From Cauchy’s determinant formula to bosonic and fermionic immanant identities
From Cauchy’s determinant formula to bosonic and fermionic immanant identities
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从柯西行列式到玻色子和费米子内在恒等式
DOI：
10.1016/j.ejc.2022.103683
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发表时间：
2023
影响因子：
1
通讯作者：
Sahi, Siddhartha
中科院分区：
文献类型：
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作者：
Khare, Apoorva;Sahi, Siddhartha
柯西行列式公式（1841）涉及det（（1− u i v j）− 1）是对称函数论中的一个基本结果。它已扩展到几个方向，包括一个行列式扩展弗罗贝纽斯（1882年）涉及的两个几何级数的总和在u i v j.这一主题也重新浮出水面的矩阵分析设置在一份文件霍恩（1969年）-其中的计算是由于Loewner-并在最近的作品贝尔顿等人。（2016）和Khare和Tao（2021）。这些公式最近在Khare（2022）中统一并扩展到任意幂级数，具有交换/玻色子变量u i，v j。在本说明中，我们制定了类似的永久恒等式，事实上，解释了所有这些结果是如何更一般恒等式的特殊情况，对于任何字符-事实上，任何复类函数-任何有限群的任何复类函数，通过有符号排列作用于玻色子变量u i和v j。(We解释为什么更大的线性群不起作用，通过一个可能是新颖的符号置换矩阵的“对称函数”特征，它在任何整数域上都成立。然后，我们提供了这些公式以及密切相关的柯西积恒等式的费米子类似物。
Abstract Cauchy’s determinant formula (1841) involving det ((1− u i v j)− 1) is a fundamental result in symmetric function theory. It has been extended in several directions, including a determinantal extension by Frobenius (1882) involving a sum of two geometric series in u i v j. This theme also resurfaced in a matrix analysis setting in a paper by Horn (1969)–where the computations are attributed to Loewner–and in recent works by Belton et al.(2016) and Khare and Tao (2021). These formulas were recently unified and extended in Khare (2022) to arbitrary power series, with commuting/bosonic variables u i, v j. In this note we formulate analogous permanent identities, and in fact, explain how all of these results are a special case of a more general identity, for any character–in fact, any complex class function–of any finite group that acts on the bosonic variables u i and on the v j via signed permutations.(We explain why larger linear groups do not work, via a–perhaps novel–“symmetric function” characterization of signed permutation matrices that holds over any integral domain.) We then provide fermionic analogues of these formulas, as well as of the closely related Cauchy product identities.