论文标题
三角洲和Theta操作员扩展
Delta and Theta Operator Expansions
论文作者
论文摘要
我们给出了$MΔ__{M_γe_1}πe_λ^{\ ast} $和$mΔ__{m_γe_1}πs_λ^{\ ast} $当$ t = 1 $在我们称之为$γ$ -parking $ lattice $ -ppark $ -ppark $ -pp -pp -pp -pp时。在这里，$δ_f$和$π$是修改后的MacDonald基础的某些特征器，$ M =（1-Q）（1-T）$。我们的主要结果依次在$ t = 1 $的基础基础扩展中，对于表格的对称功能，$mΔ_{fe_1}θ_{g} j $当$ f $以单元的扩展为基础的基础，$ j $是根据修改的基础扩展的，$ g $是按单元的扩展而扩展的。 $ \ {πe_λ^\ ast \}_λ$。即使是该总三角洲和Theta操作员表达的最特殊情况也很重要。我们重点介绍了其中一些特殊情况。我们最终给出了$ e $ - 积极性的猜想，因为$ t $不专业，建议我们的对象还可以在无专业化的对称功能中提供基础扩展。
We give an elementary symmetric function expansion for $MΔ_{m_γe_1}Πe_λ^{\ast}$ and $MΔ_{m_γe_1}Πs_λ^{\ast}$ when $t=1$ in terms of what we call $γ$-parking functions and lattice $γ$-parking functions. Here, $Δ_F$ and $Π$ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$. Our main results in turn give an elementary basis expansion at $t=1$ for symmetric functions of the form $M Δ_{Fe_1} Θ_{G} J$ whenever $F$ is expanded in terms of monomials, $G$ is expanded in terms of the elementary basis, and $J$ is expanded in terms of the modified elementary basis $\{Πe_λ^\ast\}_λ$. Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an $e$-positivity conjecture for when $t$ is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.
