论文标题
算盘和创建操作员的组合
Abacus-histories and the combinatorics of creation operators
论文作者
论文摘要
创建操作员对对称函数作用，以构建Schur函数，霍尔 - 小木多项式和相关的对称功能一次行。 Haglund，Morse，Zabrocki等人研究了通过将任何一系列Creation Operators应用于$ 1 $获得的更通用的对称功能$H_α$，$C_α$和$B_α$。我们开发了使用称为Abacus-histories的对象的对象的Schur扩展和相关对称函数的新组合模型。这些公式是通过将较小的算法融合在一起来编码单个创建算子对给定Schur函数的影响的较小公式。我们为运营商提供类似的处理，例如乘以乘以$ h_m $，$ h_m^{\ perp} $，$ω$等，它们是构建创建运营商的构建块。我们在算盘上使用介绍，为伯恩斯坦创建操作员的性质和霍尔 - 小木多项式提供了三排分区索引。
Creation operators act on symmetric functions to build Schur functions, Hall--Littlewood polynomials, and related symmetric functions one row at a time. Haglund, Morse, Zabrocki, and others have studied more general symmetric functions $H_α$, $C_α$, and $B_α$ obtained by applying any sequence of creation operators to $1$. We develop new combinatorial models for the Schur expansions of these and related symmetric functions using objects called abacus-histories. These formulas arise by chaining together smaller abacus-histories that encode the effect of an individual creation operator on a given Schur function. We give a similar treatment for operators such as multiplication by $h_m$, $h_m^{\perp}$, $ω$, etc., which serve as building blocks to construct the creation operators. We use involutions on abacus-histories to give bijective proofs of properties of the Bernstein creation operator and Hall-Littlewood polynomials indexed by three-row partitions.
