ON THE EXPANSION OF SCHUR AND SCHUBERT POLYNOMIALS INTO STANDARD ELEMENTARY MONOMIALS
ON THE EXPANSION OF SCHUR AND SCHUBERT POLYNOMIALS INTO STANDARD ELEMENTARY MONOMIALS
复制标题
论SCHUR和SHUBERT多项式扩展到标准基本单项式
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
Rudolf Winkel
中科院分区：
文献类型：
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作者：
Rudolf Winkel
摘要基于舒伯特多项式的一个简单量化过程的发现，我们研究了将舒尔多项式和舒伯特多项式展开为标准初等单项式的问题。Schur多项式的SEM展开式可以用Jacobi-Trudi公式的一个简单变体进行代数描述，也可以用基于阶梯盒图的偏序集的规则进行组合描述。这些偏序集被看作是秩对称的和序同构于介于完全对称群和各自的极大布尔子格之间的对称群的Bruhat序中的某些主序理想。我们证明并猜想了这些结果在一般舒伯特多项式上的推广。有特色的猜想是：(1)将SEM展开解释为“交替逼近”；(2)与SEM展开自然相关的不同数字的惊人性质。这暗示了舒伯特多项式的扫描电镜展开中尚未发现的更深层次的对称性。
Abstract Motivated by the recent discovery of a simple quantization procedure for Schubert polynomials we study the expansion of Schur and Schubert polynomials into standard elementary monomials (SEM). The SEM expansion of Schur polynomials can be described algebraically by a simple variant of the Jacobi–Trudi formula and combinatorially by a rule based on posets of staircase box diagrams. These posets are seen to be rank symmetric and order isomorphic to certain principal order ideals in the Bruhat order of symmetric groups ranging between the full symmetric group and the respective maximal Boolean sublattice. We prove and conjecture extensions of these results for general Schubert polynomials. The featured conjectures are: (1) an interpretation of SEM expansions as “alternating approximations” and (2) surprising properties of different numbers naturally associated to SEM expansions. This hints at as yet undiscovered deeper symmetry properties of the SEM expansion of Schubert polynomials.