10,261 research outputs found
Universal geometric cluster algebras
We consider, for each exchange matrix B, a category of geometric cluster
algebras over B and coefficient specializations between the cluster algebras.
The category also depends on an underlying ring R, usually the integers,
rationals, or reals. We broaden the definition of geometric cluster algebras
slightly over the usual definition and adjust the definition of coefficient
specializations accordingly. If the broader category admits a universal object,
the universal object is called the cluster algebra over B with universal
geometric coefficients, or the universal geometric cluster algebra over B.
Constructing universal coefficients is equivalent to finding an R-basis for B
(a "mutation-linear" analog of the usual linear-algebraic notion of a basis).
Polyhedral geometry plays a key role, through the mutation fan F_B, which we
suspect to be an important object beyond its role in constructing universal
geometric coefficients. We make the connection between F_B and g-vectors. We
construct universal geometric coefficients in rank 2 and in finite type and
discuss the construction in affine type.Comment: Final version to appear in Math. Z. 49 pages, 5 figure
Cambrian Lattices
For an arbitrary finite Coxeter group W we define the family of Cambrian
lattices for W as quotients of the weak order on W with respect to certain
lattice congruences. We associate to each Cambrian lattice a complete fan,
which we conjecture is the normal fan of a polytope combinatorially isomorphic
to the generalized associahedron for W. In types A and B we obtain, by means of
a fiber-polytope construction, combinatorial realizations of the Cambrian
lattices in terms of triangulations and in terms of permutations. Using this
combinatorial information, we prove in types A and B that the Cambrian fans are
combinatorially isomorphic to the normal fans of the generalized associahedra
and that one of the Cambrian fans is linearly isomorphic to Fomin and
Zelevinsky's construction of the normal fan as a "cluster fan." Our
construction does not require a crystallographic Coxeter group and therefore
suggests a definition, at least on the level of cellular spheres, of a
generalized associahedron for any finite Coxeter group. The Tamari lattice is
one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are
identified and characterized in terms of signed pattern avoidance. We also show
that open intervals in Cambrian lattices are either contractible or homotopy
equivalent to spheres.Comment: Revisions in exposition (partly in response to the suggestions of an
anonymous referee) including many new figures. Also, Conjecture 1.4 and
Theorem 1.5 are replaced by slightly more detailed statements. To appear in
Adv. Math. 37 pages, 8 figure
Generic rectangulations
A rectangulation is a tiling of a rectangle by a finite number of rectangles.
The rectangulation is called generic if no four of its rectangles share a
single corner. We initiate the enumeration of generic rectangulations up to
combinatorial equivalence by establishing an explicit bijection between generic
rectangulations and a set of permutations defined by a pattern-avoidance
condition analogous to the definition of the twisted Baxter permutations.Comment: Final version to appear in Eur. J. Combinatorics. Since v2, I became
aware of literature on generic rectangulations under the name rectangular
drawings. There are results on asymptotic enumeration and computations
counting generic rectangulations with n rectangles for many n. This result
answers an open question posed in the rectangular drawings literature. See
"Note added in proof.
Hong Kong – The new offence of fraud
Letter from Hong Kong by John Reading SC (Senior Assistant Director of Public Prosecutions, Commercial Crime Unit, Department of Justice, Hong Kong Special Administrative Region) describing how Hong Kong’s legislature enacted a statutory offence of fraud by inserting a new section (16A) in the Theft Ordinance. Jean reading prosecutes fraud and corruption cases and is a Senior Counsel. Published in the Letter from … section of Amicus Curiae - Journal of the Institute of Advanced Legal Studies and its Society for Advanced Legal Studies. The Journal is produced by the Society for Advanced Legal Studies at the Institute of Advanced Legal Studies, University of London
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