Related papers: Positroids, Dressian and stable polynomials

Consider the ring $R:=\Q[\tau,\tau^{-1}]$ of Laurent polynomials in the variable $\tau$. The Artin's Pure Braid Groups (or Generalized Pure Braid Groups) act over $R,$ where the action of every standard generator is the multiplication by…

We develop the theory of matroids over one-dimensional algebraic groups, with special emphasis on positive characteristic. In particular, we compute the Lindstr\"om valuations and Frobenius flocks of such matroids. Building on work by Evans…

The aim of this paper is to make a systematical study on the stability of polynomials in combinatorics. Applying the characterizations of Borcea and Br\"and\'en concerning linear operators preserving stability, we present criteria for real…

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin…

Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Bia{\l}ynicki-Birula type partitions of $P$…

This was the basis of two lectures in the Current Developments in Mathematics conference in 2011. These lectures survey the theory of hyperbolic and stable polynomials, from their origins in the theory of linear PDE's to their present uses…

We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic…

We prove a formula for the degrees of Ikeda and Naruse's $P$-Grothendieck polynomials using combinatorics of shifted tableaux. We show this formula can be used in conjunction with results of Hamaker, Marberg, and Pawlowski to obtain an…

We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally…

The multiple root loci among univariate polynomials of degree $n$ are indexed by partitions of $n$. We study these loci and their conormal varieties. The projectively dual varieties are joins of such loci where the partitions are hooks. Our…

In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We…

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave: 1. The matroid complex of independent subsets of E. 2. The broken circuit complex of…

We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges,…

We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of Orlik and Solomon…

In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order)…

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with…

In this article, a new approach based on linear algebra is adopted to study a hybrid Sheffer polynomial sequences. The recurrence relations and differential equation for these polynomials are derived by using the properties and…

The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of…

In this paper, we introduce fundamental notions of homotopy theory, including homotopy excision and the Freudenthal suspension theorem. We then explore framed cobordism and its connection to stable homotopy groups of spheres through the…

The {\em Dressian} of a matroid $M$ is the set of all valuations of $M$. This Dressian is the support of a polyhedral complex $\mathcal{Dr}(M)$ whose open cells correspond 1-1 with matroid subdivisions of the matroid polytope of $M$. We…