KdV and mKdV Hierarchies and Schur Q-functions

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED Mathematical Physics, Analysis and Geometry Pub Date : 2024-10-12 DOI:10.1007/s11040-024-09493-w

Sumitaka Tabata

引用次数: 0

Abstract

We prove two conjectures on the Korteweg-de Vries (KdV) and modified KdV (mKdV) hierarchies and Schur Q-functions presented by Yamada. The first one is that the functions defined by Sato and Mori in 1980 coincide with Schur Q-functions indexed by even or odd strict partitions. Mizukawa, Nakajima, and Yamada gave an expression for this function using symmetric functions and Littlewood-Richardson coefficients. We prove that this expression coincides with the Schur Q-function by using the formula of Lascoux, Leclerc, and Thibon. The second one is that Schur Q-functions indexed by strict partitions which have odd parts form a basis for the space of Hirota polynomials of the KdV hierarchy, and that Schur Q-functions indexed by strict partitions which have even parts form a basis for the space of Hirota polynomials of the mKdV hierarchy. This conjecture is verified by rewriting the generating series of the KdV and mKdV hierarchies using the techniques of symmetric functions.

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KdV 和 mKdV 层次和舒尔 Q 函数

我们证明了山田提出的关于 Korteweg-de Vries（KdV）和修正 KdV（mKdV）层次和舒尔 Q 函数的两个猜想。第一个猜想是，佐藤和森在 1980 年定义的函数与以偶数或奇数严格分区为索引的舒尔 Q 函数重合。水川（Mizukawa）、中岛（Nakajima）和山田（Yamada）利用对称函数和利特尔伍德-理查森系数给出了该函数的表达式。我们利用 Lascoux、Leclerc 和 Thibon 的公式证明了这个表达式与舒尔 Q 函数重合。第二个猜想是，以具有奇数部分的严格分区为索引的舒尔 Q 函数构成 KdV 层次的 Hirota 多项式空间的基础，而以具有偶数部分的严格分区为索引的舒尔 Q 函数构成 mKdV 层次的 Hirota 多项式空间的基础。通过使用对称函数技术重写 KdV 和 mKdV 层次的产生序列，验证了这一猜想。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

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