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We present a uniform construction of level 1 perfect crystals $\mathcal B$ for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine…
Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of…
We present explicit descriptions of the crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(\lambda)$ over special linear Lie algebras in the language of \emph{extended Nakajima monomials}. There is a natural correspondence between the monomial…
We investigate a class of polyhedral convex cones, with $R^k_+$ (the nonegative orthant in $\mathbb{R}^k$) as a special case. We start with the observation that for convex cones contained in $\mathbb{R}^k$, the respective cone efficiency is…
For each poset $P$, we construct a polytope $A(P)$ called the $P$-associahedron. Similarly to the case of graph associahedra, the faces of $A(P)$ correspond to certain nested collections of subsets of $P$. The Stasheff associahedron is a…
We define the_hive ring_, which has a basis indexed by dominant weights for GL(n), and structure constants given by counting hives [KT1] (or equivalently honeycombs, or Berenstein-Zelevinsky patterns [BZ1]). We use the octahedron rule from…
We define a family of convex polytopes called constrainahedra, which index collisions of horizontal and vertical lines. Our construction proceeds by first defining a poset $C(m,n)$ of good rectangular preorders, then proving that $C(m,n)$…
Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we…
For every infinite cardinal number $\kappa$, $\kappa$-monoids and their realization have recently been introduced and studied by Nazemian and Smertnig. A $\kappa$-monoid $H$ has a realization to a ring $R$ if there exists an element $x \in…
This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of Operations Research and Management Science) describing parts of the theory of convex polyhedra that are particularly important for optimization. The topics include…
Nestohedra are a family of convex polytopes that includes permutohedra, associahedra, and graph associahedra. In this paper, we study an extension of such polytopes, called extended nestohedra. We show that these objects are indeed the…
The length polyhedron $Q_P$ of an interval order $P$ is the convex hull of integral vectors representing the interval lengths in interval representations of $P$. This polyhedron has been studied by various authors, including Fishburn and…
We present a combinatorial isomorphism between Stasheff associahedra and an inductive cone construction of those complexes given by Loday. We give an alternate description of certain polytopes, known as multiplihedra, which arise in the…
The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for…
The choice of a homogeneous ideal in a polynomial ring defines a closed subscheme $Z$ in a projective space as well as an infinite sequence of cones over $Z$ in progressively higher dimension projective spaces. Recent work of Aluffi…
The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete…
It has been a long-standing challenge to find a geometric object underlying the cosmological wavefunction for Tr($\phi^3$) theory, generalizing associahedra and surfacehedra for scattering amplitudes. In this note we describe a new class of…
Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new…
Rotation-reversal symmetry was recently introduced to generalize the symmetry classification of rigid static rotations in crystals such as tilted octahedra in perovskite structures and tilted tetrahedral in silica structures. This operation…
Polyhedral semantics is a recently introduced branch of spatial modal logic, in which modal formulas are interpreted as piecewise linear subsets of an Euclidean space. Polyhedral semantics for the basic modal language has already been well…