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bibkey: faribaultschuricht2012gaudin authors: Alexandre Faribault, Dirk Schuricht year: 2012 title: “On the determinant representations of Gaudin models’ scalar products and form factors” doi: 10.1088/1751-8113/45/48/485202 url: https://arxiv.org/abs/1207.2352v2 claim: “A Cauchy permanent can be expressed as a determinant with Gaudin-type entries.” strata_touched:

• D5/S3/Combinatorics/Permanental/CycleGeodesic/GaudinPermanent license: citation-only triage: anchor

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DOI: 10.1088/1751-8113/45/48/485202

Source: https://arxiv.org/abs/1207.2352v2

Identity and conventions

Journal of Physics A: Mathematical and Theoretical 45, article 485202, Section 3. The permanent of the reciprocal-difference matrix $C_{ij}=1/(x_i-y_j)$ equals $\det G$, where

$$G_{ii}=\sum _j\frac{1}{x_i-y_j}-\sum _{k\ne i}\frac{1}{x_i-x_k},G_{ij}=-\frac{1}{x_i-x_j}(i\ne j).$$

The row parameters $x_i$ must be distinct and avoid every $y_j$; the column parameters may repeat. Reversing all reciprocal-difference signs changes both sides by $(-1{)}^n$. This specifies the sign conversion from the paper’s Gaudin convention.

The formal proof differentiates Lagrange interpolation to build a nonzero kernel vector for a rectangular Gaudin matrix. A determinant polynomial consequently loses its leading coefficient, and interpolation and induction recover the permanent’s column expansion.