Related papers: Push-pull operators on convex polytopes
By applying a Gr\"{o}bner-Shirshov basis of the symmetric group $S_{n}$, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert…
We generalize classical triangular Schubert puzzles to puzzles with convex polygonal boundary. We give these puzzles a geometric Schubert calculus interpretation and derive novel combinatorial commutativity statements, using purely…
We employ the pinching theorem, ensuring that some operators A admit any sequence of contractions as an operator diagonal of A, to deduce/improve two recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for conditional expectations onto…
We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in…
We study a class of convex bodies called operatopes that are obtained by taking Minkowski sums of affine images of an operator norm ball. This notion generalizes that of zonotopes which are Minkowksi sums of line segments. Taking the limit…
The Shapley-Folkman theorem is a statement about the Minkowski sum of (non-convex) sets, expressing the closeness of the Minkowski sum to convexity in a quantitative manner. This paper establishes similar theorems for integrally convex…
There exist natural generalizations of the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration…
One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the…
In this note the Choquet type operators are introduced, in connection to Choquet's theory of integrability with respect to a not necessarily additive set function. Based on their properties, a quantitative estimate for the nonlinear…
Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in $\mathbb{R}^d$. We prove central limit theorems for continuous motion invariant valuations including the Will's functional…
The decompositions of an element of a finite von Neumann algebra into the sum of a normal operator plus an s.o.t.-quasinilpotent operator, obtained using the Haagerup--Schultz hyperinvariant projections, behave well with respect to…
We show that some matrices are Schur multipliers and this is applied to obtain classes of operator-valued Foguel-Hankel operators similar to contractions. This provides partial answers to a problem of K. Davidson and the second author…
In this paper we study the ring $\mathcal{P}$ of combinatorial convex polytopes. We introduce the algebra of operators $\mathcal{D}$ generated by the operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all its…
An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{\xi\mu}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more…
In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the…
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…
Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show…
In the present paper we introduce and study the push pull operators on the formal affine Demazure algebra and its dual. As an application we provide a non-degenerate pairing on the dual of the formal affine Demazure algebra which serves as…
It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is…
Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived,…