231 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
Subgroups of free idempotent generated semigroups: full linear monoid
We develop some new topological tools to study maximal subgroups of free
idempotent generated semigroups. As an application, we show that the rank 1
component of the free idempotent generated semigroup of the biordered set of a
full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal
subgroup isomorphic to the multiplicative subgroup of Q.Comment: We hope to use similar methods to study the higher rank component
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