论文标题
Schur和Schubert多项式的差异操作员
Differential operators on Schur and Schubert polynomials
论文作者
论文摘要
本文涉及后背稳定的舒伯特多项式上的运算符。我们研究了两个运营商$ξ$和$ \ nabla $ -1 $，它们满足了莱布尼兹规则。此外，我们表明所有其他此类操作员都是$ξ$和$ \ nabla $的线性组合。对于Schur函数的情况，这两个操作员充分确定了Schur功能的乘积，即，只能定义Littlewood-Richardson系数仅从$ξ$和$ \ nabla $中。关于Schur功能的新观点为我们提供了Giambelli身份和Jacobi-Trudi身份的基本证明。对于舒伯特多项式的情况，我们以$ξ$和$ \ nabla $的表达方式构建了更大的减少操作员，这些表达式由Young Difargrams索引。令人惊讶的是，这些操作员与Stanley对称功能有关。尤其是，我们将骨气操作员从Schur扩展到Schubert多项式。
This paper deals with decreasing operators on back stable Schubert polynomials. We study two operators $ξ$ and $\nabla$ of degree $-1$, which satisfy the Leibniz rule. Furthermore, we show that all other such operators are linear combinations of $ξ$ and $\nabla$. For the case of Schur functions, these two operators fully determine the product of Schur functions, i.e., it is possible to define the Littlewood-Richardson coefficients only from $ξ$ and $\nabla$. This new point of view on Schur functions gives us an elementary proof of the Giambelli identity and of Jacobi-Trudi identities. For the case of Schubert polynomials, we construct a bigger class of decreasing operators as expressions in terms of $ξ$ and $\nabla$, which are indexed by Young diagrams. Surprisingly, these operators are related to Stanley symmetric functions. In particular, we extend bosonic operators from Schur to Schubert polynomials.
