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.     

The Representation Type of the Full Transforamtion Semigroup T 4

Let Tn be the semigroup of all transformations of a set of n elements and k a field of characteristic 0. According to Ponizovskii, the semigroup algebra kTn is of finite representation type if n h 3. According to Putcha, kTn is of infinite representation type if n S 5. Here, we deal with the remaining case n=4 and show that kT4 is also of finite representation type. Note that the quiver of kT4 already has been exhibited by Putcha, here we determine the relations. It turns out that kT4 is a string algebra and its global dimension is 3.     

Minimal Presentations and Efficiency of Semigroups

A finite semigroup S is said to be efficient if it can be defined by a presentation (A | R) with |R| -|A|=rank(H2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D2n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.     

Symmetric Designs Attached to Four-Weight Spin Models

H. Guo and T. Huang studied the four-weight spin models (X, W ₁, W ₂, W ₃, W ₄;D) with the property that the entries of the matrix W ₂ (or equivalently W ₄) consist of exactly two distinct values. They found that such spin models are always related to symmetric designs whose derived design with respect to any block is a quasi symmetric design. In this paper we show that such a symmetric design admits a four-weight spin model with exactly two values on W ₂ if and only if it has some kind of duality between the set of points and the set of blocks. We also give some examples of parameters of symmetric designs which possibly admit four-weight spin models with exactly two values on W ₂.     

Minimal and Maximal Semigroups Related to Causal Symmetric Spaces

Let G/H be an irreducible globally hyperbolic semisimple symmetric space, and let S ³ G be a subsemigroup containing H not isolated in S. We show that if S^o p 0 then there are H-invariant minimal and maximal cones Cmin ³ Cmax in the tangent space at the origin such that H exp Cmin ³ S ³ HZK(a)expCmax. A double coset decomposition of the group G in terms of Cartan subspaces and the group H is proved. We also discuss the case where G/H is of Cayley type.     

Invariant characters and coprime actions on finite nilpotent groups

Abstract. Let S be a subgroup of SLn(R), where R is a commutative ring with identity and n \geqq 3n \geqq 3. The order of S, o(S), is the R-ideal generated by x_ij, x_ii - x_jj (i 1 j)x_{ij},\ x_{ii} - x_{jj}\ (i \neq j), where (x_ij) ? S(x_{ij}) \in S. Let En(R) be the subgroup of SLn(R) generated by the elementary matrices. The level of S, l(S), is the largest R-ideal \frak q\frak {q} with the property that S contains all the \frak q\frak {q}-elementary matrices and all conjugates of these by elements of En(R). It is clear that l(S) \leqq o(S)l(S) \leqq o(S). Vaserstein has proved that, for all R and for all n \geqq 3n \geqq 3, the subgroup S is normalized by En(R) if and only if l(S) = o(S)     

On Free Inverse Semigroups

Using techniques of Rewriting Theory, we present a new proof of the known theorem of Munn that FIX , the free inverse semigroup on X, is isomorphic to birooted word-trees on X.     

The semigroup of singular endomorphisms

Let V be a vector space of dimension n over a field K. Here we denote by Sn the set of all singular endomorphisms of V. Erdos [5], Dawlings [4] and Thomas J. Laffey [6] have shown that Sn is an idempotent generated regular semigroup. In this paper we apply the theory of inductive groupoids, in particular the construction of the idempotent generated regular semigroup given in §6 of [8] to detemine some combinatorial properties of the semigroup Sn.     

Topology of the Semigroup of Singular Endomorphisms

The topological interpretations of some of the algebraic properties of the semigroup Sn of singular endomorphisms of an n-dimensional vector space over K are discussed here. Since Sn is known to be an idempotent generated regular semigroup, we pay more attention to the topological properties of the set En of idempotents in Sn. The local structure of En is shown to be that of a C^infinity-manifold and of a finite-dimensional vector bundle over the Grassmann manifolds. The topology of the biorder relations and sandwich sets are also discussed.     

The Semigroup of Hall Matrices over Distributive Lattices

In this paper, the semigroup Hn(L) of Hall matrices over a complete and completely distributive lattice L is studied. A Hall matrix is a matrix which is greater (for the order associated with the lattice structure) than an invertible matrix. Some necessary and sufficient conditions for a Hall matrix to be regular in the semigroup Hn(L) are given and Green's relations of the semigroup Hn(L) are described. Also, the sandwich semigroup of Hall matrices over the lattice L is studied.     

Bose-Mesner Algebras Associated with Four-Weight Spin Models

It is known that for each matrix W _i and it's transpose ^t W _i in any four-weight spin model (X, W ₁, W ₂, W ₃, W ₄; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(W _i ) and N( ^t W _i ) = N′(W _i ) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W ₁) = N′(W ₁) = N(W ₃) = N′(W ₃) holds. In this paper we show that all of them coincide, that is, N(W _i ), N′(W _i ), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra. Received: June 17, 1999 Final version received: Januray 17, 2000     

On Spin Models, Modular Invariance, and Duality

A spin model is a triple (X, W ⁺, W ^–), where W ⁺ and W ^– are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from(X, W ⁺, W ^–) ). We show that these equations imply the existence of a certain isomorphism between two algebras and associated with (X, W ⁺, W ^–) . When is the Bose-Mesner algebra of some association scheme, and is a duality of . These results had already been obtained in [15] when W ⁺, W ^– are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of spin models at the algebraic level. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.     

Centralizers and Central Idempotents of Semigroup Rings

. Let A be a nonempty subset of an associative ring R . Call the subring CR(A)={r] R\mid ra=ar \quadfor all\quad a] A} of R the centralizer of A in R . Let S be a semigroup. Then the subsemigroup S'= {s] S\mid sa=sb \quador\quad as=bs \quadimplies\quad a=b \quadfor all a,b] S} of S is called the C -subsemigroup. In this paper, the centralizer CR[S](R[M]) for the semigroup ring R[S] will be described, where M is any nonempty subset of S' . An non-zero idempotent e is called the central idempotent of R[S] if e lies in the center of R[S] . Assume that S\backslash S' is a commutative ideal of S and Annl(R)=0 . Then we show that the supporting subsemigroup of any central idempotent of R[S] must be finite.     

Geometry and analysis of spin equations

We introduce W‐spin structures on a Riemann surface Σ and give a precise definition to the corresponding W‐spin equations for any quasi‐homogeneous polynomial W. Then we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of W‐spin equations when W = W(x₁, …, x_t) is a nondegenerate, quasi‐homogeneous polynomial with fractional degrees (or weights) q_i < ½ for all i. In particular, the compactness theorem holds for the superpotentials E₆, E₇, E₈ or A_{n ? 1}, D_{n + 1} for n ≥ 3. © 2008 Wiley Periodicals, Inc.     

Representations for partially exchangeable arrays of random variables

Consider an array of random variables (X_i,j), 1 ≤ i,j < ∞, such that permutations of rows or of columns do not alter the distribution of the array. We show that such an array may be represented as functions f(α, ξ_i, η_j, λ_i,j) of underlying i.i.d, random variables. This result may be useful in characterizing arrays with additional structure. For example, we characterize random matrices whose distribution is invariant under orthogonal rotation, confirming a conjecture of Dawid.     

Twisted Extensions of Spin Models

A spin model is one of the statistical mechanical models which were introduced by V.F.R. Jones to construct invariants of links. In this paper, we give a new construction of spin models of size 4n from a given spin model of size n. The process is similar to taking tensor product with a spin model of size four, but we add some sign exchange. This construction also gives symmetric four-weight spin models of the type introduced by E. Bannai and E. Bannai.     

Wavelets with Frame Multiresolution Analysis

A frame multiresolution (FMRA for short) orthogonalwavelet is a single-function orthogonal wavelet such that theassociated scaling space V0 admits a normalized tight frame(under translations). In this article, we prove that for anyexpansive matrix A with integer entries, there existA-dilation FMRA orthogonal wavelets. FMRA orthogonal waveletsfor some other expansive matrix with non integer entries are also discussed.     

Quermassintegrals of a random polytope in a convex body

Let K be a convex body in \mathbbRⁿ \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x₁,?, x_N x_1,\ldots, x_N independently and uniformly from K, and write C(x₁,?, x_N) C(x_1,\ldots, x_N) for their convex hull. Let f : \mathbbR⁺ ? \mathbbR⁺ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 ￡ i ￡ n-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °W_i) = ò_K ?ò_K f[W_i(C(x₁, ?, x_N))]dx_N ?dx₁ E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (W_i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 ￡ i ￡ n-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x₁,?, x_N) C(x_1,\ldots, x_N) .     

The Monoid of the Random Graph

We investigate properties of the endomorphism monoid of the countable random graph R. We show that End(R) is not regular and is not generated by its idempotents. The Rees order on the idempotents of End(R) has 2^N0 many minimal elements. We also prove that the order type of Q is embeddable in the Rees order of End(R).