Joyal's notion of species constitutes a good framework for the study of certain algebraic structures associated to combinatorial objects. We briefly review the notion of "Hopf monoid" in the category of species and then illustrate it with several examples. We introduce the Hopf monoid of generalized permutahedra (the latter are certain polytopes recently studied by Postnikov, Reiner and Williams). Our main result is an explicit antipode formula for this Hopf monoid. We explain how reciprocity theorems of Stanley on graphs and of Billera, Jia and Reiner on matroids can be deduced from this result.

The talk borrows from joint works with Swapneel Mahajan and with Federico Ardila.

Note: this talk may serve as an introduction to the working algebra seminar that will meet on Fridays at 2.

We discuss several algebraic structures that can be placed on the multiplihedra {M_n}, a family of polytopes defined by Stasheff in the study of higher categories and homotopy theory. The structures we find are best understood by considering the multiplihedra's relationship to two other families of polytopes more familiar to combinatorists: the permutahedra {P_n} and the associahedra {A_n}. These objects were given the structure of Hopf algebra by Malvenuto-Reutenauer and Loday-Ronco, respectively. In this talk, we give M the structure of P-module and A-Hopf module algebra in a manner respecting the cellular maps from P to M to A. We also give a basis of coinvariants for a second Hopf module structure over A. (This is joint work with F. Sottile and S. Forcey.)

We give an overview of the present state of this problem, illustrated by the main examples coming from the theory of quantum groups.

Let X be a finite set acted upon by a cyclic group C. The cyclic sieving phenomenon (CSP), due to Reiner, Stanton, and White, is the name given to the special situation when there exists a polynomial f that precisely encodes the C-orbit structure of X. (Usually f is a certain q-enumertor for the elements of X.) As a toy example, let X be all 2-subsets of a 4 element set, and let C act by shifting mod 4. There are two orbits: {2,3} → {3,4} → {1,4} → {1,2}, and {1,3} → {2,4}. Now let f = [{4}choose{2}]_q = 1 + q + 2q² + q³ + q⁴. This is the standard q-analogue of the binomial coefficient: it counts lattice paths in a 2-by-2 box by area. If we plug in powers of 4th roots of unity, that is, powers of i=√-1, we recover the fixed-point set sizes. We get f(i) = 0, f(-1) = 2, f(i³) = 0, f(1) = 6, which tells us there are 2 subsets fixed by two shifts and 6 fixed by four shifts, whereas one or three shifts fix nothing.
I will present several examples of the CSP related to Schützenberger's action of jeu de taquin promotion. This can be visualized nicely on rectangular Young tableaux with two or three rows, as shown in joint work with Pylyavskyy and Rhoades. Very recently, in joint work with Serrano, instances of CSPs for certain shifted tableaux and for reduced expressions of the longest element of the hyperoctahedral group have emerged.

Let ν be a valuation dominating a noetherian local domain, and let S = {ν(r) | r ∈ R} be its value semi-group and Γ = S – S its value group. The possible value groups Γ have been extensively studied and classified classically, the value semi-group S however is much less understood. In this talk we will briefly look at the known results classifying value groups and some well known constraints on the value semi-groups then we will look at growth rate of the value semi-group and its asymptotic behavior to obtain new constraints on possible value semi-groups.
This is joint work with S.D. Cutkosky and O. Kashcheyeva.

It is a classical result by MacMahon that the inversion number and the major index have the same distribution over the set of rearrangements of a fixed multiset. Recently, a major index was defined for matchings and set partitions and it was shown that it has the same distribution with the number of crossings. I will present a definition of a major index for fillings of moon polyominoes with zeros and ones. This statistic, when specialized to certain shapes, reduces to the major index for words and set partitions. Moreover, I will show that it is equally distributed with the number of north-east chains, which are a natural generalization of inversions and crossings.

In this talk we will discuss the hyperpfaffian, which is an extension of the pfaffian to either multidimensional arrays or tensor algebras. We will spend the first part of the talk examining the different definitionsused for the hyperpfaffian, as there are several non-equivalent forms. Then we will examine uses of the hyperpfaffian in combinatorics. The first is the Hyperpfaffian-Cactus theorem, which is related to the classical matrix-tree theorem and the later pfaffian-tree theorem. Second we will look at hyperpfaffian orientations, which can be used to count perfect matchings on a hyperpfaffian. Finally we will look at hyperpfaffian rings/ideals, and specifically look at the fact that unlike the pfaffian ideal, the hyperpfaffian ideal is not an algebra with straightening law under reasonable assumptions.

In the late 80's Beukers and Heckmann classified all algebraic hypergeometric functions in one variable. The key ingredient in their proof is the signature of a quadratic form fixed by the monodromy group. I will explain how this quadratic form is related to an old construction due to Bezout and how we may reformulate their result in terms of the p-adic valuation of Taylor coefficients. Using this, in joint work with E. Cattani and A. Dickenstein, we were able to classify rational two-variable hypergeometric functions.