Related papers: The cd-index of Bruhat intervals
An \emph{interval vector} is a $(0,1)$-vector in $\mathbb{R}^n$ for which all the 1's appear consecutively, and an \emph{interval-vector polytope} is the convex hull of a set of interval vectors in $\mathbb{R}^n$. We study three particular…
Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. Two reduced expressions are said to be braid equivalent if they are related via a sequence of braid moves. Braid…
We introduce a general technique to create an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by…
The goal of this paper is to refine some aspects of the cohomological study of Bruhat-Tits subgroups. It aims to complement the work done in the 1987 article. As an application, we obtain a result "\`a la Grothendieck-Serre" for Bruhat-Tits…
We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a…
The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural…
The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert…
In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent…
In this paper, we study the Newton polytopes of $F$-polynomials in a TSSS cluster algebra $\mathcal A$ and generalize them to a larger set consisting of polytopes $N_{h}$ associated to vectors $h\in\Z^{n}$ as well as $\widehat{\mathcal{P}}$…
In this article a first step is made towards the extension of Britto-Cachazo-Feng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown…
Mirror graphs were introduced by Bre\v{s}ar et al. in 2004 as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article…
We present a way to associate an algebra $B_G (\Upsilon) $ with every pseudo reflection group $G$. When $G$ is a Coxeter group of simply-laced type we show $B_G (\Upsilon)$ is isomorphic to the generalized Brauer algebra of simply-laced…
The spectrum of integrable models is often encoded in terms of commuting functions of a spectral parameter that satisfy functional relations. We propose to describe this commutative algebra in a covariant way by means of the extended…
The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use…
Let $G=Sp_{2r}({\mathbb C})$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $C_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the…
We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the…
Let $P$ be a finite poset. Let $L:=J(P)$ denote the lattice of order ideals of $P$. Let $b_i(L)$ denote the number of Boolean intervals of $L$ of rank $i$. We construct a simple graph $G(P)$ from our poset $P$. Denote by $f_i(P)$ the number…
We study a variety of questions centered around the computation of cohomology of line bundles on the incidence correspondence (the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane…
We derive a localization formula for the refined index of gauged quantum mechanics with four supercharges. Our answer takes the form of a residue integral on the complexified Cartan subalgebra of the gauge group. The formula captures the…
We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, as well as of regular elements. We…