Properties of Wronskian 大溪这个村的五峰山,能远眺火车站,山上还藏有一座北宋古寺

This is a note for the article Critical points of master functions and flag varieties. Most of the properties below are from the appendix of this article. Some are not written in the paper but used there. I did not write the well-known relations between differential equations and Wronskians, such as Abel’s identity.亚博下载app下载中央公园“加速度”,什么样的产品才能引爆市场?

Here we assume all functions in this page are functions of with sufficiently many derivatives.

Define the Wronskian of functions by We follow the convention that for the corresponding Wronskian is equal to . We also write for in order to stress the order of the Wronskian.

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Symmetric Functions and Representation Theory

Schur Function

For such that , define , where , for partition with , and .

RSK correspondence

Corollary [Cauchy Identity]

Consider the specialization to and , we have the well-known -duality.

Theorem [-duality] Let act on the tersor product . Then the symmetric algebraas -modules, where the summation is over all partitions of length .

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