论文标题
走向巴特勒的猜想
Toward Butler's conjecture
论文作者
论文摘要
对于分区$ν$，让$λ，μ\subseteqν$为两个不同的分区，以便$ |ν/λ| = |ν/μ| = 1 $。巴特勒（Butler）指出，分裂的差异$ \ peratatorName {i} _ {λ，μ} [x; q，t] =（t_λ\ widetilde {h}_μ[x; q，q，t]-t_μ\ wideteDe分区$λ$和$μ$是Schur阳性。通过引入一个称为列交换规则的新的LLT等效性，我们给出了$ \ operatatorAname {i} _ {λ，μ} [x; q，t] $的组合公式，这是一个积极的单元扩展。我们还证明了巴特勒在某些特殊情况下的猜想。
For a partition $ν$, let $λ,μ\subseteq ν$ be two distinct partitions such that $|ν/λ|=|ν/μ|=1$. Butler conjectured that the divided difference $\operatorname{I}_{λ,μ}[X;q,t]=(T_λ\widetilde{H}_μ[X;q,t]-T_μ\widetilde{H}_λ[X;q,t])/(T_λ-T_μ)$ of modified Macdonald polynomials of two partitions $λ$ and $μ$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $\operatorname{I}_{λ,μ}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases.
