2,680 research outputs found
Counting compositions over finite abelian groups
We find the number of compositions over finite abelian groups under two types
of restrictions: (i) each part belongs to a given subset and (ii) small runs of
consecutive parts must have given properties. Waring's problem over finite
fields can be converted to type~(i) compositions, whereas Carlitz and locally
Mullen compositions can be formulated as type~(ii) compositions. We use the
multisection formula to translate the problem from integers to group elements,
the transfer matrix method to do exact counting, and finally the
Perron-Frobenius theorem to derive asymptotics. We also exhibit bijections
involving certain restricted classes of compositions
The multisubset sum problem for finite abelian groups
In this note, we give the explicit formula for the number of multisubsets of
a finite abelian group with any given size such that the sum is equal to a
given element . This also gives the number of partitions of into a
given number of parts over a finite abelian group. An inclusion-exclusion
formula for the number of multisubsets of a subset of with a given size and
a given sum is also obtained
Random matrices, the Cohen-Lenstra heuristics, and roots of unity
The Cohen-Lenstra-Martinet heuristics predict the frequency with which a
fixed finite abelian group appears as an ideal class group of an extension of
number fields, for certain sets of extensions of a base field. Recently, Malle
found numerical evidence suggesting that their proposed frequency is incorrect
when there are unexpected roots of unity in the base field of these extensions.
Moreover, Malle proposed a new frequency, which is a much better match for his
data. We present a random matrix heuristic (coming from function fields) that
leads to a function field version of Malle's conjecture (as well as
generalizations of it).Comment: 14 page
Quantum tori, mirror symmetry and deformation theory
We suggest to compactify the universal covering of the moduli space of
complex structures by non-commutative spaces. The latter are described by
certain categories of sheaves with connections which are flat along foliations.
In the case of abelian varieties this approach gives quantum tori as a
non-commutative boundary of the moduli space. Relations to mirror symmetry,
modular forms and deformation theory are discussed.Comment: Corrections to Section 4 and references adde
Higher categorical aspects of Hall Algebras
These are extended notes for a series of lectures on Hall algebras given at
the CRM Barcelona in February 2015. The basic idea of the theory of Hall
algebras is that the collection of flags in an exact category encodes an
associative multiplication law. While introduced by Steinitz and Hall for the
category of abelian p-groups, it has since become clear that the original
construction can be applied in much greater generality and admits numerous
useful variations. These notes focus on higher categorical aspects based on the
relation between Hall algebras and Waldhausen's S-construction.Comment: 60 pages, preliminary version, comments very welcom
Homotopy types and geometries below Spec Z
After the first heuristic ideas about `the field of one element' F_1 and
`geometry in characteristics 1' (J.~Tits, C.~Deninger, M.~Kapranov, A.~Smirnov
et al.), there were developed several general approaches to the construction of
`geometries below Spec Z'. Homotopy theory and the `the brave new algebra' were
taking more and more important places in these developments, systematically
explored by B.~To\"en and M.~Vaqui\'e, among others.
This article contains a brief survey and some new results on counting
problems in this context, including various approaches to zeta--functions and
generalised scissors congruences.
The new version includes considerable extensions and revisions suggested by
I. Zakharevich.Comment: 38 page
Bost-Connes systems and Fâ-structures in Grothendieck rings, spectra, and Nori motives
We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of Fâ-geometry, in the framework of torifications, that fit into this general setting
The cobordism hypothesis
In this expository paper we introduce extended topological quantum field
theories and the cobordism hypothesis.Comment: 36 pages; v2 has a reference correctio
The structure of double groupoids
We give a general description of the structure of a discrete double groupoid
(with an extra, quite natural, filling condition) in terms of groupoid
factorizations and groupoid 2-cocycles with coefficients in abelian group
bundles. Our description goes as follows: To any double groupoid, we associate
an abelian group bundle and a second double groupoid, its frame. The frame
satisfies that every box is determined by its edges, and thus is called a
`slim' double groupoid. In a first step, we prove that every double groupoid is
obtained as an extension of its associated abelian group bundle by its frame.
In a second, independent, step we prove that every slim double groupoid with
filling condition is completely determined by a factorization of a certain
canonically defined `diagonal' groupoid.Comment: amslatex, 28 pages, revised version to appear in J. Pure Appl.
Algebr
On the Brauer constructions and generic Jordan types of Young modules
Let p be a prime number. We study the dimensions of Brauer constructions of
Young and Young permutation modules with respect to p-subgroups of the
symmetric groups. They depend only on partitions labelling the modules and the
orbits of the action of the p-subgroups, and are related to their generic
Jordan types. We obtain some reductive formulae and, in the case of two-part
partitions, make some explicit calculation
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