Given a family of combinatorial objects we often have an associated graded Hopf algebra. Such algebraic structures encode the associations and decompositions of the objects we study. The antipode is a map from the Hopf algebra into itself that is defined recursively and is difficult to compute in general. Is it worth it to find a cancelation free formula for it?

We start with the Hopf algebra of graphs and show the cancelation free formula of Humpert and Martin for its antipode. We will see that such formula gives a structural understanding of certain evaluations of the combinatorial invariants for graphs. In particular we recover very nicely a classical theorem of Stanley for the evaluation of the chromatic polynomial at -1.

We then give a general framework where we systematically obtain cancelation free formulas for antipodes. More precisely, we define the notion of strongly linearizable Hopf monoids and show how to get cancelation free formula for antipodes in those cases. This allows us to obtain a cancelation free formula many of the combinatorial Hopf algebras in the literature and more.

$\quad\bullet$ lattice morphisms from the weak order, via the Tamari lattice, to the boolean lattice;

$\quad\bullet$ normal fan coarsenings from the permutahedron, via Loday's associahedron, to the parallelepiped generated by the simple roots \(\mathbf{e}_{i+1} - \mathbf{e}_i\);

$\quad\bullet$ Hopf algebra inclusions from Malvenuto-Reutenauer's algebra, via Loday-Ronco's algebra, to Solomon's descent algebra.

In this talk, we present an extension of this framework to acyclic k-triangulations of a convex (n+2k)-gon, or equivalently to acyclic pipe dreams for the permutation $(1, \dots, k, n+k, \dots, k+1, n+k+1, \dots, n+2k)$. These objects are in bijection with the classes of the congruence of the weak order on $\mathfrak{S}_n$ defined as the transitive closure of the rewriting rule ${U ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W}$, for letters $a < b_1, \dots, b_k < c$ and words $U, V_1, \dots, V_k, W$ on $[n]$. It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of $\equiv^k$ to these acyclic pipe dreams. We will describe the cover relations in this lattice and the product and coproduct of this algebra in terms of pipe dreams. We will also recall the connection to the geometry of the brick polytope.

Motivated by this clash of types, Yang and Zelevinsky studied a particular class of reduced double Bruhat cells of a Lie group and proved that their rings of coordinates are cluster algebras with principal coefficients of the same type of the underlying group. Moreover, and arguably more interestingly, they were able to give expressions for all the cluster variables in terms of generalized minors.

In this talk I will present an ongoing work, joint with D. Rupel and H. Williams, in which we extend the results by Yang and Zelevinsky to any Kac-Moody group.

In this talk we will give two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the aforementioned version of the classical Littlewood-Richardson rule. Furthermore, both our rules have the nice property that they contain the classical version as a special case. We will then apply our rules to classify symmetric skew quasisymmetric Schur functions diagrammatically. This talk is based on joint work with Christine Bessenrodt and Vasu Tewari.

We synthesize M. Haiman's rectification, K. Purbhoo's folding, H. Thomas and A. Yong's minuscule K-theoretic Schubert calculus techniques, and a remark made by R. Proctor to give a framework for combinatorial proofs of these poset coincidences. This is joint work with Zachary Hamaker, Rebecca Patrias, and Oliver Pechenik.

This is joint work with Rosa Orellana (Dartmouth College)