Let $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$ be nonnegative integers. The Schur polynomial $s_{\lambda}(x_1, \ldots, x_n)$ can be defined as the ratio $$s_{\lambda}(x_1, x_2, \ldots, x_n) = \frac{\det\left(x_i^{\lambda_j+n-j} \right)}{\Delta(x)} \qquad (\clubsuit)$$ where $$\Delta(x) = \det \left( x_i^{n-j} \right) = \prod_{i<j} (x_i - x_j)$$ is the Vandermonde determinant. According to the historical notes in Stanley, Enumerative Combinatorics, Volume 2, Chapter 7, this ratio was first studied by Cauchy.

Nowadays, we know that Schur polynomials are important in the representation theory of $GL_n(\mathbb{C})$, $GL_n(\mathbb{F}_p)$ and $S_n$, and in Schubert calculus. But Schur's thesis, where he related Schur polynomials to representation theory, is from 1918, and Schubert developed Schubert calculus in the 1870's. Cauchy is much earlier; he introduced these ratios in 1815 (Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment). So my first question is

Why was Cauchy studying the ratio $(\clubsuit)$?

Continuing in this vein, the following formula is now known as Cauchy's identity: $$\prod_{i,j=1}^n \frac{1}{1-x_i y_j} = \sum_{\lambda} s_{\lambda}(x) s_{\lambda}(y). \qquad (\diamondsuit)$$ Stanley writes that he can not find a source where Cauchy records $(\diamondsuit)$, but that $(\diamondsuit)$ can be deduced from Cauchy's determinantal identity: $$\det \left( \frac{1}{u_i+v_j} \right) = \frac{\Delta(u) \Delta(v)}{\prod_{i,j} (u_i + v_j)} \quad (\heartsuit)$$ using the substitution $(u_i = -1/x_i, v_j = y_j)$ and the Cauchy-Binet formula. (I understand how the deduction works; you don't need to explain this part.)

The identity $(\heartsuit)$ is due to Cauchy; it appears as an exercise in an 1841 paper (Mémoire sur les fonctions alternées et sur les sommes alternées). So my questions continue:

Why was Cauchy studying $(\heartsuit)$? Did he know that $(\heartsuit)$ and $(\clubsuit)$ were related?

Note: I have linked Cauchy's papers above, and skimmed enough to make sure they are the right ones, but I haven't actually read them.

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  • $\begingroup$ I would have added tags [symmetric-polynomials] and [determinants], but I don't have enough rep here to create them. Other tag suggestions are also welcome. $\endgroup$ Sep 21 at 19:51
  • $\begingroup$ What did Cauchy himself say about it in the paper you linked? Just looking at the first page, he was interested in symmetric functions, which were studied since Newton and Lagrange. Lagrange related them to what later became Galois theory. $\endgroup$
    – Conifold
    Sep 21 at 20:58


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