In minisymposium: Tropical Linear Algebra and BeyondTue 18:00–18:30, Room AV 01.12
The faces of any real linear hyperplane arrangement form a monoid (a semigroup with identity element). The product can be viewed either geometrically (in terms of walking in a straight line between points chosen from each of the two faces) or purely combinatorially (in terms of 'sign vectors', recording for each hyperplane in turn on which 'side' the given face lies). The talk concerns some joint work with Mark Kambites on a `tropical' (i.e. min-plus) exploration of these ideas. The natural tropical analogues of either the geometric or combinatorial interpretations of the above-mentioned product do not turn out to define a monoid structure on the faces of a tropical hyperplane arrangement, however we note that implicit in the definition of a tropical oriented matroid (Ardila and Develin) is a monoid action upon the faces of the tropical arrangement. We discuss the combinatorics of this action via a connection with tropical matrix permanents.