167 research outputs found
Cohomology of idempotent braidings, with applications to factorizable monoids
We develop new methods for computing the Hochschild (co)homology of monoids
which can be presented as the structure monoids of idempotent set-theoretic
solutions to the Yang--Baxter equation. These include free and symmetric
monoids; factorizable monoids, for which we find a generalization of the
K{\"u}nneth formula for direct products; and plactic monoids. Our key result is
an identification of the (co)homologies in question with those of the
underlying YBE solutions, via the explicit quantum symmetrizer map. This
partially answers questions of Farinati--Garc{\'i}a-Galofre and Dilian Yang. We
also obtain new structural results on the (co)homology of general YBE
solutions
A two-sided analogue of the Coxeter complex
For any Coxeter system of rank , we introduce an abstract boolean
complex (simplicial poset) of dimension that contains the Coxeter
complex as a relative subcomplex. Faces are indexed by triples , where
and are subsets of the set of simple generators, and is a
minimal length representative for the parabolic double coset . There
is exactly one maximal face for each element of the group . The complex is
shellable and thin, which implies the complex is a sphere for the finite
Coxeter groups. In this case, a natural refinement of the -polynomial is
given by the "two-sided" -Eulerian polynomial, i.e., the generating function
for the joint distribution of left and right descents in .Comment: 26 pages, several large tables and figure
Regular Incidence Complexes, Polytopes, and C-Groups
Regular incidence complexes are combinatorial incidence structures
generalizing regular convex polytopes, regular complex polytopes, various types
of incidence geometries, and many other highly symmetric objects. The special
case of abstract regular polytopes has been well-studied. The paper describes
the combinatorial structure of a regular incidence complex in terms of a system
of distinguished generating subgroups of its automorphism group or a
flag-transitive subgroup. Then the groups admitting a flag-transitive action on
an incidence complex are characterized as generalized string C-groups. Further,
extensions of regular incidence complexes are studied, and certain incidence
complexes particularly close to abstract polytopes, called abstract polytope
complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder,
A. Deza, and A. Ivic Weiss (eds), Springe
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
Homogeneous compact geometries
We classify compact homogeneous geometries of irreducible spherical type and
rank at least 2 which admit a transitive action of a compact connected group,
up to equivariant 2-coverings. We apply our classification to polar actions on
compact symmetric spaces.Comment: To appear in: Transformation Group
g-elements, finite buildings and higher Cohen-Macaulay connectivity
The main result is a proof that the g-vector of a simplicial complex with a
convex ear decomposition is an M-vector. This is a generalization of similar
results for matroid complexes. We also show that a finite building has a convex
ear decomposition. This leads to connections between higher Cohen-Macaulay
connectivity and increasing h-vectors.Comment: To appear in JCT A. 20 page
Discrete isometry groups of symmetric spaces
This survey is based on a series of lectures that we gave at MSRI in Spring
2015 and on a series of papers, mostly written jointly with Joan Porti. Our
goal here is to:
1. Describe a class of discrete subgroups of higher rank
semisimple Lie groups, which exhibit some "rank 1 behavior".
2. Give different characterizations of the subclass of Anosov subgroups,
which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of
various equivalent dynamical and geometric properties (such as asymptotically
embedded, RCA, Morse, URU).
3. Discuss the topological dynamics of discrete subgroups on flag
manifolds associated to and Finsler compactifications of associated
symmetric spaces . Find domains of proper discontinuity and use them to
construct natural bordifications and compactifications of the locally symmetric
spaces .Comment: 77 page
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