Related papers: Pointed and multi-pointed partitions of type A and…
Starting from the data of an arbor, which is a rooted tree with vertices decorated by disjoint sets, we introduce a lattice polytope and a partial order on its lattice points. We give recursive algorithms for various classical invariants of…
The Motzkin numbers can be derived as coefficients of hybrid polynomials. Such an identification allows the derivation of new identities for this family of numbers and offers a tool to investigate previously unnoticed links with the theory…
Inspired by Gansner's elegant $k$-trace generating function for rectangular plane partitions, we introduce two novel operators, $\varphi_{z}$ and $\psi_{z}$, along with their combinatorial interpretations. Through these operators, we derive…
Ehrhart theory measures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, .... We compute the Ehrhart quasipolynomials of the standard Coxeter permutahedra for the classical Coxeter groups, expressing them…
In this paper we consider the generalized little Hankel and Berezin type operators on Besov spaces of holomorphic functions on polydiscs and give results about the boundedness of these operators.
In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different…
The notion of prop models the operations with multiple inputs and multiple outpus, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras…
We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a…
The object of the paper is the dependence of Koszul complexes and dependence of dual Koszul complexes of two systems of non-homogeneous polynomials, when one system is a part of other system, in connection with the duality in a Koszul…
Motivated by various developments in algebraic combinatorics and its applications, we investigate here the fine structure of a fundamental but little known theorem, the Gerstenhaber and Schack cohomology comparison theorem.The theorem…
A quantum group of type A is defined as a Hopf algebra associated to a Hecke symmetry. We show the homology of a Koszul complex associated to the Hecke symmetry is one dimensional and determines a group-like element in the Hopf algebra.…
This paper examines a systematic method to construct a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of non-standard geometric representations. This method can be employed to construct generalizations of…
A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are…
The Poincar\'e polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety.…
The aim of this article is to investigate the cohomology (l-adic as well as Betti) of schemes, and more generally of certain algebraic stacks, that are proper and smooth over the integers and have the property that there exists a polynomial…
We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras on a fixed space M. his allows us to describe a notion of prefactorization algebra up to…
We study different algebraic structures associated to an operad and their relations: to any operad $\mathbf{P}$ is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra;…
Several recent papers have examined a rational polyhedron $P_m$ whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing $m$. A combinatorial…
We compute the set of facets of the polytope which is the convex hull of the Coxeter groups $\mathsf{F}_4$ or $\mathsf{H}_4$: For the group $\mathsf{F}_4$ we found $2$ orbits of facets which contradicts previous results published in…
In previous work, we initiated the study of the cohomology of locally acyclic cluster varieties. In the present work, we show that the mixed Hodge structure and point counts of acyclic cluster varieties are essentially determined by the…