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Expand each factor as a geometric series indexed by a nonnegative matrix. The RSK correspondence preserves its row and column contents, changing the sum to pairs of Semistandard Young tableaux of the same shape and proving the displayed formal identity. Its kernel is also ${\sum }_{\lambda }{h}_{\lambda }(x)m_{\lambda }(y)={\sum }_{\lambda }{p}_{\lambda }(x)p_{\lambda }(y)/z_{\lambda }$. These are dual-basis kernels for the Hall inner product of symmetric functions, so the Schur expansion proves orthonormality. Applying the symmetric-function involution in one alphabet gives the dual identity ${\prod }_{i,j}(1+x_i{y}_j)={\sum }_{\lambda }{s}_{\lambda }(x)s_{{\lambda }^{\prime }}(y)$.

Apply the symmetric-function involution in one alphabet to the ordinary Cauchy identity. It changes complete generators to elementary ones and, by the dual Jacobi–Trudi identity, sends $s_{\lambda }$ to $s_{{\lambda }^{\prime }}$. This proves the identity as a formal series and relates pairings with transposed Young shapes to squarefree matrix weights.

Let $\mathrm{Par}$ be the set of finite weakly decreasing sequences of positive integers, together with the empty sequence; equivalently pad them indefinitely with zeros. The size is $|\lambda |={\sum }_i{\lambda }_i$, the length counts positive parts, and $\mathrm{Par}(n)$ consists of size-$n$ integer partitions. Their Young diagrams have cells $(i,j)$ with $1\le j\le {\lambda }_i$.

Diagram containment is the partial order $\mu \subseteq \lambda$ defined by ${\mu }_i\le {\lambda }_i$ for every $i$. Componentwise minimum and maximum give its meet and join, so this is Young's lattice. Its covers add one cell. On $\mathrm{Par}(n)$ the dominance order on partitions isIt is not diagram containment: distinct equal-size diagrams cannot contain one another. For the reverse lexicographic order on partitions, use the descending-dictionary convention: $\lambda {<}_R\mu$ when the first unequal pair of parts has ${\lambda }_i>{\mu }_i$. Thus for size four its increasing list is $(4),(3,1),(2,2),(2,1,1),(1,1,1,1)$. This is a total order opposite to increasing dictionary order on integer partitions, not the last-differing-exponent order on polynomial monomials.

Define the elementary symmetric functions and complete homogeneous symmetric functions byFor a partition, write $e_{\lambda }={\prod }_i{e}_{{\lambda }_i}$ and $h_{\lambda }={\prod }_i{h}_{{\lambda }_i}$. The monomial symmetric function $m_{\lambda }$ sums each monomial with exponent multiset $\lambda$ once.

Here is a proof of the Fundamental theorem of symmetric polynomials sufficient for the stable algebra. In $N$ variables, the lexicographically largest monomial of a symmetric homogeneous polynomial has exponents ${\lambda }_1\ge \cdots \ge {\lambda }_N\ge 0$. Otherwise exchanging an inverted pair of variables produces a larger monomial with the same coefficient. The polynomialhas that monomial as its largest one, with coefficient one. Subtract a suitable multiple and continue; there are only finitely many monomials of the fixed degree, so the procedure terminates and proves generation. Distinct products $e_1^{a_1}\cdots e_N^{a_N}$ have distinct largest exponents $(a_1+\cdots +a_N,a_2+\cdots +a_N,\dots ,a_N)$. The largest such monomial in any nonzero polynomial relation cannot cancel, proving algebraic independence. Stabilizing in each degree, where only $e_r$ with $r$ no greater than that degree can occur, gives

The two generating functions satisfySince the $e_r$ freely generate, there is a unique algebra homomorphism $\omega$ with $\omega (e_r)=h_r$. Apply it to the last identity:It follows that $\omega (h_r)=e_r$ and ${\omega }^2(e_r)=e_r$. Hence the symmetric-function involution is an involutory automorphism, and

For the forgotten coefficients, it is useful to derive the required duality explicitly. Put $p_r={\sum }_i{x}_i^r$. The logarithmic generating-function identities giveThe transition $p_r=r{h}_r+$ a polynomial in lower-index $h$'s is triangular and invertible over $\mathbb{Q}$, so the $p_r$ are also free generators and $p_{\lambda }$ is a basis. Define the Hall inner product of symmetric functions byExpand the same formal kernel in two ways:The last expansion is a reproducing kernel for this inner product. Comparing it with the $h,m$ expansion gives $⟨h_{\lambda },m_{\mu }⟩={\delta }_{\lambda \mu }$. One can see this without assuming any Schur orthogonality: pair either kernel with $h_{\nu }(y)$; the reproducing property gives $h_{\nu }(x)$, and uniqueness of its $h$ expansion proves the duality. The signs in $\omega (p_{\lambda })$ show that $\omega$ is an isometry; together with ${\omega }^2=1$ this makes it self-adjoint.

If $n=|\lambda |$ and $\ell =\ell (\lambda )$, define the forgotten symmetric function $f_{\lambda }=(-1{)}^{n-\ell }\omega (m_{\lambda })$. Its monomial coefficient is consequentlyIt remains to compute this coefficient rather than merely invoke duality. Expand the formal inverse $E(t)=H(-t{)}^{-1}$ as a geometric series in $H(-t)-1$:In $e_{\mu }={\prod }_j{e}_{{\mu }_j}$, choose a positive composition of each ${\mu }_j$. A term is $h_{\lambda }$ precisely when the concatenated list is a distinct rearrangement $\alpha$ of the parts of $\lambda$. Its boundaries at the end of each chosen composition are exactly the prescribed prefix sums of $\mu$. Conversely such a rearrangement uniquely determines the compositions by cutting at those prefix sums. Every one of its signs is $(-1{)}^{\sum {\mu }_j-\ell }=(-1{)}^{n-\ell }$, canceled by the prefactor in $a_{\lambda \mu }$. ThusIdentical parts are not artificially distinguished. For example $f_{(2,1)}=m_{(2,1)}+2m_{(3)}$: only the order $(2,1)$ refines $(2,1)$, while both orders refine the single block $(3)$.

For $F(t)={\prod }_i(1+x_{i}t)$ the coefficients are the elementary symmetric functions, $f_r=e_r(x)$. The specialization is evaluation in $x$ after the symmetric-function involution, because $\omega (h_r)=e_r$. HenceOne can also read this coefficient directly from the dual Cauchy identity, after exchanging its alphabets. For a general symmetric function, $u^F=(\omega u)(x)$; it is not simply $u(x)$.

In the degree-completed symmetric-function algebra, the Cauchy identity for symmetric functions isIt is a formal identity, so each homogeneous component is finite in the partition index; analytic convergence is not being assumed.

Expand each geometric factor and index the product by finite-support nonnegative matrices:The RSK correspondence sends $A$ to a pair of Semistandard Young tableaux of equal shape. Its content identities turn the matrix weight into the product of the recording tableau's $x$ weight and the insertion tableau's $y$ weight. Summing independently over both tableaux of each shape proves the identity.

Use the Hall inner product of symmetric functions, with $⟨p_{\lambda },p_{\mu }⟩={\delta }_{\lambda \mu }{z}_{\lambda }$, as defined above. The left side is the reproducing kernel ${\sum }_{\lambda }{p}_{\lambda }(x)p_{\lambda }(y)/z_{\lambda }$. Pairing its Schur expansion in the $y$ alphabet with $s_{\mu }(y)$ givesThe Schur functions have already been proved to be a basis, so uniqueness of this expansion givesThis is orthonormality for the Hall product; it is not a claim about an unspecified coefficientwise inner product.

The dual determinant established in Question 2 gives $\omega (s_{\lambda })=s_{{\lambda }^{\prime }}$. Apply the symmetric-function involution in the $y$ alphabet. Since $\omega (H_y(t))=E_y(t)$, it transforms each factor $H_y(x_i)$ into $E_y(x_i)$. The dual Cauchy identity is

For arbitrary coefficients $f_r$ in a commutative ring, use the integral symmetric-function algebra and its free complete generators, specializing $h_r\mapsto f_r$. This works over any such coefficient ring; a literal homomorphism from the rational algebra $\Lambda$ also presupposes a $\mathbb{Q}$-algebra structure on the target. Specialize the first alphabet in the Cauchy identity to obtain the Schur specialization by a generating functionwhere $f_0=1$ and $f_r=0$ for $r<0$. Indeed, the left side is a product of the specialized $H(t_i)$, and the Jacobi–Trudi identity specializes to the displayed determinant. For an arbitrary infinite $F$ the result lies in the degree completion; its coefficients in each degree are well defined.