Algebraic Structure of Cyclic Combinatorial Objects (18frg223)

The Banff International Research Station will host the "Algebraic structure of cyclic combinatorial objects" workshop in Banff from April 29 to May 6, 2018.

Discrete objects that can be decomposed into smaller pieces and that can be put together to create bigger ones give rise to well-defined algebraic structures, as long as the label of the objects is not relevant. As an example, on any matroid $M$ one can perform operations such as restriction, contraction, and direct sum. If one permutes the ground set of $M$, one obtains a matroid $M'$ isomorphic to $M$ and the operations performed in $M$ will be the same as in $M'$, up to permutation of the ground set. This allows us to define a Hopf algebra structure on isomorphism classes of matroids. Moreover, the Hopf algebra structure can be lifted to a Hopf monoid structure on Joyal's species, in such a way that each isomorphism class is encoded in a single object. This powerful point of view unifies many constructions and theorems concerning a wide variety of combinatorial objects.

As G. C. Rota wrote in relation to the theory of species: There is a common pattern to advances [of the second kind] in mathematics. They inevitably begin when someone points out that items that were formerly thought to be "the same" are not really "the same". This project studies the Hopf algebraic structure of combinatorial objects whose underlying ground set is ordered cyclically. Examples include cycles, non-crossing partitions, toric posets, and positroids. Though seemingly very similar, their Hopf algebraic structure seems to require a new notion of cyclic monoids and bialgebras. Our goal is to lay down the algebraic foundations for this theory and reap its combinatorial benefits.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).