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  • 1
    Electronic Resource
    Electronic Resource
    Distance-Regular Graphs of Valency 6 and a1 = 1 (2000)
    Springer
    Journal of algebraic combinatorics 11 (2000), S. 101-134 
    Details
    ISSN: 1572-9192
    Keywords: distance-regular graph ; association scheme ; P-polynomial scheme
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We give a complete classification of distance-regular graphs of valency 6 and a1 = 1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008776031839
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  • 2
    Electronic Resource
    Electronic Resource
    General Form of Non-Symmetric Spin Models (2000)
    Springer
    Journal of algebraic combinatorics 12 (2000), S. 59-72 
    Details
    ISSN: 1572-9192
    Keywords: spin model ; association scheme ; Bose-Mesner algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A spin model (for link invariants) is a square matrix W with non-zero complex entries which satisfies certain axioms. Recently (Jaeger and Nomura, J. Alg. Combin. 10 (1999), 241–278) it was shown that t WW −1 is a permutation matrix (the order of this permutation matrix is called the “index” of W), and a general form was given for spin models of index 2. In the present paper, we generalize this general form to an arbitrary index m. In particular, we give a simple form of W when m is a prime number.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008711618027
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  • 3
    Electronic Resource
    Electronic Resource
    Bose-Mesner Algebras Related to Type II Matrices and Spin Models (1998)
    Springer
    Journal of algebraic combinatorics 8 (1998), S. 39-72 
    Details
    ISSN: 1572-9192
    Keywords: spin model ; link invariant ; association scheme ; Bose-Mesner algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A type II matrix is a square matrixW with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebra N(W) , which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebrasN(W) . Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008691327727
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  • 4
    Electronic Resource
    Electronic Resource
    Symmetric Versus Non-Symmetric Spin Models for Link Invariants (1999)
    Springer
    Journal of algebraic combinatorics 10 (1999), S. 241-278 
    Details
    ISSN: 1572-9192
    Keywords: spin model ; link invariant ; Bose-Mesner algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study spin models as introduced in [20]. Such a spin model can be defined as a square matrix satisfying certain equations, and can be used to compute an associated link invariant. The link invariant associated with a symmetric spin model depends only trivially on link orientation. This property also holds for quasi-symmetric spin models, which are obtained from symmetric spin models by certain “gauge transformations” preserving the associated link invariant. Using a recent result of [16] which asserts that every spin model belongs to some Bose-Mesner algebra with duality, we show that the transposition of a spin model can be realized by a permutation of rows. We call the order of this permutation the index of the spin model. We show that spin models of odd index are quasi-symmetric. Next, we give a general form for spin models of index 2 which implies that they are associated with a certain class of symmetric spin models. The symmetric Hadamard spin models of [21] belong to this class and this leads to the introduction of non-symmetric Hadamard spin models. These spin models give the first known example where the associated link invariant depends non-trivially on link orientation. We show that a non-symmetric Hadamard spin model belongs to a certain triply regular Bose-Mesner algebra of dimension 5 with duality, and we use this to give an explicit formula for the associated link invariant involving the Jones polynomial.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018771332556
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  • 5
    Electronic Resource
    Electronic Resource
    An Algebra Associated with a Spin Model (1997)
    Springer
    Journal of algebraic combinatorics 6 (1997), S. 53-58 
    Details
    ISSN: 1572-9192
    Keywords: spin model ; association scheme ; Bose-Mesner algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract To each symmetric n × n matrix W with non-zero complex entries, we associate a vector space N, consisting of certain symmetric n × n matrices. If W satisfies $$\sum\limits_{x = 1}^n {\frac{{W_{a,x} }}{{W_{b,x} }} = n{\delta }_{a,b} } (a,b = 1,...,n),$$ then N becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that N is the Bose-Mesner algebra of some association scheme. If W satisfies the star-triangle equation: $$\frac{1}{{\sqrt n }}\sum\limits_{x = 1}^n {\frac{{W_{a,x} W_{b,x} }}{{W_{c,x} }} = \frac{{W_{a,b} }}{{W_{a,c} W_{b,c} }}} (a,b,c = 1,...,n),$$ then W belongs to N. This gives an algebraic proof of Jaeger's result which asserts that every spin model which defines a link invariant comes from some association scheme.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008644201287
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  • 6
    Electronic Resource
    Electronic Resource
    Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2) (2000)
    Springer
    Journal of algebraic combinatorics 12 (2000), S. 25-36 
    Details
    ISSN: 1572-9192
    Keywords: distance-regular graph ; Terwilliger algebra ; quantum group
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate a connection between distance-regular graphs and U q(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let Γ be a distance-regular graph with diameter d ≥ 3 and valency k ≥ 3, and assume Γ is not isomorphic to the d-cube. Fix a vertex x of Γ, and let $$\mathcal{T} = \mathcal{T}(x)$$ (x) denote the Terwilliger algebra of Γ with respect to x. Fix any complex number q ∉ {0, 1, −1}. Then $$\mathcal{T}$$ is generated by certain matrices satisfying the defining relations of U q(sl(2)) if and only if Γ is bipartite and 2-homogeneous.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008707417118
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  • 7
    Electronic Resource
    Electronic Resource
    Ont-homogeneous permutation sets (1985)
    Springer
    Archiv der Mathematik 44 (1985), S. 485-487 
    Details
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01193987
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  • 8
    Electronic Resource
    Electronic Resource
    Type II Matrices of Size Five (1999)
    Springer
    Graphs and combinatorics 15 (1999), S. 79-92 
    Details
    ISSN: 1435-5914
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract . A type II matrix is an n×n matrix W with non-zero entries W i,j which satisfies , i, j=1, …, n. Two type II matrices W, W′ are said to be equivalent if W′=P 1Δ1 WΔ2 P 2 holds for some permutation matrices P 1, P 2 and for some non-singular diagonal matrices Δ1, Δ2. In the present paper, it is shown that there are up to equivalence exactly three type II matrices in M 5(C).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s003730050044
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  • 9
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    Krawtchouk polynomials, the Lie algebra sl2, and Leonard pairs (2012)
    Elsevier
    Details
    Publication Date: 2012-07-01
    Print ISSN: 0024-3795
    Electronic ISSN: 1873-1856
    Topics: Mathematics
    Published by Elsevier
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  • 10
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    Tridiagonal pairs and the μ-conjecture (2009)
    Elsevier
    Details
    Publication Date: 2009-01-01
    Print ISSN: 0024-3795
    Electronic ISSN: 1873-1856
    Topics: Mathematics
    Published by Elsevier
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