65 research outputs found
Automorphism groups of root system matroids
AbstractGiven a root system R, the vector system R̃ is obtained by taking a representative v in each antipodal pair {v,−v}. The matroid M(R) is formed by all independent subsets of R̃. The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root system matroids M(R) are uniquely determined by their independent sets of size 3. As a corollary, we compute these groups explicitly, and thus complete the classification of the automorphism groups of root system matroids
Foundations of matroids -- Part 2: Further theory, examples, and computational methods
In this sequel to "Foundations of matroids - Part 1", we establish several
presentations of the foundation of a matroid in terms of small building blocks.
For example, we show that the foundation of a matroid M is the colimit of the
foundations of all embedded minors of M isomorphic to one of the matroids
, , , , , , ,
, and we show that this list is minimal. We establish similar minimal
lists of building blocks for the classes of 2-connected and 3-connected
matroids. We also establish a presentation for the foundation of a matroid in
terms of its lattice of flats. Each of these presentations provides a useful
method to compute the foundation of certain matroids, as we illustrate with a
number of concrete examples. Combining these techniques with other results in
the literature, we are able to compute the foundations of several interesting
classes of matroids, including whirls, rank-2 uniform matroids, and projective
geometries. In an appendix, we catalogue various 'small' pastures which occur
as foundations of matroids, most of which were found with the assistance of a
computer, and we discuss some of their interesting properties.Comment: 69 page
Tropicalization of classical moduli spaces
The image of the complement of a hyperplane arrangement under a monomial map
can be tropicalized combinatorially using matroid theory. We apply this to
classical moduli spaces that are associated with complex reflection
arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa
quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our
primary example is the Burkhardt quartic, whose tropicalization is a
3-dimensional fan in 39-dimensional space. This effectuates a synthesis of
concrete and abstract approaches to tropical moduli of genus 2 curves.Comment: 33 page
Special vector configurations in geometry and integrable systems
The main objects of study of the thesis are two classes of special vector configurations appeared in the geometry and the theory of integrable systems.
In the first part we consider a special class of vector configurations known as the V-systems, which appeared in the theory of the generalised Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations.
Several families of V-systems are known, but their classification is an open problem.
We derive the relations describing the infinitesimal deformations of V-systems
and use them to study the classification problem for V-systems in dimension 3.
In particular, we prove that the isolated cases in Feigin-Veselov list admit only trivial deformations.
We present the catalogue of all known 3D V-systems including graphical representations of the corresponding matroids and values of ν-functions.
In the second part we study the vector configurations, which form vertex sets for a new class of polyhedra called affine B-regular. They are defined by a 3-dimensional analogue of the Buffon procedure proposed by Veselov and Ward. The main result is the proof of existence of star-shaped affine B-regular polyhedron with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász
Topology of Arrangements and Representation Stability
The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research
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