40.03.03 · combinatorics / symmetric-functions-rsk

The Cauchy Identity, Dual Bases, and the Hall Inner Product

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Anchor (Master): Macdonald 1995 *Symmetric Functions and Hall Polynomials* 2e (Oxford) Ch. I §4-5 (the scalar product on $\Lambda$ defined by $\langle h_\lambda, m_\mu\rangle = \delta_{\lambda\mu}$, the dual-bases lemma $\sum u_\lambda(x)v_\lambda(y) = \prod(1-x_iy_j)^{-1}$, the Cauchy and dual Cauchy identities, Schur orthonormality $\langle s_\lambda, s_\mu\rangle = \delta_{\lambda\mu}$, the isometry of $\omega$, the adjoint $f^\perp$ of multiplication by $f$, and skew Schur functions); Stanley 1999 *Enumerative Combinatorics, Volume 2* Ch. 7 and Appendix 2; Cauchy 1815 (the antisymmetric-quotient identity); Littlewood 1940 *The Theory of Group Characters* (the symmetric-function form of the orthogonality of characters)

Intuition Beginner

Suppose you have two different rulers for measuring the same family of objects. One ruler reports how many of each basic pattern an object contains; the other reports the patterns directly. If the two rulers are matched correctly, then measuring one ruler's marks against the other's gives a clean answer: a basic mark on one side lines up with exactly its partner on the other side, and with nothing else. Two families of building blocks that match this cleanly are called dual to each other. The whole point of this unit is one such matching, and a single product formula that detects when any matching is correct.

Why bother? In the previous units you met several families of symmetric expressions — the monomial family, the complete family, and the all-important Schur family. They all describe the same objects, but each emphasises something different. To compare them you need a way to "multiply two expressions and read off a number," much like measuring the angle between two arrows by a dot product. That measuring device is the Hall inner product. It is set up so the complete family and the monomial family are exactly dual: pairing one against the other returns one or zero, never anything messy.

The payoff is a single miracle. The Schur family turns out to be dual to itself: pairing two Schur expressions gives one when they are the same shape and zero otherwise. So the Schur functions are like a set of arrows all at right angles and all of unit length — an orthonormal system. That fact, proved here, is the keystone that makes Schur functions the natural alphabet of the subject. There is also a compact way to package the whole matching: one product formula, built by multiplying small fractions across every pairing of an $x$ -variable with a $y$ -variable, that detects exactly when two families fit together.

Visual Beginner

The matching idea in miniature. Read each row as a pairing of one expression from the left family against one from the right family. A pairing returns $1$ when the labels agree and $0$ when they do not — the signature of dual families. The third column shows the same answers arranged as a tiny table, the pattern that "orthonormal" means.

left expression	right expression	their pairing
$h_{(2)}$	$m_{(2)}$	$1$
$h_{(2)}$	$m_{(1, 1)}$	$0$
$h_{(1, 1)}$	$m_{(1, 1)}$	$1$
$s_{(2)}$	$s_{(2)}$	$1$
$s_{(2)}$	$s_{(1, 1)}$	$0$

Notice the two distinct patterns. In the top block the complete tiles pair cleanly with the monomial tiles of the same label — that is what "dual families" looks like. In the bottom grid the Schur tiles pair cleanly with themselves: ones down the diagonal, zeros elsewhere. The second pattern is the stronger one, and it is special to the Schur family. The product detector at the bottom is the same in both cases; the unit explains why one formula governs every such matching.

Worked example Beginner

Let us watch the matching work in the smallest interesting degree, degree two, using the two-variable building blocks from the previous units.

Step 1. List the degree-two pieces. There are two partitions of $2$ , namely $(2)$ and $(1, 1)$ . The monomial pieces are $m_{(2)} = x_{1}^{2} + x_{2}^{2}$ and $m_{(1, 1)} = x_{1} x_{2}$ . The complete pieces are $h_{(2)} = h_{2} = x_{1}^{2} + x_{1} x_{2} + x_{2}^{2}$ and $h_{(1, 1)} = h_{1} h_{1} = (x_{1} + x_{2})^{2} = x_{1}^{2} + 2 x_{1} x_{2} + x_{2}^{2}$ .

Step 2. State the matching rule. The Hall pairing is built so that $h$ paired with $m$ gives $1$ when the labels match and $0$ otherwise. So we expect $⟨ h_{(2)}, m_{(2)} ⟩ = 1$ , $⟨ h_{(2)}, m_{(1, 1)} ⟩ = 0$ , $⟨ h_{(1, 1)}, m_{(1, 1)} ⟩ = 1$ , and $⟨ h_{(1, 1)}, m_{(2)} ⟩ = 0$ .

Step 3. Check it against the Schur pieces. The Schur functions in degree two are $s_{(2)} = h_{2} = m_{(2)} + m_{(1, 1)}$ and $s_{(1, 1)} = e_{2} = m_{(1, 1)}$ . Pair $s_{(2)}$ with $s_{(2)}$ : write each Schur function in the complete-and-monomial language and use the matching rule from Step 2.

Step 4. Compute the keystone pairing. We have $s_{(2)} = h_{2}$ , and $s_{(2)} = m_{(2)} + m_{(1, 1)}$ . Pairing the $h$ form against the $m$ form, $⟨ h_{2}, m_{(2)} + m_{(1, 1)} ⟩ = ⟨ h_{2}, m_{(2)} ⟩ + ⟨ h_{2}, m_{(1, 1)} ⟩ = 1 + 0 = 1$ . So $⟨ s_{(2)}, s_{(2)} ⟩ = 1$ .

Step 5. Confirm the off-diagonal. Now $s_{(1, 1)} = e_{2} = m_{(1, 1)}$ , and one can write $s_{(2)} = h_{2}$ . Pair them: $⟨ h_{2}, m_{(1, 1)} ⟩ = 0$ . So $⟨ s_{(2)}, s_{(1, 1)} ⟩ = 0$ .

What this tells us: the Schur functions of the two different shapes are at right angles, and each has unit length. The matching rule for the complete and monomial families, applied carefully, forces the Schur family to be orthonormal. That self-matching is the heart of the unit.

Check your understanding Beginner

Formal definition Intermediate+

Work in the ring $Λ$ of symmetric functions over $Z$ (or $Q$ where denominators occur), with the monomial $m_{λ}$ , elementary $e_{λ}$ , complete $h_{λ}$ , power-sum $p_{λ}$ , and Schur $s_{λ}$ bases of 40.03.01 and 40.03.02, the involution $ω$ with $ω (e_{n}) = h_{n}$ , $ω (s_{λ}) = s_{λ^{'}}$ , the conjugate $λ^{'}$ , and $z_{λ} = \prod_{i \geq 1} i^{m_{i}} m_{i}!$ where $m_{i}$ is the multiplicity of $i$ in $λ$ . All notation is recorded in _meta/NOTATION.md. Use two disjoint variable alphabets $x = (x_{1}, x_{2}, \dots)$ and $y = (y_{1}, y_{2}, \dots)$ .

Definition (Hall inner product). The Hall inner product is the $Z$ -bilinear symmetric form $⟨ \cdot, \cdot ⟩ : Λ \times Λ \to Z$ characterised by declaring the complete and monomial bases dual, $$ \langle h_\lambda, m_\mu\rangle ;=; \delta_{\lambda\mu} \qquad (\lambda, \mu \vdash n,\ n \ge 0). $$ The form is graded: $⟨ f, g ⟩ = 0$ when $f \in Λ^{d}$ , $g \in Λ^{e}$ with $d \neq = e$ . Its existence and symmetry are part of the Key theorem.

Definition (dual bases). Two homogeneous bases ${u_{λ}}$ and ${v_{λ}}$ of $Λ$ , indexed by partitions and graded so $u_{λ}, v_{λ} \in Λ^{∣ λ ∣}$ , are dual (or adjoint) with respect to the Hall inner product when $⟨ u_{λ}, v_{μ} ⟩ = δ_{λ μ}$ . By definition ${h_{λ}}$ and ${m_{μ}}$ are dual; the Key theorem shows ${p_{λ}}$ is self-dual up to the scalars $z_{λ}$ and ${s_{λ}}$ is exactly self-dual.

Definition (the Cauchy kernel). The Cauchy kernel in the two alphabets is the formal product $$ \Omega(x, y) ;=; \prod_{i, j} \frac{1}{1 - x_i y_j} ;=; \sum_{\lambda} h_\lambda(x), m_\lambda(y) ;\in; \widehat{\Lambda(x) \otimes \Lambda(y)}, $$ an element of the degree-completed tensor product, the last equality recorded below. The dual Cauchy kernel is $Ω^{'} (x, y) = \prod_{i, j} (1 + x_{i} y_{j}) = \sum_{λ} e_{λ} (x) m_{λ} (y)$ .

Definition (the skewing operator). For $f \in Λ$ , the skewing operator $f^{⊥} : Λ \to Λ$ is the adjoint of multiplication by $f$ : the unique additive map with $$ \langle f g,, h\rangle ;=; \langle g,, f^\perp h\rangle \qquad (g, h \in \Lambda). $$ On the Schur basis it acts by $s_{λ}^{⊥} s_{μ} = s_{μ / λ}$ , the skew Schur function of 40.03.02 (with $s_{μ / λ} = 0$ unless $λ \subseteq μ$ ); this identity is established in the Full proof set.

Counterexamples to common slips Intermediate+

"The $p_{λ}$ are orthonormal like the $s_{λ}$ ." They are orthogonal but not normal: $⟨ p_{λ}, p_{μ} ⟩ = z_{λ} δ_{λ μ}$ , and $z_{λ} > 1$ except for $λ = (1^{n})$ . The normalised family ${z_{λ}^{- 1/2} p_{λ}}$ is orthonormal only over a ring containing the square roots, so the $p_{λ}$ are a $Q$ -orthogonal basis, not a $Z$ -orthonormal one.
"Dual to ${h_{λ}}$ means ${e_{λ}}$ ." The dual of ${h_{λ}}$ is ${m_{μ}}$ , not ${e_{μ}}$ . The elementary functions are dual to the forgotten basis ${f_{μ}} = {ω m_{μ}}$ , since $⟨ e_{λ}, ω m_{μ} ⟩ = ⟨ ω h_{λ}, ω m_{μ} ⟩ = ⟨ h_{λ}, m_{μ} ⟩ = δ_{λ μ}$ once $ω$ is known to be an isometry.
"The Cauchy product is symmetric in $x$ and $y$ term by term in every expansion." The kernel $Ω (x, y)$ is symmetric under swapping the two alphabets, but its expansion $\sum_{λ} h_{λ} (x) m_{λ} (y)$ is not visibly so; the symmetry surfaces only in the Schur form $\sum_{λ} s_{λ} (x) s_{λ} (y)$ , where both alphabets carry the same basis.

Key theorem with proof Intermediate+

The signature theorem is the dual-bases characterisation: a single product formula detects duality, and applied to the Schur basis it yields the keystone orthonormality.

Theorem (dual bases via the Cauchy kernel; Schur orthonormality). Let ${u_{λ}}, {v_{λ}}$ be homogeneous $Q$ -bases of $Λ_{Q}$ indexed by partitions with $u_{λ}, v_{λ} \in Λ^{∣ λ ∣}$ . Then $$ \langle u_\lambda, v_\mu\rangle = \delta_{\lambda\mu} \ \text{ for all } \lambda, \mu \qquad \Longleftrightarrow \qquad \sum_{\lambda} u_\lambda(x), v_\lambda(y) = \prod_{i,j} \frac{1}{1 - x_i y_j}. $$ Consequently the Hall inner product is symmetric and well defined, $⟨ p_{λ}, p_{μ} ⟩ = z_{λ} δ_{λ μ}$ , the Cauchy identity holds in its three forms $$ \prod_{i,j}\frac{1}{1 - x_i y_j} = \sum_\lambda h_\lambda(x) m_\lambda(y) = \sum_\lambda z_\lambda^{-1} p_\lambda(x) p_\lambda(y) = \sum_\lambda s_\lambda(x) s_\lambda(y), $$ and the Schur functions are orthonormal, $⟨ s_{λ}, s_{μ} ⟩ = δ_{λ μ}$ . ^{[Stanley §7.12]}

Proof. Fix the degree- $n$ graded pieces; everything is finite-dimensional there. Write $A = (⟨ u_{λ}, v_{μ} ⟩)_{λ, μ ⊢ n}$ . Expand each basis against the dual pair ${h_{λ}}, {m_{μ}}$ : put $u_{λ} = \sum_{ρ} a_{λ ρ} h_{ρ}$ and $v_{μ} = \sum_{σ} b_{μ σ} m_{σ}$ , so that $⟨ u_{λ}, v_{μ} ⟩ = \sum_{ρ} a_{λ ρ} b_{μ ρ}$ , i.e. $A = a b^{T}$ in matrix form. On the generating-function side, $$ \sum_\lambda u_\lambda(x) v_\lambda(y) = \sum_\lambda \Big(\sum_\rho a_{\lambda\rho} h_\rho(x)\Big)\Big(\sum_\sigma b_{\lambda\sigma} m_\sigma(y)\Big) = \sum_{\rho,\sigma} \Big(\sum_\lambda a_{\lambda\rho} b_{\lambda\sigma}\Big) h_\rho(x) m_\sigma(y) = \sum_{\rho,\sigma} (a^{\mathsf T} b){\rho\sigma}, h\rho(x) m_\sigma(y). $$ This equals the kernel expansion $\sum_{ρ} h_{ρ} (x) m_{ρ} (y)$ if and only if $a^{T} b = I$ , that is $b = (a^{T})^{- 1}$ . Meanwhile $A = a b^{T} = a ((a^{T})^{- 1})^{T} = a (a^{- 1}) = I$ . Conversely $A = I$ forces $a b^{T} = I$ , hence $a^{T} b = I$ , giving the kernel. So duality and the product formula are equivalent, provided the kernel itself expands as $\sum_{ρ} h_{ρ} (x) m_{ρ} (y)$ .

That base expansion is a direct computation: in a single $y$ -variable, $\prod_{i} (1 - x_{i} y)^{- 1} = \sum_{n} h_{n} (x) y^{n}$ is the definition of the complete functions; with the full alphabet $y$ , expanding the product over $j$ and collecting the monomial $y^{α}$ groups the $x$ -coefficients into $h_{α_{1}} (x) h_{α_{2}} (x) \dots$ , and summing over the $S$ -orbit of each exponent pattern $λ$ assembles $h_{λ} (x) m_{λ} (y)$ . Hence $Ω (x, y) = \sum_{λ} h_{λ} (x) m_{λ} (y)$ , the $u = h$ , $v = m$ instance, which records that ${h_{λ}}, {m_{μ}}$ are dual and certifies the form is well defined.

Symmetry of the form: swapping the two alphabets fixes $Ω (x, y)$ , so $\sum_{λ} h_{λ} (x) m_{λ} (y) = \sum_{λ} m_{λ} (x) h_{λ} (y)$ , whence ${m_{λ}}, {h_{μ}}$ are also dual, i.e. $⟨ m_{λ}, h_{μ} ⟩ = δ_{λ μ} = ⟨ h_{μ}, m_{λ} ⟩$ , and bilinear extension gives $⟨ f, g ⟩ = ⟨ g, f ⟩$ throughout. For the power sums, the identity $h_{n} = \sum_{λ ⊢ n} z_{λ}^{- 1} p_{λ}$ of 40.03.01 applied in both alphabets, together with $\prod_{i, j} (1 - x_{i} y_{j})^{- 1} = exp \sum_{k \geq 1} \frac{1}{k} p_{k} (x) p_{k} (y)$ (take $lo g$ of the product), yields $Ω = \sum_{λ} z_{λ}^{- 1} p_{λ} (x) p_{λ} (y)$ ; matching this to the dual-bases criterion with $u_{λ} = z_{λ}^{- 1} p_{λ}$ , $v_{λ} = p_{λ}$ gives $⟨ z_{λ}^{- 1} p_{λ}, p_{μ} ⟩ = δ_{λ μ}$ , i.e. $⟨ p_{λ}, p_{μ} ⟩ = z_{λ} δ_{λ μ}$ .

Finally the Schur form. The Schur functions satisfy the Cauchy identity $\sum_{λ} s_{λ} (x) s_{λ} (y) = Ω (x, y)$ — proved bijectively by the RSK correspondence of the co-produced 40.03.04, and independently by the Jacobi-Trudi / Lindström-Gessel-Viennot route in the Full proof set. Taking $u_{λ} = v_{λ} = s_{λ}$ in the established equivalence, the product formula on the right is exactly the criterion for $⟨ s_{λ}, s_{μ} ⟩ = δ_{λ μ}$ . $□$

Bridge. This theorem is the foundational reason the Hall inner product is forced rather than chosen: the single kernel $\prod (1 - x_{i} y_{j})^{- 1}$ simultaneously defines the form (through ${h_{λ}}, {m_{μ}}$ ), measures it (a basis is dual iff its Cauchy sum reproduces the kernel), and diagonalises it (the Schur form). It builds toward the Frobenius characteristic of the co-produced 40.03.06, where this very inner product becomes the character inner product on the symmetric groups and Schur orthonormality becomes the orthogonality of irreducible characters. The dual-bases criterion is exactly the statement that the kernel is the reproducing element of a self-pairing, and the three expansions of $Ω$ are dual to one another through the change-of-basis matrices: the $h$ - $m$ form exhibits the defining duality, the $p$ -form exhibits the orthogonality with weights $z_{λ}$ , and the $s$ -form is the central insight that one basis is self-dual. Putting these together, the orthonormality $⟨ s_{λ}, s_{μ} ⟩ = δ_{λ μ}$ generalises the Gram-matrix computation of any finite inner-product space to the graded ring $Λ$ , and it appears again in the Pieri and Littlewood-Richardson adjointness of Advanced results, where $⟨ s_{λ} s_{μ}, s_{ν} ⟩ = c_{λ μ}^{ν}$ .

Exercises Intermediate+

Exercise 6 (hard, symbolic).

Prove that $ω$ is an isometry, $⟨ ω f, ω g ⟩ = ⟨ f, g ⟩$ , by applying $ω$ in one alphabet to the Cauchy kernel and identifying the result with the dual Cauchy kernel.

Hint

Apply $ω_{x}$ (acting on the $x$ -alphabet only) to $\sum_{λ} h_{λ} (x) m_{λ} (y) = \prod (1 - x_{i} y_{j})^{- 1}$ . Use $ω (h_{λ}) = e_{λ}$ and the dual Cauchy identity.

Answer

Apply $ω_{x}$ to $Ω (x, y) = \sum_{λ} h_{λ} (x) m_{λ} (y)$ : the left side becomes $\sum_{λ} e_{λ} (x) m_{λ} (y) = \prod_{i, j} (1 + x_{i} y_{j}) = Ω^{'} (x, y)$ , the dual Cauchy kernel. So $ω_{x}$ carries $Ω \mapsto Ω^{'}$ . Now expand both kernels in the Schur basis: $Ω = \sum_{λ} s_{λ} (x) s_{λ} (y)$ and $Ω^{'} = \sum_{λ} s_{λ} (x) s_{λ^{'}} (y)$ . Matching $ω_{x} Ω = Ω^{'}$ term by term in the $y$ -Schur basis forces $ω_{x} s_{λ} (x) = s_{λ^{'}} (x)$ , recovering $ω (s_{λ}) = s_{λ^{'}}$ . Since ${s_{λ}}$ is orthonormal and $ω$ permutes it (by conjugation, an involution on partitions), $⟨ ω s_{λ}, ω s_{μ} ⟩ = ⟨ s_{λ^{'}}, s_{μ^{'}} ⟩ = δ_{λ^{'} μ^{'}} = δ_{λ μ} = ⟨ s_{λ}, s_{μ} ⟩$ ; bilinear extension gives the isometry on all of $Λ$ . Rubric: full credit for $ω_{x} Ω = Ω^{'}$ , the deduction $ω (s_{λ}) = s_{λ^{'}}$ , and the permutation-of-orthonormal-basis isometry argument.

Exercise 7 (hard, symbolic).

Prove the adjointness $⟨ f g, h ⟩ = ⟨ g, f^{⊥} h ⟩$ determines $f^{⊥}$ uniquely, and show $h_{r}^{⊥} s_{μ} = s_{μ / (r)} = \sum_{ν} s_{ν}$ summed over $ν$ with $μ / ν$ a horizontal $r$ -strip. Deduce the Pieri rule.

Hint

Uniqueness: orthonormality of ${s_{λ}}$ lets you read coefficients. For the action, use $⟨ h_{r} s_{ν}, s_{μ} ⟩ = ⟨ s_{ν}, h_{r}^{⊥} s_{μ} ⟩$ and the Pieri rule of 40.03.02.

Answer

Since ${s_{λ}}$ is an orthonormal basis, any linear map $T$ is determined by the numbers $⟨ s_{λ}, T s_{μ} ⟩$ ; the requirement $⟨ f s_{ν}, s_{μ} ⟩ = ⟨ s_{ν}, f^{⊥} s_{μ} ⟩$ fixes every coefficient $⟨ s_{ν}, f^{⊥} s_{μ} ⟩ = ⟨ f s_{ν}, s_{μ} ⟩$ , so $f^{⊥}$ is unique. Taking $f = h_{r}$ : the Pieri rule 40.03.02 gives $h_{r} s_{ν} = \sum_{κ} s_{κ}$ over $κ$ with $κ / ν$ a horizontal $r$ -strip, so $⟨ h_{r} s_{ν}, s_{μ} ⟩ = 1$ iff $μ / ν$ is a horizontal $r$ -strip, else $0$ . Hence $⟨ s_{ν}, h_{r}^{⊥} s_{μ} ⟩$ equals that same indicator, i.e. $h_{r}^{⊥} s_{μ} = \sum_{ν} s_{ν}$ over $ν$ with $μ / ν$ a horizontal $r$ -strip, which is precisely $s_{μ / (r)}$ . Reading the adjoint relation in reverse recovers the Pieri rule $s_{ν} h_{r} = \sum_{μ} s_{μ}$ over horizontal strips $μ / ν$ , since the coefficient of $s_{μ}$ is $⟨ h_{r} s_{ν}, s_{μ} ⟩$ . Rubric: full credit for the orthonormality-based uniqueness, the strip computation of $h_{r}^{⊥}$ , and the Pieri deduction.

Advanced results Master

The Hall inner product turns every multiplicative identity of $Λ$ into an adjoint statement, and the Cauchy kernel is the reproducing element threading all of them.

Theorem 1 (adjointness of multiplication and the Littlewood-Richardson pairing). For every $f \in Λ$ the skewing operator $f^{⊥}$ is the adjoint of multiplication by $f$ , $⟨ f g, h ⟩ = ⟨ g, f^{⊥} h ⟩$ , and it is a coderivation-compatible map with $s_{λ}^{⊥} s_{μ} = s_{μ / λ}$ . The structure constants of the Schur basis are read off as inner products: with $s_{λ} s_{μ} = \sum_{ν} c_{λ μ}^{ν} s_{ν}$ , $$ c^\nu_{\lambda\mu} = \langle s_\lambda s_\mu,, s_\nu\rangle = \langle s_\mu,, s_\lambda^\perp s_\nu\rangle = \langle s_\mu,, s_{\nu/\lambda}\rangle, $$ so the Littlewood-Richardson coefficients of the co-produced 40.03.05 are the Schur-expansion coefficients of skew Schur functions, $s_{ν / λ} = \sum_{μ} c_{λ μ}^{ν} s_{μ}$ . The Pieri rules are the cases $μ = (r)$ and $μ = (1^{r})$ ^{[Macdonald §5]}.

Theorem 2 (the Cauchy kernel as reproducing element; coproduct duality). Equip $Λ$ with the coproduct $Δ f (x) = f (x^{'}, x^{''})$ (separating a single alphabet into two), making it the self-dual Hopf algebra of 40.03.01. The Cauchy kernel $Ω (x, y) = \sum_{λ} s_{λ} (x) s_{λ} (y)$ is the canonical element of $Λ (x) \otimes Λ (y)$ dual to the inner product: for any $f$ , $$ \big\langle f(x),, \Omega(x, y)\big\rangle_x = f(y), $$ the pairing taken in the $x$ -alphabet. Thus $Ω$ reproduces every symmetric function, and the adjoint of multiplication is dual to the coproduct: $⟨ f g, h ⟩ = ⟨ f \otimes g, Δ h ⟩$ . The skewing operator $f^{⊥}$ is the image of $f$ under the convolution action, and $ω$ is the Hopf isometry exchanging $Ω \leftrightarrow Ω^{'}$ .

Theorem 3 (two proofs of Cauchy and their meeting point). The Cauchy identity $\sum_{λ} s_{λ} (x) s_{λ} (y) = \prod_{i, j} (1 - x_{i} y_{j})^{- 1}$ admits a bijective proof and a determinantal proof. The RSK proof of the co-produced 40.03.04 is a content-preserving bijection between $N$ -matrices $(a_{ij})$ with $\sum a_{ij} < \infty$ — whose generating function is $\prod_{i, j} (1 - x_{i} y_{j})^{- 1}$ — and pairs $(P, Q)$ of semistandard tableaux of the same shape, whose generating function is $\sum_{λ} s_{λ} (x) s_{λ} (y)$ . The Jacobi-Trudi / Lindström-Gessel-Viennot proof expands $s_{λ} (x) = det (h_{λ_{i} - i + j} (x))$ and resums the determinantal series against the geometric kernel, the antisymmetrisation collapsing to the product. The two proofs meet at the same cancellation seen in 40.03.02: the sign-reversing involution of the LGV lemma is the determinantal shadow of the RSK insertion, so the bijective and the determinantal routes compute one identity.

Theorem 4 (the inner product is the character inner product). Under the Frobenius characteristic $ch : ⨁_{n} R (S_{n}) \to Λ$ of the co-produced 40.03.06, carrying the class function $f$ on $S_{n}$ to $ch (f) = \sum_{λ ⊢ n} z_{λ}^{- 1} f (λ) p_{λ}$ , the Hall inner product pulls back to the symmetric-group character inner product: $$ \langle \mathrm{ch}(f), \mathrm{ch}(g)\rangle = \frac{1}{n!}\sum_{\sigma \in S_n} f(\sigma), g(\sigma^{-1}) = \frac{1}{|S_n|}\sum_\sigma f(\sigma)\overline{g(\sigma)}. $$ Schur orthonormality $⟨ s_{λ}, s_{μ} ⟩ = δ_{λ μ}$ becomes the first orthogonality relation $⟨ χ^{λ}, χ^{μ} ⟩ = δ_{λ μ}$ for the irreducible characters $χ^{λ} = ch^{- 1} (s_{λ})$ , and $⟨ p_{λ}, p_{μ} ⟩ = z_{λ} δ_{λ μ}$ is the column orthogonality weighted by centraliser orders. The Cauchy kernel becomes the regular-representation character, and the dual Cauchy identity the sign-twisted version.

Synthesis. Putting these together, the Hall inner product is the structure that makes $Λ$ a self-dual object, and the Cauchy kernel is the element that witnesses the self-duality at every level. The dual-bases theorem is exactly the statement that $Ω (x, y)$ is the reproducing kernel of the pairing, so the foundational reason the Schur functions are orthonormal is that they are the basis in which the kernel is diagonal — $Ω = \sum_{λ} s_{λ} (x) s_{λ} (y)$ with a single shared index. The central insight is that three identifications coincide: the duality of ${h_{λ}}$ and ${m_{μ}}$ that defines the form, the self-duality of ${s_{λ}}$ that diagonalises it, and the orthogonality of the irreducible $S_{n}$ -characters under Frobenius — and the involution $ω$ , which is conjugation of diagrams in 40.03.01, is the isometry threading all three, sending $Ω$ to the dual kernel $Ω^{'}$ and $s_{λ}$ to $s_{λ^{'}}$ . This is exactly why the Littlewood-Richardson coefficients of 40.03.05 are simultaneously multiplication constants $⟨ s_{λ} s_{μ}, s_{ν} ⟩$ and comultiplication constants $⟨ s_{μ}, s_{ν / λ} ⟩$ : the adjointness of multiplication and skewing is dual to the Hopf coproduct of 40.03.01. The bridge is the RSK correspondence of the co-produced 40.03.04, whose bijective proof of Cauchy is the combinatorial mechanism beneath the orthonormality, and which appears again as the Robinson-Schensted shadow of the character theory; the same kernel governs Pieri, the skewing operator, and the Frobenius characteristic of 40.03.06, so the entire metric geometry of $Λ$ generalises the orthogonality of characters from a single finite group to all symmetric groups at once.

Full proof set Master

Proposition 1 (the two normalisations of the Hall inner product agree). The bilinear form fixed by $⟨ h_{λ}, m_{μ} ⟩ = δ_{λ μ}$ satisfies $⟨ p_{λ}, p_{μ} ⟩ = z_{λ} δ_{λ μ}$ , and is symmetric.

Proof. From $h_{n} = \sum_{λ ⊢ n} z_{λ}^{- 1} p_{λ}$ of 40.03.01, the generating identity $\sum_{n} h_{n} (x) t^{n} = \prod_{i} (1 - x_{i} t)^{- 1}$ gives, with $t$ replaced by the alphabet $y$ , $$ \Omega(x,y) = \prod_{i,j}(1 - x_i y_j)^{-1} = \exp\Big(\sum_i \sum_{k\ge 1} \tfrac1k (x_i y_j)^k \text{ summed over } j\Big) = \exp\Big(\sum_{k\ge 1} \tfrac1k p_k(x) p_k(y)\Big). $$ Expanding the exponential and collecting by cycle type as in the proof of $h_{n} = \sum z_{λ}^{- 1} p_{λ}$ yields $Ω = \sum_{λ} z_{λ}^{- 1} p_{λ} (x) p_{λ} (y)$ . The kernel also equals $\sum_{λ} h_{λ} (x) m_{λ} (y)$ (base expansion, Key theorem). Both are the canonical element of the pairing, so for the form with $⟨ h_{λ}, m_{μ} ⟩ = δ_{λ μ}$ the dual-bases lemma applied to $u_{λ} = z_{λ}^{- 1} p_{λ}$ , $v_{μ} = p_{μ}$ gives $⟨ z_{λ}^{- 1} p_{λ}, p_{μ} ⟩ = δ_{λ μ}$ , i.e. $⟨ p_{λ}, p_{μ} ⟩ = z_{λ} δ_{λ μ}$ . Symmetry: $Ω (x, y) = Ω (y, x)$ shows $\sum_{λ} h_{λ} (x) m_{λ} (y) = \sum_{λ} m_{λ} (x) h_{λ} (y)$ , so $⟨ m_{λ}, h_{μ} ⟩ = δ_{λ μ} = ⟨ h_{μ}, m_{λ} ⟩$ , and bilinearity propagates symmetry to all of $Λ$ . The $p$ -form, being diagonal with real entries $z_{λ}$ , is manifestly symmetric, confirming consistency. $□$

Proposition 2 (dual-bases lemma). Homogeneous partition-indexed bases ${u_{λ}}, {v_{λ}}$ of $Λ_{Q}$ satisfy $⟨ u_{λ}, v_{μ} ⟩ = δ_{λ μ}$ iff $\sum_{λ} u_{λ} (x) v_{λ} (y) = Ω (x, y)$ .

Proof. Restrict to degree $n$ . Write $u_{λ} = \sum_{ρ} a_{λ ρ} h_{ρ}$ , $v_{μ} = \sum_{σ} b_{μ σ} m_{σ}$ with invertible matrices $a, b$ over $Q$ (both are bases). Then $⟨ u_{λ}, v_{μ} ⟩ = \sum_{ρ} a_{λ ρ} b_{μ ρ} = (a b^{T})_{λ μ}$ , so $⟨ u_{λ}, v_{μ} ⟩ = δ_{λ μ}$ iff $a b^{T} = I$ . Separately, $$ \sum_\lambda u_\lambda(x) v_\lambda(y) = \sum_{\rho,\sigma}\Big(\sum_\lambda a_{\lambda\rho} b_{\lambda\sigma}\Big) h_\rho(x) m_\sigma(y) = \sum_{\rho,\sigma} (a^{\mathsf T} b){\rho\sigma} h\rho(x) m_\sigma(y), $$ which equals $Ω = \sum_{ρ} h_{ρ} (x) m_{ρ} (y)$ iff $a^{T} b = I$ . The conditions $a b^{T} = I$ and $a^{T} b = I$ are equivalent for square matrices (each says $b^{T} = a^{- 1}$ ), so the two characterisations coincide. $□$

Proposition 3 (Cauchy identity by RSK and by Jacobi-Trudi). $λ \sum s_{λ} (x) s_{λ} (y) = i, j \prod (1 - x_{i} y_{j})^{- 1}$ .

Proof. (RSK route, cross-referenced to 40.03.04.) The product $\prod_{i, j} (1 - x_{i} y_{j})^{- 1} = \prod_{i, j} \sum_{a_{ij} \geq 0} (x_{i} y_{j})^{a_{ij}}$ is the generating function $\sum_{A} x^{row (A)} y^{col (A)}$ over finitely-supported matrices $A = (a_{ij}) \in N^{\infty \times \infty}$ , where $row (A)_{i} = \sum_{j} a_{ij}$ and $col (A)_{j} = \sum_{i} a_{ij}$ . The RSK correspondence is a bijection $A \leftrightarrow (P, Q)$ with $P, Q \in SSYT$ of a common shape $λ$ , content of $P$ equal to $col (A)$ and content of $Q$ equal to $row (A)$ . Hence $x^{row (A)} = x^{wt (Q)}$ and $y^{col (A)} = y^{wt (P)}$ , so summing over $A$ equals $\sum_{λ} (\sum_{Q} x^{wt (Q)}) (\sum_{P} y^{wt (P)}) = \sum_{λ} s_{λ} (x) s_{λ} (y)$ .

(Jacobi-Trudi route.) Work in $n$ $x$ -variables and $n$ $y$ -variables; the stable limit follows by letting $n \to \infty$ . Using the bialternant $s_{λ} (x) = a_{λ + δ} (x) / a_{δ} (x)$ of 40.03.02, the sum $\sum_{λ} s_{λ} (x) s_{λ} (y)$ multiplied by $a_{δ} (x) a_{δ} (y)$ becomes $\sum_{λ} a_{λ + δ} (x) a_{λ + δ} (y)$ , a sum over strictly decreasing exponent vectors $α = λ + δ$ of $det (x_{i}^{α_{j}}) det (y_{i}^{α_{j}})$ . By the Cauchy-Binet expansion this equals $det (\sum_{α} x_{i}^{α} y_{k}^{α})$ restricted appropriately, which resums to $det ((1 - x_{i} y_{k})^{- 1})_{i, k}$ . The latter determinant is evaluated by the Cauchy determinant formula $det ((1 - x_{i} y_{k})^{- 1}) = \frac{\prod _{i < i^{'}} ( x _{i} - x _{i^{'}} ) \prod _{k < k^{'}} ( y _{k} - y _{k^{'}} )}{\prod _{i, k} ( 1 - x _{i} y _{k} )} = \frac{a _{δ} ( x ) a _{δ} ( y )}{\prod _{i, k} ( 1 - x _{i} y _{k} )}$ . Dividing by $a_{δ} (x) a_{δ} (y)$ gives $\sum_{λ} s_{λ} (x) s_{λ} (y) = \prod_{i, k} (1 - x_{i} y_{k})^{- 1}$ . $□$

Proposition 4 (Schur orthonormality and the isometry of $ω$ ). $⟨ s_{λ}, s_{μ} ⟩ = δ_{λ μ}$ , and $⟨ ω f, ω g ⟩ = ⟨ f, g ⟩$ with $ω (s_{λ}) = s_{λ^{'}}$ .

Proof. By Proposition 3, $\sum_{λ} s_{λ} (x) s_{λ} (y) = Ω (x, y)$ ; by Proposition 2 with $u_{λ} = v_{λ} = s_{λ}$ this is equivalent to $⟨ s_{λ}, s_{μ} ⟩ = δ_{λ μ}$ . For $ω$ : apply $ω_{x}$ to $Ω (x, y) = \sum_{λ} h_{λ} (x) m_{λ} (y)$ , giving $\sum_{λ} e_{λ} (x) m_{λ} (y) = \prod_{i, j} (1 + x_{i} y_{j}) = Ω^{'} (x, y)$ . The dual Cauchy identity $Ω^{'} (x, y) = \sum_{λ} s_{λ} (x) s_{λ^{'}} (y)$ (Proposition 5) and $Ω = \sum_{λ} s_{λ} (x) s_{λ} (y)$ then give $ω_{x} (\sum_{λ} s_{λ} (x) s_{λ} (y)) = \sum_{λ} s_{λ} (x) s_{λ^{'}} (y)$ ; comparing coefficients in the orthonormal $y$ -Schur basis yields $ω_{x} s_{λ} (x) = s_{λ^{'}} (x)$ . As conjugation $λ \mapsto λ^{'}$ is an involution on partitions, $ω$ permutes the orthonormal basis ${s_{λ}}$ , so $⟨ ω s_{λ}, ω s_{μ} ⟩ = ⟨ s_{λ^{'}}, s_{μ^{'}} ⟩ = δ_{λ^{'} μ^{'}} = δ_{λ μ}$ ; bilinear extension gives the isometry. $□$

Proposition 5 (dual Cauchy identity). $i, j \prod (1 + x_{i} y_{j}) = λ \sum s_{λ} (x) s_{λ^{'}} (y) = λ \sum e_{λ} (x) m_{λ} (y)$ .

Proof. The $e$ - $m$ form is the base expansion read for the elementary functions: $\prod_{i} (1 + x_{i} t) = \sum_{n} e_{n} (x) t^{n}$ defines the $e_{n}$ , and expanding over the alphabet $y$ and collecting orbits assembles $\sum_{λ} e_{λ} (x) m_{λ} (y)$ . For the Schur form, apply $ω_{x}$ to the Cauchy identity $\sum_{λ} s_{λ} (x) s_{λ} (y) = \prod_{i, j} (1 - x_{i} y_{j})^{- 1}$ . On the right, $ω_{x}$ in the $h$ - $m$ expansion sends $h_{λ} (x) \mapsto e_{λ} (x)$ , turning $\prod (1 - x_{i} y_{j})^{- 1} = \sum_{λ} h_{λ} (x) m_{λ} (y)$ into $\sum_{λ} e_{λ} (x) m_{λ} (y) = \prod_{i, j} (1 + x_{i} y_{j})$ . On the left, $ω_{x} s_{λ} (x) = s_{λ^{'}} (x)$ (Proposition 4), so $ω_{x} (\sum_{λ} s_{λ} (x) s_{λ} (y)) = \sum_{λ} s_{λ^{'}} (x) s_{λ} (y) = \sum_{μ} s_{μ} (x) s_{μ^{'}} (y)$ after reindexing $μ = λ^{'}$ . Equating the two images gives the dual Cauchy identity. $□$

Proposition 6 (adjointness of $f^{⊥}$ and the LR pairing). For all $f$ there is a unique $f^{⊥}$ with $⟨ f g, h ⟩ = ⟨ g, f^{⊥} h ⟩$ ; it satisfies $s_{λ}^{⊥} s_{μ} = s_{μ / λ}$ and $c_{λ μ}^{ν} = ⟨ s_{μ}, s_{ν / λ} ⟩$ .

Proof. Orthonormality of ${s_{λ}}$ makes $Λ$ a (graded) inner-product space, so multiplication-by- $f$ , a linear map, has a unique adjoint $f^{⊥}$ defined by $⟨ s_{ν}, f^{⊥} s_{μ} ⟩ := ⟨ f s_{ν}, s_{μ} ⟩$ ; bilinear extension gives $⟨ f g, h ⟩ = ⟨ g, f^{⊥} h ⟩$ for all $g, h$ . For $f = s_{λ}$ , the structure constants $s_{λ} s_{ν} = \sum_{κ} c_{λ ν}^{κ} s_{κ}$ give $⟨ s_{λ} s_{ν}, s_{μ} ⟩ = c_{λ ν}^{μ}$ , so $⟨ s_{ν}, s_{λ}^{⊥} s_{μ} ⟩ = c_{λ ν}^{μ}$ . The skew Schur function expands as $s_{μ / λ} = \sum_{ν} c_{λ ν}^{μ} s_{ν}$ (the defining property of the LR coefficients in 40.03.05, equivalently the dual-basis expansion of the coproduct $Δ s_{μ} = \sum_{λ, ν} c_{λ ν}^{μ} s_{λ} \otimes s_{ν}$ ), so $⟨ s_{ν}, s_{μ / λ} ⟩ = c_{λ ν}^{μ} = ⟨ s_{ν}, s_{λ}^{⊥} s_{μ} ⟩$ for every $ν$ . By orthonormality $s_{λ}^{⊥} s_{μ} = s_{μ / λ}$ , and reading $c_{λ μ}^{ν} = ⟨ s_{λ} s_{μ}, s_{ν} ⟩ = ⟨ s_{μ}, s_{λ}^{⊥} s_{ν} ⟩ = ⟨ s_{μ}, s_{ν / λ} ⟩$ gives the stated pairing. $□$

Connections Master

This unit is the metric layer on the Schur functions of 40.03.02: the orthonormality $⟨ s_{λ}, s_{μ} ⟩ = δ_{λ μ}$ promotes the unitriangular $Z$ -basis built there into an orthonormal basis, the Jacobi-Trudi determinants and the bialternant of that unit supply the determinantal proof of the Cauchy identity (via the Cauchy determinant $det ((1 - x_{i} y_{k})^{- 1})$ ), and the Pieri rules proved there are recovered here as the action of the skewing operator $h_{r}^{⊥}$ . The involution $ω$ and the bases $h_{λ}, m_{λ}, p_{λ}, e_{λ}, z_{λ}$ are all imported from 40.03.01, where $ω$ is conjugation of diagrams and the $z_{λ}$ are the cycle-centralizer orders; here $ω$ becomes an isometry and the $z_{λ}$ become the square-norms $⟨ p_{λ}, p_{λ} ⟩$ .
The Cauchy identity $\prod (1 - x_{i} y_{j})^{- 1} = \sum_{λ} s_{λ} (x) s_{λ} (y)$ is proved bijectively by the RSK correspondence of the co-produced 40.03.04, which is the combinatorial heart of the orthonormality; the Littlewood-Richardson coefficients $c_{λ μ}^{ν} = ⟨ s_{λ} s_{μ}, s_{ν} ⟩ = ⟨ s_{μ}, s_{ν / λ} ⟩$ of the co-produced 40.03.05 are exactly the multiplication-versus-skewing constants made visible by the adjointness developed here. The Frobenius characteristic of the co-produced 40.03.06 carries the Hall inner product to the symmetric-group character inner product $\frac{1}{n !} \sum_{σ} χ (σ) ψ (σ^{- 1})$ , so Schur orthonormality becomes the orthogonality of irreducible characters and the $z_{λ}$ -weighting becomes column orthogonality.
The same inner product is the bilinear form underlying the representation theory of 07.05.02 (Young diagrams and the irreducible $S_{n}$ -modules) and 07.05.03 (Specht modules), where $⟨ χ^{λ}, χ^{μ} ⟩ = δ_{λ μ}$ is the first orthogonality relation; the power-sum expansion of $s_{λ}$ governed here by $⟨ s_{λ}, p_{μ} ⟩ = χ^{λ} (μ)$ is the Murnaghan-Nakayama rule of 07.05.10. The Cauchy kernel $\prod (1 - x_{i} y_{j})^{- 1}$ is the symmetric-function avatar of the reproducing kernel in the theory of the resultant and of the Cauchy determinant, linking to the alternant computations of 40.03.02 and to the generating-function methods of 40.08.01; the dual Cauchy identity $\prod (1 + x_{i} y_{j}) = \sum_{λ} s_{λ} (x) s_{λ^{'}} (y)$ is the sign-twist that recurs in the Schur-Weyl duality of 07.05.04.

Historical & philosophical context Master

The product $\prod_{i, j} (1 - x_{i} y_{j})^{- 1}$ and the antisymmetric quotients that become Schur functions both appear in Cauchy's 1815 memoir on alternating functions ^{[Cauchy 1815]}; the Cauchy determinant $det ((x_{i} + y_{j})^{- 1})$ and its relatives, from which the Schur-form Cauchy identity descends, are named for that work. The bilinear pairing that organises the modern theory — the inner product making the complete and monomial functions dual and the Schur functions orthonormal — is due to Philip Hall, who introduced it in the 1950s in connection with the Hall algebra of finite abelian $p$ -groups, where the structure constants are the Hall polynomials; the name Hall inner product and the systematic development are codified in Macdonald's Symmetric Functions and Hall Polynomials (1979; 2nd ed. 1995) ^{[Macdonald §4]}.

The identification of the symmetric-function inner product with the orthogonality of symmetric-group characters runs through Frobenius's 1900 determination of the characters of $S_{n}$ and was given its symmetric-function form by Littlewood in The Theory of Group Characters ^{[Littlewood 1940]}, where the Cauchy identity is read as the decomposition of the tensor algebra and the orthonormality of the $s_{λ}$ as the orthonormality of the irreducible characters. The bijective proof of the Cauchy identity by the Robinson-Schensted-Knuth correspondence — Robinson (1938), Schensted (1961), and Knuth (1970) — supplied the combinatorial mechanism beneath the orthogonality, and the adjointness of multiplication and skewing was placed in its Hopf-algebraic setting by Geissinger and Zelevinsky (1981), who identified $Λ$ as the universal positive self-dual Hopf algebra in which the Cauchy kernel is the canonical pairing element.

Bibliography Master

@book{macdonald1995symmetric,
  author    = {Macdonald, Ian G.},
  title     = {Symmetric Functions and Hall Polynomials},
  series    = {Oxford Mathematical Monographs},
  edition   = {2},
  publisher = {Oxford University Press},
  year      = {1995}
}

@book{stanley1999ec2,
  author    = {Stanley, Richard P.},
  title     = {Enumerative Combinatorics, Volume 2},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {62},
  publisher = {Cambridge University Press},
  year      = {1999}
}

@book{sagan2001symmetric,
  author    = {Sagan, Bruce E.},
  title     = {The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions},
  series    = {Graduate Texts in Mathematics},
  volume    = {203},
  edition   = {2},
  publisher = {Springer},
  year      = {2001}
}

@book{littlewood1940group,
  author    = {Littlewood, Dudley E.},
  title     = {The Theory of Group Characters and Matrix Representations of Groups},
  publisher = {Oxford University Press},
  year      = {1940}
}

@article{cauchy1815memoire,
  author  = {Cauchy, Augustin-Louis},
  title   = {M\'{e}moire sur les fonctions qui ne peuvent obtenir que deux valeurs \'{e}gales et de signes contraires par suite des transpositions op\'{e}r\'{e}es entre les variables qu'elles renferment},
  journal = {Journal de l'\'{E}cole Polytechnique},
  volume  = {10},
  pages   = {29--112},
  year    = {1815}
}

@article{frobenius1900charaktere,
  author  = {Frobenius, Ferdinand Georg},
  title   = {\"{U}ber die Charaktere der symmetrischen Gruppe},
  journal = {Sitzungsberichte der K\"{o}niglich Preussischen Akademie der Wissenschaften zu Berlin},
  pages   = {516--534},
  year    = {1900}
}

@article{knuth1970permutations,
  author  = {Knuth, Donald E.},
  title   = {Permutations, matrices, and generalized Young tableaux},
  journal = {Pacific Journal of Mathematics},
  volume  = {34},
  pages   = {709--727},
  year    = {1970}
}

@book{zelevinsky1981representations,
  author    = {Zelevinsky, Andrey V.},
  title     = {Representations of Finite Classical Groups: A Hopf Algebra Approach},
  series    = {Lecture Notes in Mathematics},
  volume    = {869},
  publisher = {Springer},
  year      = {1981}
}

Prerequisites

40.03.02

Tier anchors

beginner: Stanley 1999 *Enumerative Combinatorics, Volume 2* (Cambridge) §7.9, §7.12 opening (the inner product on symmetric functions presented through dual families of building blocks, and the Cauchy product $\prod 1/(1-x_iy_j)$ expanded by example in a few variables); Sagan 2001 *The Symmetric Group* 2e (Springer) §4.5-4.6 (the bilinear form making the complete and monomial functions dual, and the Schur functions an orthonormal family)
intermediate: Stanley 1999 *Enumerative Combinatorics, Volume 2* 2e (Cambridge) §7.9-7.12 (the Hall inner product $\langle h_\lambda, m_\mu\rangle = \delta_{\lambda\mu}$ and $\langle p_\lambda, p_\mu\rangle = z_\lambda\delta_{\lambda\mu}$, the dual-bases characterisation via the Cauchy kernel, the Cauchy and dual Cauchy identities, Schur orthonormality, and the self-adjointness of $\omega$ and the skewing operator $f^\perp$); Macdonald 1995 *Symmetric Functions and Hall Polynomials* 2e (Oxford) Ch. I §4 (the scalar product, dual bases, Cauchy's formulae, and the adjoint of multiplication)
master: Macdonald 1995 *Symmetric Functions and Hall Polynomials* 2e (Oxford) Ch. I §4-5 (the scalar product on $\Lambda$ defined by $\langle h_\lambda, m_\mu\rangle = \delta_{\lambda\mu}$, the dual-bases lemma $\sum u_\lambda(x)v_\lambda(y) = \prod(1-x_iy_j)^{-1}$, the Cauchy and dual Cauchy identities, Schur orthonormality $\langle s_\lambda, s_\mu\rangle = \delta_{\lambda\mu}$, the isometry of $\omega$, the adjoint $f^\perp$ of multiplication by $f$, and skew Schur functions); Stanley 1999 *Enumerative Combinatorics, Volume 2* Ch. 7 and Appendix 2; Cauchy 1815 (the antisymmetric-quotient identity); Littlewood 1940 *The Theory of Group Characters* (the symmetric-function form of the orthogonality of characters)

References

Macdonald, I. G. — Symmetric Functions and Hall Polynomials · Oxford Mathematical Monographs, 2nd edition (1995). Chapter I §4 defines the Hall (scalar) inner product on $\Lambda$ as the $\mathbb{Z}$-bilinear, symmetric form for which the complete and monomial bases are dual, $\langle h_\lambda, m_\mu\rangle = \delta_{\lambda\mu}$, and proves the equivalent normalisation on the power sums $\langle p_\lambda, p_\mu\rangle = z_\lambda \delta_{\lambda\mu}$ with $z_\lambda = \prod_i i^{m_i} m_i!$. The dual-bases lemma (4.6) states that two graded bases $\{u_\lambda\}, \{v_\lambda\}$ are dual under this form iff $\sum_\lambda u_\lambda(x) v_\lambda(y) = \prod_{i,j}(1 - x_i y_j)^{-1}$, the Cauchy kernel. The Cauchy identity $\prod_{i,j}(1-x_iy_j)^{-1} = \sum_\lambda s_\lambda(x) s_\lambda(y) = \sum_\lambda h_\lambda(x) m_\lambda(y) = \sum_\lambda z_\lambda^{-1} p_\lambda(x) p_\lambda(y)$ and its dual $\prod_{i,j}(1 + x_i y_j) = \sum_\lambda s_\lambda(x) s_{\lambda'}(y) = \sum_\lambda e_\lambda(x) m_\lambda(y)$ then give the orthonormality $\langle s_\lambda, s_\mu\rangle = \delta_{\lambda\mu}$, the isometry $\langle \omega f, \omega g\rangle = \langle f, g\rangle$ with $\omega(s_\lambda) = s_{\lambda'}$, and (4.8, ex. 5.3) the adjoint $f^\perp$ of multiplication by $f$, with $\langle f g, h\rangle = \langle g, f^\perp h\rangle$ and $s_\lambda^\perp s_\mu = s_{\mu/\lambda}$.
Stanley, R. P. — Enumerative Combinatorics, Volume 2 · Cambridge Studies in Advanced Mathematics 62 (1999). Chapter 7 §7.9 introduces the standard scalar product on $\Lambda$ by declaring $\langle h_\lambda, m_\mu\rangle = \delta_{\lambda\mu}$ and proves $\langle p_\lambda, p_\mu\rangle = z_\lambda\delta_{\lambda\mu}$, so the $p_\lambda$ are orthogonal with square-norm $z_\lambda$; §7.12 proves the Cauchy identity $\prod_{i,j}(1-x_iy_j)^{-1} = \sum_\lambda s_\lambda(x)s_\lambda(y)$ and Theorem 7.12.1 the dual-basis characterisation that $\sum_\lambda u_\lambda(x)v_\lambda(y) = \prod(1-x_iy_j)^{-1}$ iff $\{u_\lambda\},\{v_\lambda\}$ are dual bases, deducing Corollary 7.12.2 the Schur orthonormality $\langle s_\lambda, s_\mu\rangle = \delta_{\lambda\mu}$. The dual Cauchy identity $\prod(1+x_iy_j) = \sum_\lambda s_\lambda(x)s_{\lambda'}(y)$ is §7.14. The two proofs of Cauchy — the RSK bijection (Ch. 7.11, the co-produced 40.03.04) and the Jacobi-Trudi / Lindström-Gessel-Viennot determinantal route — are given, together with the adjointness of the skewing operator $f^\perp$ and $\langle s_\lambda s_\mu, s_\nu\rangle = c^\nu_{\lambda\mu}$.
Sagan, B. E. — The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions · Springer Graduate Texts in Mathematics 203, 2nd edition (2001). Chapter 4 §4.5-4.7 builds the Hall inner product on the symmetric functions, proving $\langle h_\lambda, m_\mu\rangle = \delta_{\lambda\mu}$ and $\langle p_\lambda, p_\mu\rangle = z_\lambda\delta_{\lambda\mu}$ equivalent, deriving the Cauchy identity and Schur orthonormality, and connecting the inner product to the orthogonality of the irreducible characters of the symmetric group via the Frobenius characteristic map (developed further in the co-produced 40.03.06), under which the symmetric-function inner product is exactly the character inner product $\frac{1}{n!}\sum_{\sigma} \chi(\sigma)\overline{\psi(\sigma)}$.

Estimated time

beginner: 20m
intermediate: 50m
master: 90m