Group Association Schemes Research Articles

The Terwilliger algebras of the group association schemes of three metacyclic groups

AbstractFor any finite group , the Terwilliger algebra of the group association scheme satisfies the following inclusions: , where is a specific vector space and is the centralizer algebra of the permutation representation of induced by the action of conjugation. The group is said to be triply transitive if . In this paper, we determine the dimensions of and for being , and , and show that and are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of , and when they are triply transitive.

On the Terwilliger algebra of the group association scheme of Cn⋊C2

In 1992, Terwilliger introduced the notion of the Terwilliger algebra in order to study association schemes. The Terwilliger algebra of an association scheme A is the subalgebra of the complex matrix algebra, generated by the Bose-Mesner algebra of A and its dual idempotents with respect to a point x.In Bannai and Munemasa (1995) [3] determined the dimension of the Terwilliger algebra of abelian groups and dihedral groups, by showing that they are triply transitive (i.e., triply regular and dually triply regular). In this paper, we give a generalization of their results to the group association scheme of semidirect products of the form Cn⋊C2, where Cm is a cyclic group of order m≥2. Moreover, we will give the complete characterization of the Wedderburn components of the Terwilliger algebra of these groups.

Joint D2D Group Association and Channel Assignment in Uplink Multi-Cell NOMA Networks: A Matching-Theoretic Approach

This paper studies joint device-to-device (D2D) group association and channel assignment in uplink multi-cell non-orthogonal multiple-access (NOMA) networks. Particularly, the goal is to assign D2D groups to cellular user channels at each base-station, while accounting for negative network externality due to the interference caused by pairing a user with a D2D group. To that end, a multi-objective signal-to-interference-plus-noise ratio (SINR)-maximizing power allocation solution procedure is proposed to determine the optimal power allocation for each (D2D group, user) pair, while meeting quality-of-service (QoS) requirements. After that, the joint D2D group association and channel assignment problem is modeled as a student-project allocation with preferences over (student, project) pairs matching problem. More specifically, two polynomial-time complexity stable matching algorithms are proposed to pair D2D groups with users, and associate them with base-stations. Simulation results are presented to evaluate the proposed matching algorithms when combined with the devised solution procedure, and compare them to a joint D2D group association, channel assignment and power allocation (J-GA-CA-PA) scheme. More importantly, the proposed algorithms are shown to efficiently yield comparable SINR-per user and D2D receiver-to the J-GA-CA-PA scheme, while maintaining QoS requirements.

Quantum stabilizer codes from Abelian and non-Abelian groups association schemes

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

The provision of optimal entanglement on underlying networks of group associationschemes

We investigate the creation of optimal entanglement in axially symmetric networks. Theproposed network is one in which the connections are organized using a group table—inother words, using group association schemes. We show that optimal entanglement can beprovided on such networks by adjusting couplings and the initial state. We find that theoptimal initial state is a simple locally excited state. Moreover we obtain an analyticformula for the entanglement achievable between vertex pairs located in the same orconjugate strata and calculate their relevant optimal couplings. For those pairs located innon-conjugate strata, two upper bounds are presented. These results are also checkednumerically.

Perfect state transfer of a qudit over underlying networks of group associationschemes

As generalizations of results of Christandl et al (2004 Phys. Rev. Lett. 92 187902; 2005Phys. Rev. A 71 032312) and Facer et al (2008 Phys. Rev. A 77 012334), Bernasconi et al (2008 0808.0510 [quant-ph]; 2008 0806.2074 [math.CO]) studied perfect state transfer(PST) between two particles in quantum networks modeled by a large class of cubelikegraphs (e.g. the hypercube) which are the Cayley graphs of the elementary Abelian groupZ2n. In Jafarizadeh and Sufiani (2008 Phys. Rev. A 77 022315) Jafarizadehet al (2008 J. Phys. A: Math. Theor. 41 475302) respectively, PST of a qubitover distance regular spin networks and optimal state transfer (ST) of ad-level quantum state (qudit) over pseudo-distance regular networks were discussed, wherethe networks considered there were not, in general, related to a certain finite group. In thispaper, PST of a qudit over antipodes of more general networks, called underlying networksof association schemes, is investigated. In particular, we consider the underlying networksof group association schemes in order to employ the group properties (such as irreduciblecharacters) and use the algebraic structure of these networks (such as Bose–Mesneralgebra) in order to give an explicit analytical formula for coupling constants in theHamiltonians so that the state of a particular qudit initially encoded on one site willperfectly evolve to the opposite site without any dynamical control. It is shownthat the only necessary condition in order for PST over these networks to beachieved is that the centers of the corresponding groups be non-trivial. Therefore,PST over the underlying networks of the group association schemes over all thegroups with non-trivial centers such as the Abelian groups, the dihedral groupD2n with evenn, the Cliffordgroup CL(n) andall of the p-groups can be achieved.

Investigation of continuous-time quantum walk via modules of Bose–Mesner and Terwilliger algebras

The continuous-time quantum walk on the underlying graphs of association schemes have been studied, via the algebraic combinatorics structures of association schemes, namely semi-simple modules of their Bose-Mesner and (reference state dependent) Terwilliger algebras. By choosing the (walk) starting site as a reference state, the Terwilliger algebra connected with this choice turns the graph into the metric space, hence stratifies the graph into a (d+1) disjoint union of strata, where the amplitudes of observing the continuous-time quantum walk on all sites belonging to a given stratum are the same. In graphs of association schemes with known spectrum, the transition amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes. As most of association schemes arise from finite groups, hence the continuous-time walk on generic group association schemes have been studied in great details, where the transition amplitudes are given in terms of characters of groups. Further investigated examples are the walk on graphs of association schemes of symmetric $S_n$, Dihedral $D_{2m}$ and cyclic groups. Also, following Ref.\cite{js}, the spectral distributions connected to the highest irreducible representations of Terwilliger algebras of some rather important graphs, namely distance regular graphs, have been presented. Then using spectral distribution, the amplitudes of continuous-time quantum walk on graphs such as cycle graph $C_n$, Johnson and normal subgroup graphs have been evaluated. {\bf Keywords: Continuous-time quantum walk, Association scheme, Bose-Mesner algebra, Terwilliger algebra, Spectral distribution, Distance regular graph.} {\bf PACs Index: 03.65.Ud}

A characterization of the group association scheme of An

In this paper, we investigate a certain fusion scheme X˜(An) of the group association scheme X(An) of the alternating group of arbitrary degree n. In particular, under some extra assumption on 'geometry of maximal cliques', we characterize X˜(An) by parameters.

Association schemes and permutation groups

Every permutation group which is not 2-transitive acts on a non-trivial coherent configuration, but the question of which permutation groups G act on non-trivial association schemes (symmetric coherent configurations) is considerably more subtle. A closely related question is: when is there a unique minimal G-invariant association scheme? We examine these questions, and relate them to more familiar concepts of permutation group theory (such as generous transitivity) and association scheme theory (such as stratifiability). Our main results are the determination of all regular groups having a unique minimal association scheme, and a classification of groups with no non-trivial association scheme. The latter must be primitive, and are 2-homogeneous, or almost simple, or of diagonal type. The diagonal groups have some very interesting features, and we examine them further. Among other things we show that a diagonal group with non-abelian base group cannot be stratifiable if it has ten or more factors, or generously transitive if it has nine or more; and we characterise the quaternion group Q 8 as the unique non-abelian group T such that a diagonal group with eight factors T is generously transitive.

A Pair of Families of Non-isomorphic Group Association Schemes with the same Parameters

We show that for the split and non-split extensions ofFq2bySL (2, q) (q= 2e,e≥ 3), the group association schemes have the same parameters but are not isomorphic. For the split and non-split extensions ofFq2by the standard Borel subgroup of SL(2,q ) (q= 2e, e≥ 3), the group association schemes are shown to be isomorphic.

Character Products and Q-Polynomial Group Association Schemes

We study a finite group having a faithful character whose square has a small number of irreducible characters as constituents. Let Irr(G) be the set of absolutely irreducible ordinary characters of a finite group G. For each φ∈Irr(G), let φ̂=φ if φ is real valued and φ̂=φ+φ̄ otherwise, where φ̄ denotes the complex conjugate of φ. Let RIrr(G)={φ̂|φ∈Irr(G)}. For χ̂∈RIrr(G), letχ̂2=b·1+a·χ̂+Ψsuch that Ψ is a character of G which does not contain χ nor the principal character 1 as a constituent. We study the case when Ψ is a scalar multiple of a sum of the characters in IRrr(G), which are in a single orbit with respect to the action of the Galois group Gal(Q̄/Q(χ̂)). Here Q̄ denotes the algebraic closure of Q in C and Q(χ̂) is the field generated by the values of χ̂. As an application, we give a classification of Q-polynomial group association schemes.

On Four-Weight Spin Models and their Gauge Transformations

We study the four-weight spin models (W1, W2, W3, W4) introduced by Eiichi and Etsuko Bannai (Pacific J. of Math, to appear). We start with the observation, based on the concept of special link diagram, that two such spin models yield the same link invariant whenever they have the same pair (W1, W3), or the same pair (W2, W4). As a consequence, we show that the link invariant associated with a four-weight spin model is not sensitive to the full reversal of orientation of a link. We also show in a similar way that such a link invariant is invariant under mutation of links.Next, we give an algebraic characterization of the transformations of four-weight spin models which preserve W1, W3 or preserve W2, W4. Such “gauge transformations” correspond to multiplication of W2, W4 by permutation matrices representing certain symmetries of the spin model, and to conjugation of W1, W3 by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W1, W3 are symmetric.Finally we apply these results to two-weight spin models obtained as solutions of the modular invariance equation for a given Bose-Mesner algebra B and a given duality of B. We show that the set of such spin models is invariant under certain gauge transformations associated with the permutation matrices in B. In the case where B is the Bose-Mesner algebra of some Abelian group association scheme, we also show that any two such spin models (which generalize those introduced by Eiichi and Etsuko Bannai in J. Alg. Combin. 3 (1994), 243–259) are related by a gauge transformation. As a consequence, the link invariant associated with such a spin model depends only trivially on the link orientation.

Characterization of the Group Association Scheme of PSL(2, 7)

The association scheme having the same intersection numbers as those of the group association scheme X(PSL(2, 7)) is shown to be isomorphic to X(PSL(2, 7)), where PSL(2, 7) is the 2-dimensional projective special linear group over the field of order 7.

Characterization of the Group Association Scheme of the Symmetric Group

Letnbe a non-zero positive integer and Λ(n) the set of all partitions ofn. There is a one-to-one correspondence between Λ(n) and the set of the conjugacy classes ofSn, the symmetric group of degreen. Let X(Sn)= (Sn, {R*λ}λ∈Λ(n)) be the group association scheme ofSnand X=(X, {Rλ}λ∈Λ(n)) be an association scheme having intersection numbers identical to those of X(Sn). Suppose there exists no set of four vertices {x1,x2,x3,x4} withx1,x2,x3,x4∈Xsatisfying (x1,x2), (x2,x3), (x3,x4), (x4,x1) ∈R(2), (x1,x3) ∈R(3)and (x2,x4) ∈R(2,2). Then X is shown to be isomorphic to X (Sn). (In [17], the authors show that ifn≥ 5, X does not possess four vertices of this type.)

Characterization of the group association scheme of A5 by its intersection numbers

Characterization of the group association scheme of A5 by its intersection numbers

An Example of Non-isomorphic Group Association Schemes with the same Parameters

It is shown that the group association scheme for the split extension of 23bySL3(2) is not isomorphic to that for the non-split extension, although they have the same set of parameters (intersection numbers).

Modular invariance property and spin models attached to cyclic group association schemes

Several examples of generalized spin models are constructed using the modular invariance property of self-dual association schemes (Bannai and Bannai, J. Algebraic Combin. (to appear); Pacific J. Math. (to appear)). Conversely, it is shown in Bannai et al. (in preparation) that if there is a spin model in the Bose-Mesner algebra of an association scheme, then with some additional conditions the association scheme is self-dual and satisfies a modular invariance property. In this paper we give a classification of some kind of spin models attached to cyclic group association schemes and their symmetrizations.

THE TERWILLIGER ALGEBRAS OF GROUP ASSOCIATION SCHEMES

The Terwilliger algebra of an association scheme was introduced by Paul Terwilliger [7] in order to study P-and Q-polynomial association schemes. The purpose of this paper is to discuss in detail properties of the Terwilliger algebra of the group association scheme of a finite group. We shall give bounds on the dimension of the Terwilliger algebra, and define triple regularity. Let G be a finite group, C0 = {e}, C1, . . . , Cd the conjugacy classes of G. For i = 0, 1, . . . , d, define

Spin Models on Finite Cyclic Groups

The concept of spin model is due to V. F. R. Jones. The concept of nonsymmetric spin model, which generalizes that of the original (symmetric) spin model, is defined naturally. In this paper, we first determine the diagonal matrices T satisfying the modular invariance or the quasi modular invariance property, i.e., (PT)^3 = \sqrt{m}P^2 or (PT)^3=m^{3 \over 2} I (respectively), for the character table P of the group association scheme of a cyclic group G of order m. Then we show that a (symmetric or nonsymmetric) spin model on G is constructed from each of the matrices T satisfying the modular or quasi modular invariance property.

THE TERWILLIGER ALGEBRAS OF THE GROUP ASSOCIATION SCHEMES OF S5 AND A5

Terwilliger proposed a method for studying commutative association schemes by introducing a non-commutative, semi-simple C-algebra, whose structure reflects the combinatorial nature of the corresponding scheme, and applied the method to the P and Q polynomial schemes. In this paper, we continue the initial investigation of Bannai and Munemasa of the Terwilliger algebras of group association schemes. In particular, we determine the structure of the Terwilliger algebras of the group schemes of S 5 and A 5.