# On Combinatorial Algebras of Multi-Complexes

## Miodrag Iovanov, University of Iowa

## Date and time: 2pm (New York time), Tuesday, April 27, 2021

## Place: On Zoom (details provided on the seminar mailing list)

A fruitful idea in combinatorics is to consider the totality of objects
of a certain type, and encode natural operations on these objects in
operations of some algebra with coefficients in a field of
characteristic 0. We introduce a general class of combinatorial
objects, which we call multi-complexes, which simultaneously
generalizes graphs, multigraphs, hypergraphs and simplicial and delta
complexes. A natural algebra of multi-complexes is defined as the
vector space which has a formal basis C consisting of all
isomorphism types of multi-complexes, and multiplication the disjoint
union. This is a Hopf algebra with an operation encoding assembly and
disassembly information for such objects, and extends and generalizes
the Hopf algebra of graphs. Such Hopf algebras are connected, graded
commutative and cocommutative, and by general results of
Cartier-Kostant-Milnor-Moore, are just a polynomial algebra. However,
it comes with additional structure which encodes combinatorial
information, and the main goal is to describe its structure in a way
relating to combinatorics. Such structures have been of high interest
in literature, both intrinsically and due to applications to
combinatorial problems.

We give a solution to the problem in this generality and find an
explicit basis B of the space of primitives, and which is of
combinatorial relevance: it is such that each multi-complex is a
polynomial with non-negative integer coefficients of the elements of
B, and each element of B is a polynomial with integer coefficients in
C. The coefficients appearing in all these polynomials are, up to
signs, numbers counting multiplicities of sub-multi-complexes in a
multi-complex. Using this, we also solve the antipode formula
problem, and find the cancellation and grouping free formula for the
antipode. Such formulas are usually hard to obtain, and constitute a
main question for such algebras

The main method is to use certain Mobius type formulas for posets of
sub-objects of multi-complexes. As examples, we will explicitly
illustrate how our results specialize to the formulas for graphs or
simplicial complexes, and relations of these to open questions such
as the graph reconstruction conjecture.