从柯西行列式公式到玻色子与费米子immanant恒等式
From Cauchy's determinant formula to bosonic and fermionic immanant identities
AI总结：
本文将柯西行列式公式及其推广统一为适用于任意有限群特征标的一般恒等式，并进一步构造了相应的玻色子与费米子immanant恒等式及柯西乘积恒等式的费米子类比。
摘要：
柯西行列式公式（1841年）涉及 $\det ((1-u_i v_j)^{-1})$，是对称函数理论中的一个基本结果。它已在多个方向上得到推广，包括Frobenius [J. reine angew. Math. 1882] 给出的行列式推广，其中涉及 $u_i v_j$ 的两个几何级数之和。这一主题后来也在矩阵分析背景下重新出现，见于Horn [Trans. Amer. Math. Soc. 1969] 的论文——其中计算被归于Loewner——以及Belton-Guillot-Khare-Putinar [Adv. Math. 2016] 和Khare-Tao [Amer. J. Math. 2021] 的近作。这些公式最近在 [Trans. Amer. Math. Soc. 2022] 中被统一并推广到任意幂级数，使用交换的玻色子变量 $u_i, v_j$。在本文中，我们构造了类似的permanent恒等式，并且事实上解释了所有这些结果都是一个更一般恒等式的特殊情形，该恒等式适用于任何有限群的任何特征标——事实上，任何复类函数——只要该有限群通过带符号置换作用于玻色子变量 $u_i$ 和 $v_j$。（我们解释了为什么更大的线性群不适用，借助一个——也许是新的——带符号置换矩阵的“对称函数”刻画，该刻画在任何整环上都成立。）然后我们给出这些公式以及密切相关的柯西乘积恒等式的费米子类比。
英文摘要：
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 [J. reine angew. Math. 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 [Trans. Amer. Math. Soc. 1969] - where the computations are attributed to Loewner - and in recent works by Belton-Guillot-Khare-Putinar [Adv. Math. 2016] and Khare-Tao [Amer. J. Math. 2021]. These formulas were recently unified and extended in [Trans. Amer. Math. Soc. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
