630 research outputs found
Poset topology and homological invariants of algebras arising in algebraic combinatorics
We present a beautiful interplay between combinatorial topology and
homological algebra for a class of monoids that arise naturally in algebraic
combinatorics. We explore several applications of this interplay. For instance,
we provide a new interpretation of the Leray number of a clique complex in
terms of non-commutative algebra.
R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie
combinatoire et l'alg\`ebre homologique d'une classe de mono\"ides qui figurent
naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs
applications de cette interaction. Par exemple, nous introduisons une nouvelle
interpr\'etation du nombre de Leray d'un complexe de clique en termes de la
dimension globale d'une certaine alg\`ebre non commutative.Comment: This is an extended abstract surveying the results of arXiv:1205.1159
and an article in preparation. 12 pages, 3 Figure
Support Sets in Exponential Families and Oriented Matroid Theory
The closure of a discrete exponential family is described by a finite set of
equations corresponding to the circuits of an underlying oriented matroid.
These equations are similar to the equations used in algebraic statistics,
although they need not be polynomial in the general case. This description
allows for a combinatorial study of the possible support sets in the closure of
an exponential family. If two exponential families induce the same oriented
matroid, then their closures have the same support sets. Furthermore, the
positive cocircuits give a parameterization of the closure of the exponential
family.Comment: 27 pages, extended version published in IJA
Hopf algebras for matroids over hyperfields
Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields
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