Extremal bases for finite cyclic groups

AbstractLet m and h be positive integers. A set A of integers is called a basis of orderh for Z(m) if every integer n is congruent to a sum of h elements in A modulo m. Let m(h, A) denote the greatest positive integer m such that A is a basis of order h for Z(m). For any k ≥ 1, define m(h, k) = max∥A∥ = k + 1 m(h, A). This generalizes a function of Graham and Sloane. In this paper, it is proved that, for fixed k ≥ 4 as h → ∞, m(h, k) ≥ αk (256125)⌞k4⌟ (hk)k + O(hk − 1), where αk = 1 if k ≡ 0 or 1 (mod 4), 43 if k ≡ 2 (mod 4), and 2716 if k ≡ 3 (mod 4). A lower bound for m(h, k) is also obtained for fixed h. Using these results, new lower bounds are proved for the order of subsets of asymptotic bases

Minimal Seifert manifolds for higher ribbon knots

We show that a group presented by a labelled oriented tree presentation in which the tree has diameter at most three is an HNN extension of a finitely presented group. From results of Silver, it then follows that the corresponding higher dimensional ribbon knots admit minimal Seifert manifolds.Comment: 33 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper12.abs.htm

Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces

Orbifolds of Lattice Vertex Operator Algebras at d=48d=48 and d=72d=72

Motivated by the notion of extremal vertex operator algebras, we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self-dual lattices in $d=48$ and $d=72$. In this way we construct about one hundred new examples of holomorphic VOAs with a small number of low weight states.Comment: 18 pages, LaTe

Quantum extremal loop weight modules and monomial crystals

In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, we get finite-dimensional modules of quantum toroidal algebras at roots of unity. In general, we give a conjectural process to construct extremal loop weight modules from monomial realizations of crystals.Comment: 49 pages. Accepted for publication in Pacific Journal of Mathematic

Representation theory for subfactors, λ\lambda-lattices and C*-tensor categories

We develop a representation theory for $\lambda$-lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $\lambda$-lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with property (T) standard invariant and that are not constructed from property (T) groups.Comment: v3: minor changes, final version to appear in Communications in Mathematical Physics. v2: improved exposition; permanence of property (T) under quotients adde