论文标题
从凯奇的决定因素公式到骨气和费米尼的内在身份
From Cauchy's determinant formula to bosonic and fermionic immanant identities
论文作者
论文摘要
Cauchy的决定符公式（1841）涉及$ \ det（（（1-u_i v_j）^{ - 1}）$是对称函数理论的基本结果。它已朝多个方向扩展，包括Frobenius的决定扩展[J. Reine Angew。数学。 [1882]涉及$ u_i v_j $中的两个几何系列的总和。这个主题还以horn [trans。阿米尔。数学。 Soc。 1969年] - 计算归因于Loewner的其中 - 在Belton-Guillot-Khare-Putinar的最新作品中[Adv。数学。 2016]和Khare-tao [Amer。 J. Math。 2021]。这些公式最近统一并扩展了[trans。阿米尔。数学。 Soc。 2022]到任意功率系列，并带有通勤/玻感变量$ u_i，v_j $。 在本说明中，我们制定了类似的永久身份，实际上，解释了所有这些结果是如何具有更一般身份的特殊情况，对于任何在玻色子变量$ u_i $和$ v_j $上的有限群体的任何角色（实际上，任何复杂的类功能）。 （我们解释了为什么较大的线性基团通过一个新颖的“对称函数”对签名的置换矩阵的表征不起作用，该矩阵在任何积分域上都存在。）然后，我们提供了这些公式的费米子类似物以及密切相关的cauchy产品标识。
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.
