Related papers: Skew shapes, Ehrhart positivity and beyond

We generalize a theorem of Littlewood concerning the factorization of Schur polynomials when their variables are twisted by roots of unity. We show that a certain family of flagged skew Schur polynomials admit a similar factorization. These…

Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It…

We show that plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) and transversal structures can be set in correspondence to certain (weighted) models of quadrant walks, via suitable specializations of a bijection…

In this work we establish a connection between copositivity, that is, nonnegativity on the positive orthant, of sparse real Laurent polynomials and discriminants. Specifically, we consider Laurent polynomials in the positive orthant with…

We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux…

Let $k$ and $m$ be positive integers and $\lambda/\mu$ a skew partition. We compute the principal specialization of the skew Schur polynomials $s_{\lambda /\mu}(x_1, \ldots, x_{k})$ modulo $q^m-1$ under suitable conditions. We interpret the…

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these…

We prove a skew generalization of the Newton-Puiseux theorem for the field $F = \bigcup_{n=1}^\infty \mathbb{C}((x^\frac{1}{n}))$ of Puiseux series: For any positive real number $\alpha$, we consider the $\mathbb{C}$-automorphism $\sigma$…

We reflect on the notions of positivity and square roots. We review many examples which underline our thesis that square roots of positive maps related to *-algebras are Hilbert modules. As a result of our considerations we discuss…

A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact…

Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the…

The order complex of inclusion poset $PF_n$ of Borel orbit closures in skew-symmetric matrices is investigated. It is shown that $PF_n$ is an EL-shellable poset, and furthermore, its order complex triangulates a ball. The rank-generating…

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincar\'e-extended ab-index, which generalizes both the ab-index and the Poincar\'e polynomial.…

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their…

Eberhard proved that for every sequence $(p_k), 3\le k\le r, k\ne 5,7$ of non-negative integers satisfying Euler's formula $\sum_{k\ge3} (6-k) p_k = 12$, there are infinitely many values $p_6$ such that there exists a simple convex…

Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper…

We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the…

Each degree $n+k$ polynomial of the form $(x+1)^k(x^n+c_1x^{n-1}+\cdots +c_n)$, $k\in \mathbb{N}$, is representable as Schur-Szeg\H{o} composition of $n$ polynomials of the form $(x+1)^{n+k-1}(x+a_j)$. We study properties of the affine…

For a permutation $u\in S_n$, let $N\ast u\in S_{Nn}$ be the permutation with scaled Lehmer code. For given $u,v,w\in S_n$ and integer $N$, the stretched Schubert coefficients are defined as $f_{u,v,w}(N):=c_{N*u,N*v}^{N*w}$. Our main…

We prove Stanley's conjecture that, if delta_n is the staircase shape, then the skew Schur functions s_{delta_n / mu} are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted…