Skew Divided Difference Operators and Schubert Polynomials

Skew divided difference operators and Schubert polynomials

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform Schubert polynomials into polynomials with positive integer coefficients.

Positive expressions for skew divided difference operators

For permutations $v,w \in \mathfrak S_n$, Macdonald defines the skew divided difference operators $\partial_{w/v}$ as the unique linear operators satisfying $\partial_w(PQ) = \sum_v v(\partial_{w/v}P) \cdot \partial_vQ$ for all polynomials $P$ and $Q$. We prove that $\partial_{w/v}$ has a positive expression in terms of divided difference operators $\partial_{ij}$ for $i<j$. In fact, we prove that the analogous result holds in the Fomin-Kirillov algebra $\mathcal E_n$, which settles a conjecture of Kirillov.

Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $α\in R$, $v,w\in W$, we denote by $\partial_α$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew divided difference operators acting on the coinvariant algebra of $W$. Generalizing the work of Liu, we prove that $\partial_{w/v}$ can be written as a polynomial with nonnegative coefficients in $\partial_α$ where $α\in R^+$. In fact, we prove the stronger and analogous statement in the Nichols-Woronowicz algebra model for Schubert calculus on $W$ after Bazlov. We draw …

The Taylor resolution over a skew polynomial ring

Let $\Bbbk$ be a field and let $I$ be a monomial ideal in the polynomial ring $Q=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex which provides a finite free resolution for $Q/I$ as a $Q$-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring $R$. Under the hypothesis that the skew commuting parameters defining $R$ are roots of unity, we prove as an application that as …

Self-self-dual spaces of polynomials

A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V. A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self-self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V. We show that the self-self-dual spaces have a natural non-degenerate skew-symmetric 3-form defined in terms of Wronskians. We show that the self-self-dual spaces correspond to G_2-populations related to the Bethe Ansatz of the Gaudin model of …

四面体L-算子、张量Schur多项式和q-变形的环初等对称函数

Tetrahedral $L$-operators, tensor Schur polynomials and $q$-deformed loop elementary symmetric functions

本文基于四面体L-算子研究三维分区函数，提出可表示为张量Schur多项式类乘积的分区函数，推导Schur多项式洗牌公式及多个恒等式的统一，并展示其在多物种完全非对称排斥过程稳态中的应用，同时确定q-变形环初等对称函数的显式形式。