entries are then

(N) (i1,j1),(i2,j2) 1 if (i1, j1,i2, j2,i3, j3) can be paired, [T ](i ,j ) = 3 3 (0 otherwise.

Note that due to the antisymetrization, we need to also assume i1 6= j1, i2 6= j2, (n+1) and i3 6= j3. We claim that this exactly matches the intertwiner T˘ . Indeed, the latter is just a symmetrization of T˘(n+1) = T (n+1). It is clear that the only way how to pair a tuple of indices i1 6= j1, i2 6= j2, i3 6= j3 is according to the partition (up to symmetrization, i.e. permuting neighbouring pairs of points). This finishes the proof for the case n 6=6, 8. For the cases n =6, 8, consider the intertwiner ofu ˆ(1) induced by . Again, the entries of T are zeros and ones depending on whether the indices can be 2222 (n+1) (n+1) paired or not. That is, T = NT˘p , where T˘p is given by Equation (5.6). 2222 Thus, the result follows from Corollary 5.12. 

5.4. Open problem. Let us again finish with some open problem. The hyperoc- tahedral group Hn can be seen not only as the symmetry group of the hypercube Qn, but also as the symmetry group of n copies of a segment K2 ⊔···⊔ K2. While −1 the quantum symmetry of the former is On , the quantum symmetry of the latter + −1 is Hn , which are two distinct quantum groups. We have just proven that P On is the quantum symmetry of the folded hypercube FQn. This result suggests the following question: Question 5.13. Is there some graph, whose quantum symmetry is described by + + the quantum group PHn for some n? Does PHn act on the set of N points for + + some N at all? (That is, do we have PHn ⊆ SN for some N?) This is related to a big question on whether there is a quantum analogue of the Frucht theorem, which was discussed recently in [BM21]. 26 DANIEL GROMADA

6. Hamming graphs n Hamming graph H(n,m) is the Cayley graph of the group Γ = Zm with respect to the generating set S = {aεi | i = 1,...,n, a = 1,...,m − 1}, where εi = (0,..., 0, 1, 0,..., 0) is the generator of the i-th copy of Zm. In other words, vertices n of H(n,m) are n-tuples of numbers a =0,...,m − 1 (that is, indeed, V = Zm) and two such tuples are connected with an edge if and only if they differ in exactly one coordinate. Another possible description is using the Cartesian product of graphs (see Sec- tion 6.3): Hamming graph H(n,m) is the n-fold Cartesian product of the full graph Km, that is, H(n,m)= Km  ···  Km. The classical automorphism group of H(n,m) is known to be the wreath product Sm ≀ Sn. About the quantum automorphism group, only partial results are known so far: + + • n = 1: H(1,m) is the full graph Km, so Aut H(1,m)= Sm. • m = 1: H(n, 1) contains just a single vertex. + −1 • m = 2: H(n, 2) is the hypercube Qn, so Aut H(n, 2) = On . • m = 3: H(n, 3) was proven to have no quantum symmetries [Sch20a], so + Aut H(n, 3) = Aut H(n, 3) = S3 ≀ Sn. • m > 3: H(n,m) was proven to have some quantum symmetries [Sch20a], but the explicit quantum group was not known. We are going to answer the question about the quantum automorphism group of Hamming graphs in full generality in the following theorem (by which we also answer Question 8.2(i) of Simon Schmidt’s PhD thesis [Sch20b]):

+ + Theorem 6.1. Consider m ∈ N \{1, 2}. Then Aut H(n,m) ≃ Sm ≀ Sn. Before formulating the proof itself, we would like to explain some important ingredients more in detail. 6.1. Full graph. As we just mentioned, a special case for n = 1 is the full graph Km. Of course, we know that the quantum symmetry group of the full graph is + the free symmetric quantum group Sm. Nevertheless, we would like to use this simple example to point out a certain subtlety that one needs to keep in mind when working with cyclic groups Zm for m 6= 2. So, the full graph Km is the Cayley graph corresponding to the group Zm and the generating set consisting of all elements of the group except for identity, so Km = Cay(Zm, Zm \{0}). We denote simply by a = 0,...,m − 1 the elements of ab Zm and by τa ∈ Irr Zm the characters τa(b)= γ , where γ is some primitive m-th root of unity. The spectrum of Km is hence computed as m−1 m−1 ab m − 1 a =0, λa = τa(b)= γ = b b (−1 otherwise. X=1 X=1 Now comes the important point we wanted to make in this subsection: The Fourier transform F on Zm, that is, the transformation of the bases (δa) → (τa) is a ab unitary, but not orthogonal! Its entries are [F]b = γ , so they are obviously not real (unless m = 2). To be more concrete, the basis elements τa, which are the ∗ columns of F are not self-adjoint, but satisfy τa = τm−a. ˆ −1 ˆ+ −1 + Consequently, if we denote Sm := F SmF and Sm := F SmF the symmetric group and the free symmetric quantum group represented by the Fourier transform QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 27 of the standard permutation matrices, then those matrix (quantum) groups are not ˆ ˆ+ + + orthogonal. Instead, they satisfy Sm ⊆ Sm ⊆ O (F ) ⊆ Um with F ∈ Mm(C) a a ∗ m−a defined by Fb = δa+b,m (indices modulo m). That is, ub = um−b . + Therefore, if we study the intertwiners of Sˆm ≃ Aut Km in this basis, then instead of the familiar maps such as T (m), T (m), T (m), we discover their Fourier transforms, which may look rather exotic.

Observation 6.2. The category C is generated by Tˆ(m) and Tˆ(m), where Sˆm

ˆ(m) ab ˆ(m) c [T ] = δa+b,m, [T ]ab = δa+b,c.

6.2. Wreath product of quantum groups. We should also explain, what does the ≀ sign in the formulation of Theorem 6.1 means. It is not the free wreath product of quantum groups, but the classical one:

Definition 6.3. Let G be a compact matrix quantum group with m×m fundamen- a + tal representation v = (vb ) and let H ⊆ Sm be a quantum permutation group with i n × n fundamental representation w = (wj ). Then we define the wreath product of G and H to be the quantum group O ⊗n O ai ai i G ≀ H := ( (G) ⊗ (H),u), where ubj = vb wj ,

i ai where we denote by v = (vb )a,b the fundamental representations of the n copies of G occurring in the definition of G ≀ H.

Remark 6.4. Let us state a few important remarks to this definition. (a) The wreath product G ≀ H is just a quantum subgroup of the free wreath product G ≀∗ H defined in [Bic04] given simply by passing from the free product O(G)∗n ∗ O(H) to the tensor product O(G)⊗n ⊗ O(H). (b) Although we defined the wreath product G≀H for compact matrix quantum groups using their fundamental representations, the construction works in general for arbitrary compact quantum groups and it does not depend on the particular fundamental representations (see [Bic04] for details). In par- ticular, if we take some other matrix realization G′ ≃ G of the quantum group G, then G′ ≀ H is isomorphic to G ≀ H. (c) As in the classical case, the wreath product G ≀ H has a semidirect product structure G×n ⋊ H. In particular, notice the following two facts: (d) The matrix w is a representation of G ≀ H. Therefore, H can be seen as a quantum subgroup of G ≀ H. (e) On the other hand, the matrices vi are not representations of G ≀ H – ai O a O the coproduct does not act on vb ⊆ (G ≀ H) as it does on vb ∈ (G). Therefore, the copies of G are not quantum subgroups of G ≀ H. (f) We can express

i ai∗ ai ai∗ ai ai ai wj = ubj ubj = ubj ubj , vb = ubj b i j X X X ai O ⊗n O That is, the entries ubj indeed generate the whole algebra (G) ⊗ (H). This remark is essential to notice that the definition above is a good defi- nition of G ≀ H as a compact matrix quantum group. 28 DANIEL GROMADA

6.3. Cartesian product of graphs. Given two graphs X1 and X2, we define their Cartesian product to be the graph X1  X2 with the vertex set V (X1  X2) = V (X1) × V (X2) and with an edge ((v1, v2), (w1, w2)) ∈ E(X1  X2) if and only if (v1, w1) ∈ E(X1) and v2 = w2 or if (v2, w2) ∈ E(X2) and v1 = w1. Alternatively, we can describe the Cartesian product by its adjacency matrix AX1X2 = AX1 ⊗IX2 +

IX1 ⊗ AX2 , where IXi denotes the identity matrix. The Cartesian product of graphs is associative and we can conveniently describe the product of n given graphs by the adajcency matrix

AX1···Xn = AX1 ⊗ IX2 ⊗ IX3 ⊗···⊗ IXn + IX1 ⊗ AX2 ⊗ IX3 ⊗···⊗ IXn + ···

b It is well known that if G acts on a finite space or a graph X by δa 7→ b δb ⊗va, then G ≀ S acts on the n-fold disjoint union X ⊔···⊔ X by δ 7→ δ ⊗ ubj. n ai b,jPbj ai (Notice that C(X ⊔···⊔ X)= C(X) ⊕···⊕ C(X); the indices i, j are indexing the P n copies of X or C(X) here.) Now, consider the Cartesian product X  ···  X. In this case, we have   m−1 C(X ··· X) = C(X) ⊗···⊗ C(X). Consider a basis (xi)i=0 of C(X) such that x0 = 1C(X) (if X is a regular graph, then we can consider the basis of a eigenvectors of AX ). Denote byv ˆb the entries of the action of G on X in this basis, b so xa 7→ b xb ⊗ vˆa. Denote xai := 1C(X) ⊗···⊗ 1C(X) ⊗ xa ⊗ 1C(X) ⊗···⊗ 1C(X), where the xa is on the i-th place. In the following we are going to prove that P bj   xai 7→ b,j xbj ⊗ uˆai extends to an action of G ≀ Sn on X ··· X, where bj bj j uˆai =v ˆaPwi . First, assume for a moment that this action really exists. Then it is easy to   determine, how it must act on the basis xa1,...,an := xa1 ⊗···⊗ xan of C(X ··· X):

m−1 n b1k1 bnkn xa1,...,an 7→ xb1k1 ··· xbnkn ⊗ uˆa11 ··· uˆann b ,...,b k ,...,k 1 Xn=0 1 Xn=1 m−1 b1σ(1) bnσ(n) = xb1σ(1) ··· xbnσ(n) ⊗ uˆa11 ··· uˆann b ,...,b σ∈S 1 Xn=0 Xn m−1 b1σ(1) bnσ(n) = x − ··· x − ⊗ uˆ ··· uˆ bσ 1(1) 1 bσ 1(n)n a11 ann b ,...,b σ∈S 1 Xn=0 Xn m−1 bσ(1)σ(1) bσ(n)σ(n) = xb11 ··· xbnn ⊗ uˆa11 ··· uˆann b ,...,b σ∈S 1 Xn=0 Xn m−1 ˆb1,...,bn = xb1,...,bn ⊗ u˜a1,...,an , b ,...,b 1 Xn=0

b σ(1) b σ(n) ˆb1,...,bn σ(1) σ(n) where u˜a1,...,an = σ∈Sn uˆa11 ··· uˆann . We can also change the basis to b ,...,bn the standard one and obtain δa ,...,a 7→ δb ,...,b ⊗ u˜ 1 , whereu ˜ is P 1 n b1,...,bn 1 n a1,...,an ˆ given by a formula analogous to the one forPu˜. QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 29

Lemma 6.5. Let G be a compact matrix quantum group with fundamental repre- sentation v. Then n b σ b σ n b k b1,...,bn σ(1) (1) σ(n) ( ) k1 1 bkn kn u˜a1,...,an := ua11 ··· uann = ua11 ··· uann σ∈S k ,...,k Xn 1 Xn=1 + is a representation of G ≀ Sn. If G ⊆ Sm, then u˜ is faithful. Proof. First, we should prove the second equality in the formula. To see this, it is bk1 k1 bkn kn enough to notice that the product ua11 ··· uann equals to zero whenever ki = kj ki kj for some i 6= j (since wi wj = 0). Proving thatu ˜ is a representation of G ≀ Sn is a straightforward computation: b σ b n b1,...,bn σ(1) (1) σ(n) ∆(ua1,...,an )= ∆(ua11 ) ··· ∆(uann ) σ∈S Xn b σ(1) b σ(n) = u σ(1) ··· u σ(n) ⊗ uc1k1 ··· ucnkn c1k1 cnkn a11 ann σ∈Sn c1,...,cn X k1X,...,kn b ρ(1) b ρ(n) d π = u ρ(1) ··· u ρ(n) ⊗ u 1 (1) ··· udnπ(n) d11 dnkn a11 ann d ,...,d π,ρ∈S 1X n X n = ub1,...,bn ⊗ ud1,...,dn d1,...,dn a1,...,an d ,...,d 1X n To get from the second to the third line, we need to notice several things: First, as c1k1 cnkn we already mentioned, the product ua11 ··· uann equals to zero unless k1,...,kn is a permutation of 1,...,n. Hence, we can denote this permutation by π, so ki = π(i). Secondly, the terms of the product mutually commute, so we can reorder the −1 −1 − − bσ(1)σ(1) bσ(n)σ(n) bσ(π 1(1)) σ(π (1)) bσ(π 1(n)) σ(π (n)) first product as u ··· u = u − ··· uc −1 n . c1k1 cnkn cπ 1(1)1 π (n) −1 Finally, we denote ρ := σ ◦ π and di := cπ−1(i). Assume now that G ⊆ Sm for some m. The proof of the last statement – that m−1 u˜ is faithful – gets a bit easier if we work in the basis (xa)a=0 of C(X) such that a x0 =1C(X) since x0 is an invariant vector of v. So denote byv ˆb the entries of v in ai ai i 0 0 a the basis (xa) and similarlyu ˆbj :=v ˆb wj . We have thenv ˆ0 =1 andv ˆb =0=ˆv0 for every a, b. We need to show that the entries of u˜ˆ already generate the whole algebra O(G ≀ Sn). Of course, it is enough to show that it generates the generators ai i ˆ0,...,0,a,0,...,0 ai vˆb and wj . We claim that u˜0,...,0,b,0,...,0 =u ˆbj , where the a is on the i-th position and the b is on the j-th position on the left-hand side. Indeed, we get n ˆ0,...,0,a,0,...,0 ai k1 kn ai i ai  (6.1) u˜0,...,0,b,0,...,0 = vˆb w1 ··· wn =ˆvb wj =u ˆbj . k1,...,kn=1 exceptX for kj := i

+ + Proposition 6.6. Let Γ be a graph. Then Aut (Γ  ···  Γ) ⊇ (Aut Γ) ≀ Sn.   More precisely, G≀Sn acts faithfully on the n-fold product Γ ··· Γ by δa1,...,an 7→ δ ⊗ u˜b1,...,bn . b1,...,bn b1,...,bn a1,...,an PProof. Notice that we can expressu ˜ in a more “matricial way”

σ(1) σ(n) u˜ = Tσ(v ⊗···⊗ v )δσ, σ∈S Xn 30 DANIEL GROMADA

m ⊗n m ⊗n where Tσ : (C ) → (C ) is the linear operator permuting the tensor factors σ(1) σ(n) according to σ and δσ := w1 ··· wn (it is actually indeed the delta function Sn → C mapping δσ(π) = δσπ). We will use this matrix approach throughout the proof, but one could of course also rewrite the computation in terms of the matrix entries. + We want to show that (Aut Γ) ≀ Sn represented by the faithful representation u˜ is a quantum subgroup of Aut+(Γ  ···  Γ). To do this, we need to show that the representation category associated tou ˜ contains the generators of the category C (mn) (mn) ˜ ˜ Aut+(Γ···Γ), which are T , T , and A, where A is the adjacency matrix of   Γ ··· Γ. n We start with the singleton T (m ), which is the easiest one. Notice first that n T (m ) = T (m) ⊗···⊗ T (m). Consequently,

n n (m ) σ(1) σ(n) (m ) uT˜ = Tσ(v ⊗···⊗ v )T δσ σ∈S Xn σ(1) (m) σ(n) (m) = Tσ(v T ⊗···⊗ v T )δσ σ∈S Xn (m) (m) (mn) = Tσ(T ⊗···⊗ T )δσ = T , σ∈ S Xn where we used the fact that δ =1 . σ∈Sn σ C(Sn) To show the second intertwiner relation, denote by R the natural “disentangling m m ⊗n Pm ⊗n m ⊗n operator” (C ⊗C ) → (C ) ⊗(C ) mapping (x1 ⊗y1)⊗···⊗(xn ⊗yn) 7→ (mn) (m) (m) (x1 ⊗···⊗ xn) ⊗ (y1 ⊗···⊗ yn). Then we can write T = (T ⊗···⊗ T )R. So,

n n (m ) σ(1) σ(n) (m ) uT˜ = Tσ(v ⊗···⊗ v )T δσ σ∈S Xn σ(1) (m) σ(n) (m) = Tσ(v T ⊗···⊗ v T )Rδσ σ∈S Xn (m) σ(1) σ(1) (m) σ(n) σ(n) = Tσ(T (v ⊗ v ) ⊗···⊗ T (v ⊗ v ))Rδσ

σ∈Sn X n (m ) σ(1) σ(n) σ(1) σ(n) = TσT ((v ⊗···⊗ v ) ⊗ (v ⊗···⊗ v ))δσ

σ∈Sn X n (m ) σ(1) σ(n) σ(1) σ(n) = T (Tσ ⊗ Tσ)((v ⊗···⊗ v ) ⊗ (v ⊗···⊗ v ))δσ

σ∈Sn Xn = T (m )(˜u ⊗ u˜).

˜ n Finally, we prove thatu ˜ commutes with A := i=1 id ⊗···⊗id ⊗A⊗id ⊗···⊗id, where A is the adjacency matrix of Γ and in each summand it appears at the i-th P QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 31 factor of the tensor product. n σ(1) σ(i) σ(n) u˜A˜ = Tσ(v ⊗···⊗ v A ⊗···⊗ v )

σ∈Sn i=1 X Xn σ(1) σ(i) σ(n) = Tσ(v ⊗···⊗ Av ⊗···⊗ v )= A˜u˜ σ∈S i=1 Xn X 

Remark 6.7. The inclusion in Proposition 6.6 may and may not be strict. Ham- ming graphs H(n,m) provide examples for both. Taking m = 2 and n ≥ 2, we +   −1 + have Aut (K2 ··· K2) = On ) S2 ≀ Sn = (Aut K2) ≀ Sn. By [Sch20a], we + + have equality for m = 3: Aut (K3  ···  K3) = S3 ≀ Sn = (Aut K3) ≀ Sn. We are going to prove the equality for any m> 2 in the case of Hamming graphs.

m−1 Remark 6.8. Let (xa)a=0 be a basis of C(X) such that x0 = 1C(X) and denote xai = 1C(X) ⊗···⊗ 1C(X) ⊗ xa ⊗ 1C(X) ⊗···⊗ 1C(X) as we already did once. Equation (6.1) shows that the action from Proposition 6.6 indeed maps xai 7→ bj b,j xbj uˆai. n 6.4.P Proof of Theorem 6.1. Recall that H(n,m) is the Cayley graph of Zm with respect to the generating set {aεi | i = 1,...,n, a = 1,...,m − 1}. We denote n n α1µ1+···+αnµn by τµ, µ ∈ Zm the irreducible characters of Zm defined by τµ(α)= γ , where γ is some primitive m-th root of unity. As usual, we start with determining the spectrum using Proposition 3.1:

n m−1 µia λµ = γ = mlµ − n, i=1 a=1 X X where lµ = #{i | µi = 0}. So, the spectrum contains n + 1 distinct eigen- values n(m − 1), n(m − 2),..., −n. Denote by V0,...,Vn+1 the corresponding eigenspaces Vi = span{τµ | lµ = n − i}. So, for instance V0 = span{τ0,...,0}, a V1 = span{τai | i =1,...,n, a =1,...,m − 1}, where we denote τai := τi = τµ for µ = (0,..., 0, a, 0,..., 0) – the a being on the i-th place. Those must be invariant subspaces of the fundamental representation of Aut+ H(n,m). +   In Proposition 6.6, we showed that Sm ≀ Sn acts on H(n,m)= Km ··· Km bj + + via τai 7→ b,j τbj ⊗ uai (see Remark 6.8), so Aut H(n,m) ⊇ Sm ≀ Sn. It remains to show the opposite inclusion. DenoteP by u the fundamental representation of Aut+ H(n,m). Denote byu ˆ := −1 F uF the Fourier transform of u, that is, the matrix u expressed in the basis of τµ. This matrix decomposes into a direct sum with respect to the invariant subspaces (0) (1) (n) ai (1) V0, V1,...,Vn asu ˆ =u ˆ ⊕ uˆ ⊕···⊕ uˆ . We denote byu ˆbj the entries ofu ˆ . It is enough to show that this matrixu ˆ(1) satisfies the relations of the fundamental ˆ+ representation of Sm ≀ Sn. So let us study its intertwiners. Recall the formula (3.2) for computing the Fourier transform of intertwiners cor- ˆ ν1,...,νl 1−k responding to block partitions [Tbk,l ]µ1,...,µk = N δµ1+···+µk ,ν1+···+νl . We start by taking p = and focus on the entries of Tˆ(N) corresponding to the invariant sub- ˆ(N) bj ˆ(N) space V1 and see that [T ]a1i1,a2i2 = δi1i2j δa1+a2,b. Let us denote R := T 1 1 ∈ 1 32 DANIEL GROMADA

(1) (1) (1) (N) Mor(ˆu ⊗ uˆ , uˆ ) the restriction/projection of Tˆ onto V1. Let us also denote ∗ (1) ⊗2 (1) ⊗2 b1j1,b2j2 R := R R ∈ Mor(ˆu , uˆ ), so that [R ]a1i1,a2i2 = δi1i2j1j2 δa1+a2,b1+b2 . (N) Next, let us study the intertwiner Tˆ . Its projection onto V1 can be expressed as

ˆ(N) NT 1 1 = R + R + R + R . 1 1

Here, we use the following notation

b1j1,b2j2 [R ]a1i1,a2i2 = δa1+a2,0δb1+b2,0δi1=i2=6 j1=j2 ,

b1j1,b2j2 [R ]a1i1,a2i2 = δa1,b2 δa2b1 δi1=j2=6 i2=j1 , [R ]b1j1,b2j2 = δ δ δ , a1i1,a2i2 a1b1 a2b2 i1=j1=6 i2=j2 where we use slightly more general notation for the deltas, which is hopefully self- descriptive: For instance, δi1=j1=6 i2=j2 equals to one if i1 = j1 6= i2 = j2 and otherwise it equals to zero. The idea behind the diagrams is that the dashed and dotted blocks idicate the fact that the i, j indices corresponding to the different blocks must not coincide. We already know that the map R is an intertwiner, which implies that also the sum R + R + R must be an intertwiner. We are going to show that actually each term of this sum is an intertwiner: First, compute the square of the sum. Obviously, R R =0= R R , so it is actually quite easy: (R + R + R )2 = 2(m − 1)R + 2R + 2R . Here, [R ]a1i1,a2,i2 = δ δ δ (both blocks are dashed so all of b1j1,b2j2 a1+a2,m b1+b2,m a1a2b1b2 the a1,a2,b1,b2 do have to coincide). That was not very helpful actually, but in a similar manner, we can compute the third power: (R + R + R )3 = 4(m − 1)2R +4R +4R . Subtracting four times the original sum, we get 4m(m − 2)R , so R is an intertwiner unless m = 2. But now R is just a rotation of R , so it must also be an intertwiner. Consequently, also R must be an intertwiner and also R must be an intertwiner. Those are all intertwiners we need. Now, we just look at the relations they imply. First, the intertwiner R implies the following relation:

m−1 m−1 (6.2) uˆa1i1 uˆa2i2 δ δ = uˆdi1 uˆm−d,i2 δ δ cj1 m−c,j2 j1j2 b1+b2,m b1j1 b2j2 i1i2 a1+a2,m c=1 d X X=1

i ai ai di m−d,i So, we can define wj := c uˆcjuˆm−c,j = d uˆbj ubj (thanks to the relation above, all the sums are equal regardless of the choice of a,b). Let us also define ai ai P P vˆb := j ubj (compare with Remark 6.8(f)). Now it remains to derive the following relations: P QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 33

i (6.3) j wj =1 w satisfies the relations of S (we use the inter- i k i k n (6.4) w w δ = w w δ (m) (m) (m) Pj l jl j l ik  twiner relations of T , T , and T ), i k k i  (6.5) wj wl = wl wj  ak i i ak i (6.6) vb wj = wj vb  wj commute with everything, i v satisfy the relations of Sˆm (we use the inter- (6.7) va1iva2i = va1iδ b b b a1+a2,n twiner relation of T (n)),

ai cj cj ai i j (6.8) vb vd = vd vb entries of v commute with entries of v for i 6= j, ai ai i (6.9) ubj = vb wj u indeed has the correct structure.

The twisted orthogonality relation corresponding to the intertwiner Tˆ(N) looks as follows: a1i1 a2i2 ubj um−b,j = δa1+a2,mδi1i2 b,j X This, in particular, implies Relation (6.3). We write down the relation corresponding to R : ua1i1 ua2i2 δ = ua2i2 ua1i1 δ b2j2 b1i1 j1=6 j2 b1j1 b2i2 i1=6 i2 This implies two things. First, ai ck ck ai (6.10) ubjudl = udl ubj whenever i 6= k or j 6= l. Secondly, (and for this we might as well use the relation corresponding to R ),

ai ck ai ci (6.11) ubjudj =0 for i 6= k, ubj udl =0 for j 6= l. The latter remark allows us to check Relation (6.4). Assume j 6= l, then

i i ai ai ai ai wj wl = uˆcj uˆm−c,juˆdluˆm−d,l =0. c,d X i k Similarly, we derive that wj wj = 0 for i 6= k. i k k i Relation (6.10) then implies Relation (6.5), i.e. the commutativity wj wl = wl wj . i Indeed, notice that wj obviously commutes with itself, so we can assume that i 6= k or j 6= l and then it is a direct application of (6.10). We can also use Relation (6.11) to check (6.9):

ai i ai ai ai ai ai ai ai vb wj = ubk ucjum−c,j = ubj ucl um−c,l = ubj . k c c,l X X X ai i ai Similarly, we can derive ubj = wj vb , which proves part of Relation (6.6). To finish the proof of this relation, assume that j 6= k and compute ak i ak ai ai ak ai ai i ak vb wj = ubl ucjum−c,j = ubl ucjum−c,j = wj vb . l c l c X X X X Relation (6.8) goes the same way. Finally, Relation (6.7) follows directly from the relation corresponding to R , which reads a1i1 a2i2 a1+a2,i1  ubj ubj = ubj δb1b2 . 34 DANIEL GROMADA

References

[Bab79] L´aszl´oBabai. Spectra of Cayley graphs. Journal of Combinatorial Theory, Series B, 27(2):180–189, 1979. doi:10.1016/0095-8956(79)90079-0. [Ban96] Teodor Banica. Th´eorie des repr´esentations du groupe quantique compact libre O(n). Comptes rendus de l’Acad´emie des sciences. S´erie 1, Math´ematique, 322:241–244, 1996. [Ban99] Teodor Banica. Symmetries of a generic coaction. Mathematische Annalen, 314:763–780, 1999. [Ban02] Teodor Banica. Quantum groups and Fuss–Catalan algebras. Communications in Math- ematical Physics, 226:221–232, 2002. doi:10.1007/s002200200613. [Ban05] Teodor Banica. Quantum automorphism groups of homogeneous graphs. Journal of Functional Analysis, 224(2):243–280, 2005. doi:10.1016/j.jfa.2004.11.002. [BB07] Teodor Banica and Julien Bichon. Quantum automorphism groups of vertex-transitive graphs of order ≤ 11. Journal of Algebraic Combinatorics, 26:83–105, 2007. [BBC07] Teodor Banica, Julien Bichon, and Benoˆıt Collins. The hyperoctahedral quantum group. Journal of the Ramanujan Mathematical Society, 22:345–384, 2007. [Bic03] Julien Bichon. Quantum automorphism groups of finite graphs. Proceedings of the Amer- ical Mathematical Society, 131(3):665–673, 2003. doi:10.1090/S0002-9939-02-06798-9. [Bic04] Julien Bichon. Free wreath product by the quantum permutation group. Algebras and Representation Theory, 7:343–362, 2004. doi:10.1023/B:ALGE.0000042148.97035.ca. [BM21] Teodor Banica and J.P. McCarthy. The Frucht property in the quantum group setting. 2021, arXiv:2106.04999. [Bra37] Richard Brauer. On algebras which are connected with the semisimple continuous groups. Annals of Mathematics, 38(4):857–872, 1937. doi:10.2307/1968843. [BS09] Teodor Banica and Roland Speicher. Liberation of orthogonal Lie groups. Advances in Mathematics, 222(4):1461–1501, 2009. doi:10.1016/j.aim.2009.06.009. [Cha19] Arthur Chassaniol. Study of quantum symmetries for vertex-transitive graphs using intertwiner spaces. 2019, arXiv:1904.00455. [Gro20a] Daniel Gromada. Compact matrix quantum groups and their representation categories. PhD thesis, Saarland University, 2020. doi:10.22028/D291-32389. [Gro20b] Daniel Gromada. Free quantum analogue of Coxeter group D4. 2020, arXiv:2011.13242. [GW21] Daniel Gromada and Moritz Weber. Generating linear categories of partitions. To appear in Kyoto Journal of Mathematics, 2019–21, arXiv:1904.00166v3. [GW20] 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. [Jon94] Vaughan F.R. Jones. The Potts model and the symmetric group. In Subfactors: Pro- ceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993), pages 259–267, 1994. [KS97] Anatoli Klimyk and Konrad Schm¨udgen. Quantum Groups and Their Representations. Springer-Verlag, Berlin, 1997. [Lov75] L´aszl´oLov´asz. Spectra of graphs with transitive groups. Periodica Mathematica Hun- garica, 6:191–195, 1975. doi:10.1007/BF02018821. [LZ19] Xiaogang Liu and Sanming Zhou. Eigenvalues of Cayley graphs. 2019, arXiv:1809.09829. [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. [Mir16] Seyed Morteza Mirafzal. Some other algebraic properties of folded hypercubes. Ars Combinatoria, 124:154–159, 2016. URL http://www.combinatorialmath.ca/ArsCombinatoria/ACgetit.php?vol=124&paper=11. [NT13] Sergey Neshveyev and Lars Tuset. Compact Quantum Groups and Their Representation Categories. Soci´et´eMath´ematique de France, Paris, 2013. [Sch20a] Simon Schmidt. On the quantum symmetry of distance-transitive graphs. Advances in Mathematics, 368:107150, 2020. doi:10.1016/j.aim.2020.107150. [Sch20b] Simon Schmidt. Quantum automorphism groups of finite graphs. PhD thesis, Saarland University, 2020. doi:10.22028/D291-31806. [Sch20c] Simon Schmidt. Quantum automorphisms of folded cube graphs. Annales de l’Institut Fourier, 70(3):949–970, 2020. doi:10.5802/aif.3328. QUANTUM SYMMETRIES OF CAYLEY GRAPHS OF ABELIAN GROUPS 35

[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. [Web17] Moritz Weber. Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups. Proceedings – Mathematical Sciences, 127(5):881–933, 2017. doi:10.1007/s12044-017-0362-3. [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.

Czech Technical University in Prague, Faculty of Electrical Engineering, Depart- ment of Mathematics, Technicka´ 2, 166 27 Praha 6, Czechia Email address: [email protected]