Related papers: A quasisymmetric function for matroids
We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane…
Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic…
The theory of integrals is used to analyse the structure of Hopf algebroids, introduced in math.QA/0302325. We prove that the total algebra of the Hopf algebroid is a separable extension of the base algebra if and only if it is a…
In this paper, we extend the iterated integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated integrals on a digraph naturally give rise to the iterated path algebra and the…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a…
A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension $\leq 1$, and for a finite dimensional Hopf algebra, this dimension is…
We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions…
In this manuscript we lift the theory of r-quasisymmetric functions to the theory of Hopf monoids. We provide a general method of interpolating between two Hopf monoids, one being the free monoid on a positive comonoid and the other being…
We study in detail the Hopf algebra of noncommutative symmetric functions in superspace sNSym, introduced by Fishel, Lapointe and Pinto. We introduce a family of primitive elements of sNSym and extend the noncommutative elementary and power…
Generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization. We prove that they are the universal family of polyhedra with a certain Hopf algebraic structure. Their antipode…
We study the coefficients of a natural basis for the space of weak harmonic Maass forms of weight $5/2$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition…
Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the…
Cho and van Willigenburg (arXiv:1508.07670) and Alinaeifard, Wang, and van Willgenburg (arXiv:2010.00147) introduce multiplicative chromatic bases for the ring $\Lambda$ of symmetric functions, consisting of the chromatic symmetric…
In this paper we construct an algebraic invariant attached to Galois representations over number fields. This invariant, which we call an Artin symmetric function, lives in a certain ring we introduce called the ring of arithmetic symmetric…
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
We give a graph-sum algorithm that expresses any genus-$g$ Gromov-Witten invariant of the symmetric product orbifold $\mathrm{Sym}^d\mathbb{P}^r:=[(\mathbb{P}^r)^d/S_d]$ in terms of "Hurwitz-Hodge integrals" -- integrals over (compactified)…
We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of…
In this paper, we first discuss some properties of the Galois linear maps. We provide some equivalent conditions for Hopf algebras and Hopf (co)quasigroups as its applications. Then let $H$ be a Hopf quasigroup with bijective antipode and…
We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R=$\mathbb{Z}$, and when R is a DVR, we get…
Duchamp--Hivert--Thibon introduced the construction of a right $H_n(0)$-module, denoted as $M_P$, for any partial order $P$ on the set $[n]$. This module is defined by specifying a suitable action of $H_n(0)$ on the set of linear extensions…