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Sublattices of associahedra and permutohedra equations satisfied by associahedra
 
 
Description: 

The set of permutations on n-elements is a poset when endowed with weak Bruhat ordering.  Actually, this poset is a lattice since each finite subset of permutations has both a supremum and infimum. Such a lattice is known as the Permutohedron on n-elements.
A strictly related lattice is Associahedron on n+1 letters. Its elements are all the ways of parenthesizing a word of length n+1 (i.e. binary trees with n+1 leaves). The order is the transitive closure of the operation that replaces a sub-parenthesized word (uv)w with u(vw).
In this talk I shall describe undergoing work on Associahedra and Permutohedra.
It was conjectured in 1992 by Geyer that every bounded image of a free lattice embeds into an Associahedron. We disprove the conjecture, and go further to state an analogous statement for the Permutohedra: not every bounded image of a free lattice embeds into a Permutohedron.
While the first result leads to discover non-trivial lattice-theoretic identities holding in all the Associahedra, we do not yet know any non-trivial identity that holds in all the Permutohedra.
Joint work with Fred Wehrung.

Date:  2013-02-26
Start Time:   16:00
Speaker:  Luigi Santocanale (Univ. de Provence, France)
Institution:  Université de Provence
Place:  Sala 5.5
Research Groups: -Algebra, Logic and Topology
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