Reflection matrices from Hadamard-type Temperley-Lieb R-matrices

We classify nonoperatorial matrices K solving the Skylanin quantum reflection equation for all R-matrices obtained from the newly defined general rank-n Hadamard-type representations of the Temperley-Lieb algebra TL_N(√n). They are characterized by a universal set of algebraic equations in a specific canonical basis uniquely defined by the “master matrix” associated with the chosen realization of the Temperley-Lieb algebra.     

Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices

In answer to “Research Problem 16” in Horadam's recent book Hadamard matrices and their applications, we provide a construction for generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 456–458, 2009     

On symmetric nets and generalized Hadamard matrices from affine designs

Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q³, q², q²–1/q–1) designs without symmetric (q, q)-subnets.     

On generalized Hadamard matrices of minimum rank

Generalized Hadamard matrices of order qⁿ⁻¹ (q—a prime power, n2) over GF(q) are related to symmetric nets in affine 2-(qⁿ,qⁿ⁻¹,(qⁿ⁻¹−1)/(q−1)) designs invariant under an elementary abelian group of order q acting semi-regularly on points and blocks. The rank of any such matrix over GF(q) is greater than or equal to n−1. It is proved that a matrix of minimum q-rank is unique up to a monomial equivalence, and the related symmetric net is a classical net in the n-dimensional affine geometry AG(n,q).     

A construction for modified generalized Hadamard matrices using QGH matrices

Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U ₁,U ₂, . . . ,U _m } be the set of cosets of U in G. We say a matrix H = [h _ij ] of order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if \({\sum_{1\le t \le k} h_{it}h_{jt}^{-1} = \lambda_{ij1}U_1+\cdots+\lambda_{ijm}U_m (\exists\lambda_{ij1},\ldots, \exists \lambda_{ijm} \in \mathbb{Z})}\) for any i ≠ j. On the other hand, in our previous article we defined a modified generalized Hadamard matrix GH(s, u, λ) over a group G, from which a TD _λ (uλ, u) admitting G as a semiregular automorphism group is obtained. In this article, we present a method for combining quasi-generalized Hadamard matrices and semiregular relative difference sets to produce modified generalized Hadamard matrices.     

Modified generalized Hadamard matrices and constructions for transversal designs

It is well known that there exists a transversal design TD_λ[k; u] admitting a class regular automorphism group U if and only if there exists a generalized Hadamard matrix GH(u, λ) over U. Note that in this case the resulting transversal design is symmetric by Jungnickel’s result. In this article we define a modified generalized Hadamard matrix and show that transversal designs which are not necessarily symmetric can be constructed from these under a modified condition similar to class regularity even if it admits no class regular automorphism group.     

Murphy bases of generalized Temperley-Lieb algebras

In this article we show how to use some results of G. E. Murphy on the so-called standard basis of Hecke-Algebras of Type A to derive a similar basis for generalized Temperley-Lieb algebras. This standard basis is compared to the usual diagrammatic basis of the original Temperley-Lieb algebra used in knot theory and statistical physics.     

Spin models constructed from Hadamard matrices

A spin model (for link invariants) is a square matrix W which satisfies certain axioms. For a spin model W, it is known that W ^T W ^?1 is a permutation matrix, and its order is called the index of W. Jaeger and Nomura found spin models of index?2, by modifying the construction of symmetric spin models from Hadamard matrices. The aim of this paper is to give a construction of spin models of an arbitrary even index from any Hadamard matrix. In particular, we show that our spin models of indices a power of 2 are new.     

Hadamard matrices generated by an adaptation of generalized quaternion type array

The purpose of this paper is to prove that ifq 1 (mod 4) andq – 2 are both prime powers, then there exists an Hadamard matrix of order 4q. We rely on relative Gauss sums and generalized quaternion type array. Under the same assumption onq, E. Spence has obtained an Hadamard matrix of order 4q by using a relative difference set and the Goethals-Seidel array. We believe that the matrix constructed here is inequivalent to Spence's matrix, in general.Notation q a power of a primep - Z the rational integer ring - F = GF(q) a finite field withq elements - K = GF(q ^s ) an extension ofF of degrees 2 - F ^× multiplicative group ofF - a primitive element ofK - S _F absolute trace fromF - S _K/F relative trace fromK toF - N _K/F relative norm fromK toF - I _m the unit matrix of orderm - J _m the matrix of orderm with every element + 1 - e the column vector of ordern with every element + 1 - A ^* the transpose of a matrixA - J _m (x) 1 +x + x ² + ... +x ^m-1     

Embedding matrices in Hadamard matrices

It is shown that if A is any n×n matrix of zeros and ones, and if k is the smallest number not less than n which is the order of an Hadamard matrix, then A is a submatrix of an Hadamard matrix of order k².     

Deciding Hadamard equivalence of Hadamard matrices

Equivalence of Hadamard matrices can be decided inO(log² n) space, and hence in subexponential time. These resource bounds follow from the existence of small distinguishing sets.     

Enumeration of generalized Hadamard matrices of order 16 and related designs

We investigate signings of symmetric GDD( , 16, )s over for . Beginning with , at each stage of this process a signing of a GDD( , 16, ) produces a GDD( , 16, ). The initial GDDs ( ) correspond to Hadamard matrices of order 16. For , the GDDs are semibiplanes of order 16, and for the GDDs are semiplanes of order 16 which can be extended to projective planes of order 16. In this article, we completely enumerate such signings which include all generalized Hadamard matrices of order 16. We discuss the generation techniques and properties of the designs obtained during the search. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 119–135, 2009     

Power Hadamard matrices

We introduce power Hadamard matrices, in order to study the structure of (group) generalized Hadamard matrices, Butson (generalized) Hadamard matrices and other related orthogonal matrices, with which they share certain common characteristics. The new objects turn out to be as interesting, and perhaps as useful, as the objects that motivated them.We develop a basic theory of power Hadamard matrices, explore these relationships, and offer some new insights into old results. For example, we show that all 4×4 Butson Hadamard matrices are equivalent to circulant ones, and how to move between equivalence classes.We provide, among other new things, an infinite family of circulant Butson Hadamard matrices that extends a known class to include one of each positive integer order.Dedication: In 1974 Jennifer Seberry (Wallis) introduced what was then a totally new structure, orthogonal designs, in order to study the existence and construction of Hadamard matrices. They have proved their worth for this purpose, and have also become an object of interest for their own sake and in applications (e.g., [H.J.V. Tarok, A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory 45 (1999) 1456-1467. [26]]). Since then many other generalizations of Hadamard matrices have been introduced, including some discussed herein. In the same spirit we introduce a new object showing this kind of promise.Seberry's contributions to this field are not limited to her own work, of which orthogonal designs are but one example—she has mentored many young mathematicians who have expanded her legacy by making their own marks in this field. It is fitting, therefore, that our contribution to this volume is a collaboration between one who has worked in this field for over a decade and an undergraduate student who had just completed his third year of study at the time of the work.     

Complex Hadamard matrices

R. J. Turyn introduced complex Hadamard matrices and showed that if there is a complex Hadamard matrix of order c and a real Hadamard matrix of order h> > 1, then there is a real Hadamard matrix of order he. Previously, complex Hadamard matrices were only known for a few small orders and the orders for which symmetric conference matrices were known. These latter are known only to exist for orders which can be written as 1+a² +b² where a, b are integers. We give many constructions for new infinite classes of complex Hadamard matrices and show that they exist for orders 306,650, 870,1406,2450 and 3782: for the orders 650, 870, 2450 and 3782, a symmetric conference matrix cannot exist.