Related papers: The Tchebyshev transforms of the first and second …
We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks.…
The notion of Rees product of posets was introduced by Bj\"orner and Welker, where they study connections between poset topology and commutative algebra. Bj\"orner and Welker conjectured and Jonsson proved that the dimension of the top…
Given a finite graded poset with labeled Hasse diagram, we construct a quasi- symmetric generating function for (saturated) chains whose labels have fixed descents. This is a common generalization of a generating function for the flag…
Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z:\mathcal{B}\to \widehat{\mathbb{A}}$, where $\mathcal{B}$ is the…
We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an $R$-labeling, imply the…
We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions $H = \mb{C}[\bar{G}] \bowtie \mb{C}[G]$ ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly…
According to a statement by Pavel Etingof, in the special case of an affine variety $X$ with a faithful action by a finite group $G$, the sheaf of (twisted) Cherednik algebras $\mathcal{H}_{1, c, \psi, X, G}$ with formal parameters $c,…
To an extended permutohedron we associate the weighted integer points enumerator, whose principal specialization is the $f$-polynomial. In the case of poset cones it refines Gessel's $\mathsf{P}$-partitions enumerator. We show that this…
In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of…
We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible $\zed$-valued…
We propose a general method to construct symmetric tensor polynomials in the D-dimensional Euclidean space which are orthonormal under a general weight. The D-dimensional Hermite polynomials are a particular case of the present ones for the…
In this paper we consider two topological transforms that are popular in applied topology: the Persistent Homology Transform (PHT) and the Euler Characteristic Transform (ECT). Both of these transforms are of interest for their mathematical…
We make explict a description in terms of convex geometry of the higher Bruhat orders B(n,d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher Stasheff-Tamari poset S_1(n,d). We show that the map f sketched…
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary…
Chebyshev polynomials in one variable are typical chaotic maps on the complex 1-space. Chebyshev endomorphisms f on the complex n-space A are also chaotic. The endomorphisms f induce mappings on the quotient space A/G, where G is the…
The peculiarity of the Eckart potential problem on ${\bf H}^2_+$ (the upper sheet of the two-sheeted two-dimensional hyperboloid), to preserve the $(2l+1)$-fold degeneracy of the states typical for the geodesic motion there, is usually…
The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise…
This paper is the second part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey…
Let $P$ be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of $Z$-polynomials associated with $P$-kernels, providing a unified framework for various intersection cohomology Poincar\'e polynomials…
We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of…