论文标题
Haglund的积极性猜想一对
Haglund's positivity conjecture for multiplicity one pairs
论文作者
论文摘要
Haglund的猜想指出，$ \ dfrac {\ langlej_λ（q，q^k），s_μ\ rangle} {（1- q） $J_λ$是麦克唐纳对称功能的积分形式，而$s_μ$是schur函数。本文证明了Haglund的猜想在$ $（λ，μ）$满足$ k_ {λ，μ} = 1 $或$ k_ {μ'，λ'} = 1 $的情况下，其中$ k $表示kostka编号。我们还获得了有关MacDonald对称函数和Schur函数之间过渡矩阵的一些一般结果。
Haglund's conjecture states that $\dfrac{\langle J_λ(q,q^k),s_μ\rangle}{(1-q)^{|λ|}} \in \mathbb{Z}_{\geq 0}[q]$ for all partitions $λ,μ$ and all non-negative integers $k$, where $J_λ$ is the integral form Macdonald symmetric function and $s_μ$ is the Schur function. This paper proves Haglund's conjecture in the cases when the pair $(λ,μ)$ satisfies $K_{λ,μ}=1$ or $K_{μ',λ'}=1$ where $K$ denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.
