Power-sum, complete, and Schur expansions of the Cauchy kernel

For $N\ge 1$, put $K_N=((1-x_i{y}_j{)}^{-1}{)}_{i,j=1}^N$, $P_N={\prod }_{i,j=1}^N(1-x_i{y}_j)$, and $D_N=P_N\det K_N$. Clearing denominators gives $D_N={\sum }_{\sigma \in S_N}\mathrm{sgn}(\sigma ){\prod }_i{\prod }_{j\mathrm{\ne }\sigma (i)}(1-x_i{y}_j)$. This polynomial is alternating separately in the $x$ and $y$ variables. Vandermonde divisibility gives ${\mathrm{\Delta }}_N(x){\mathrm{\Delta }}_N(y)\mid D_N$ in $\mathbb{Z}[x_1,\dots ,x_N,y_1,\dots ,y_N]$. Since $D_N$ has degree at most $N-1$ in each individual variable and each Vandermonde has degree $N-1$ in each of its variables, the quotient has degree zero in every variable and is an integer constant. Here ${\mathrm{\Delta }}_N(x)=\det (x_i^{N-j})$, and likewise for $y$. To determine the constant, truncate each geometric series at a common exponent bound $M\ge N-1$ and apply finite Cauchy–Binet; the least possible total $y$-degree uses the distinct exponents $0,1,\dots ,N-1$ and contributes ${\mathrm{\Delta }}_N(x){\mathrm{\Delta }}_N(y)$. Reversing exponent order changes both determinants by the same sign. This lowest-degree term is unchanged as $M$ increases, so it is the least-degree part of $\det K_N$. Since $P_N$ has constant term $1$ in the $y$ variables, the least-degree part of $D_N$ is also ${\mathrm{\Delta }}_N(x){\mathrm{\Delta }}_N(y)$, and the constant is $1$. Thus $\det K_N={\mathrm{\Delta }}_N(x){\mathrm{\Delta }}_N(y){\mathrm{\Omega }}_N$, where ${\mathrm{\Omega }}_N=P_N^{-1}$. For $N=0$ the identity holds with empty determinants and products equal to $1$.

Fix $d\ge 0$ and $N\ge d$. If $d>0$, every partition of $d$ has length at most $d\le N$. By [F3], projection carries the stable basis $m_{\lambda }$ to the finite orbit sums; by [F5] those form a basis of the rank-$N$ symmetric polynomials. Therefore ${\mathrm{\Lambda }}^d\to A_N^d$ is an isomorphism. For $d=0$, it is the unit map $\mathbb{Z}\to \mathbb{Z}$. Tensoring these projection isomorphisms in the two variables makes equality at rank $N\ge d$ sufficient to prove equality in bidegree $(d,d)$.

For $N\ge 1$, put $K_N=((1-x_i{y}_j{)}^{-1}{)}_{i,j=1}^N$, truncate each geometric series at exponent $M$, apply finite Cauchy–Binet, and let $M$ increase; every fixed bidegree receives contributions from finitely many exponent sets, giving $\det K_N={\sum }_{0\le k_1<\cdots <k_N}\det (x_i^{k_j}{)}_{i,j}\det (y_i^{k_j}{)}_{i,j}$. The assignment ${\lambda }_i=k_{N+1-i}-(N-i)$ is a bijection from these strictly increasing exponent sets to partitions of length at most $N$, since strict increase makes the parts weakly decreasing and nonnegative.

At finite rank $N$, write ${\mathrm{\Omega }}_N={\prod }_{i,j=1}^N(1-x_i{y}_j{)}^{-1}$. Expand each geometric product as ${\prod }_{i=1}^N(1-x_i{y}_j{)}^{-1}={\sum }_{a_j\ge 0}{h}_{a_j}(x_1,\dots ,x_N)y_j^{a_j}$, since multiplying the $N$ one-variable geometric series gives the coefficient formula in [F7]. Hence ${\mathrm{\Omega }}_N={\sum }_{(a_1,\dots ,a_N)\in {\mathbb{N}}^N}({\prod }_j{h}_{a_j}(x))y_1^{a_1}\cdots y_N^{a_N}$. Sort the positive entries of each exponent tuple into a partition $\lambda$. Its coefficient is $h_{\lambda }(x)$, and its distinct coordinate permutations sum to exactly $m_{\lambda }(y)$ by [F4]. Thus ${\mathrm{\Omega }}_N={\sum }_{\mathrm{\ell }(\lambda )\le N}{h}_{\lambda }(x)m_{\lambda }(y)$. Each bidegree has finitely many partitions; the kernel's stable coefficients are those finite-rank limits by [F1], so passing rank $N\ge d$ by step 1.2 proves the stable complete–monomial expansion.

At finite rank write ${\mathrm{\Omega }}_N={\prod }_{i,j=1}^N(1-x_i{y}_j{)}^{-1}$. Formal logarithms over $\mathbb{Q}$ give $\log {\mathrm{\Omega }}_N={\sum }_{i,j}{\sum }_{r\ge 1}(x_i{y}_j{)}^r\mathrm{/}r={\sum }_{r\ge 1}{p}_r^{(N)}(x)p_r^{(N)}(y)\mathrm{/}r$. In the componentwise rational completion, the positive-degree part of $\mathrm{\Omega }$ is topologically nilpotent for the bidegree filtration: each fixed bidegree receives contributions from only finitely many powers and finitely many $r$. Thus formal logarithm and exponential are defined coefficientwise. By [F2], [F7], and [F8], the finite identity lifts to $\log \mathrm{\Omega }={\sum }_{r\ge 1}{p}_r(x)p_r(y)\mathrm{/}r$. Exponentiating and multiplying the commuting series $\exp (p_r(x)p_r(y)\mathrm{/}r)={\sum }_{m_r\ge 0}(p_r(x)p_r(y){)}^{m_r}\mathrm{/}(r^{m_r}{m}_r!)$ over $r\ge 1$ gives one term for each finitely supported multiplicity sequence $(m_r)$, equivalently each partition $\lambda$. Its coefficient is $1\mathrm{/}{z}_{\lambda }$. By [F8] these are the rational power-sum basis elements in the two degree-$d$ factors, proving the stated expansion.

Reversing the exponent columns in both determinants of step 1.3 changes each by $(-1{)}^{N(N-1)\mathrm{/}2}$, so their product becomes $a_{\lambda +{\delta }_N}(x)a_{\lambda +{\delta }_N}(y)$. By [F9], each alternant is ${\mathrm{\Delta }}_N$ times its finite Schur quotient. Combine the determinant expansion of step 1.3 with step 1.1 and cancel the nonzero polynomial ${\mathrm{\Delta }}_N(x){\mathrm{\Delta }}_N(y)$ in each homogeneous bidegree of the integral-domain polynomial ring; this gives ${\mathrm{\Omega }}_N={\sum }_{\mathrm{\ell }(\lambda )\le N}{s}_{\lambda }(x_1,\dots ,x_N)s_{\lambda }(y_1,\dots ,y_N)$.

For each bidegree $(d,d)$, take $N\ge d$ and use the projection isomorphisms of step 1.2 to pass the finite Schur identity of step 2.3 to the stable completion. The empty rank $N=0$ has empty determinants and products equal to $1$, and the empty partition gives the constant term in all three expansions. Setting either alphabet to zero leaves only that term; all off-diagonal components are zero by [F1]. At degree one, rank-one projection sends $h_{(1)},m_{(1)},p_{(1)}$, and $s_{(1)}$ to $x_1$, so every expansion has coefficient one. These arguments include the threshold rank $N=d$ and the first allowed bialternant rank $N=\mathrm{\ell }(\lambda )$.