通过流多面体实现 $s$-置换多面体

Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements

Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements a…

The equivariant Ehrhart theory of the permutahedron

Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case.

关于排列多面体的细分和格路矩阵的旗帜

On subdivisions of the permutahedron and flags of lattice path matroids

本文研究了将排列多面体分成两个子多面体的细分，对应于正则旗的格路矩阵。通过超平面分割得到最细的细分，对应于BIPs。

Lifted generalized permutahedra and composition polynomials

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a "lifting" construction for these polytopes, which turns an $n$-dimensional generalized permutahedron into an $(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff's multiplihedron from homotopy theory, and to the more general "nestomultiplihedra," answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pie…

Newton Polytopes in Algebraic Combinatorics

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. Fo…

Hochschild多面体

Hochschild polytopes

通过删除$(m,n)$-multiplihedron刻面描述中的不等式，构造了$(m,n)$-Hochschild多面体，并证明其定向骨架为Hochschild格的Hasse图，同时揭示了相关的交半格态射。