Schur Functions: the Combinatorial Definition and the Jacobi-Trudi Determinant

Intuition Beginner

Picture a staircase of empty boxes — a few boxes in the top row, no more in the second row, fewer still further down. This staircase shape is called a Young diagram, and its row lengths spell out a partition of a number, met in the previous unit. Now fill each box with a positive whole number, following two house rules: numbers never decrease as you read a row left to right, and they strictly increase as you read a column top to bottom. A filling obeying both rules is a semistandard tableau. The Schur function is the bookkeeping device that records every legal filling at once.

Why build such a thing? Each filling is a recipe for a single product of variables: a box holding the number $3$ contributes one factor of the third variable $x_3$. Multiply across all boxes and you get one term; add up the terms over every legal filling and you get the Schur function of that shape. It is a sum that remembers, for each pattern of variable-counts, how many fillings produce it. The shape is the input; the symmetric expression is the output.

The pay-off is that these expressions are the single most important family in the whole subject. They are symmetric — relabelling the variables leaves them unchanged — even though the filling rules look lopsided, treating rows and columns differently. That hidden symmetry is a small miracle, and proving it is the first real theorem here. Once you have it, the Schur functions become the natural alphabet: every symmetric expression spells out uniquely in them, and they are the ones that count fillings, measure dimensions, and translate into the language of group symmetry later in the curriculum.

Visual Beginner

A semistandard filling of the staircase shape with rows of length $2$ and $1$. Reading rule one (rows): left-to-right values do not decrease. Reading rule two (columns): top-to-bottom values strictly increase. The variable-record of a filling multiplies one $x$ per box, indexed by the box's number.

shape (boxes)	a legal filling	the record it contributes
row of 2, row of 1	top row $11$, bottom row $2$	$x_1\cdot x_1\cdot x_2=x_1^2{x}_2$
row of 2, row of 1	top row $12$, bottom row $2$	$x_1\cdot x_2\cdot x_2=x_1{x}_2^2$
row of 2, row of 1	top row $12$, bottom row $3$	$x_1{x}_2{x}_3$

Notice the two rules pull in opposite directions: rows allow repeats, columns forbid them. The forbidding is what keeps the count finite in each shape and degree. If you collect every legal filling using only the variables $x_1,x_2,x_3$ and add their records, the messy-looking sum turns out to be symmetric: swapping the roles of $x_1$ and $x_2$ throughout permutes the fillings among themselves and leaves the total alone.

Worked example Beginner

Let us compute the Schur function of the shape with a single row of two boxes, using only two variables $x_1$ and $x_2$, and watch the symmetry appear.

Step 1. List the legal fillings. One row of two boxes needs values that do not decrease left to right, with no column rule to worry about (each column has one box). With entries from $\{1,2\}$ the choices are: $11$, then $12$, then $22$. That is three fillings.

Step 2. Record each filling. The filling $11$ gives $x_1\cdot x_1=x_1^2$. The filling $12$ gives $x_1\cdot x_2$. The filling $22$ gives $x_2^2$.

Step 3. Add. The Schur function of this shape is $x_1^2+x_1{x}_2+x_2^2$.

Step 4. Test the symmetry. Swap the names $x_1$ and $x_2$ everywhere. The sum becomes $x_2^2+x_2{x}_1+x_1^2$, which is the same three terms in a different order. The total did not change, so this expression is symmetric.

Step 5. Recognise it. The expression $x_1^2+x_1{x}_2+x_2^2$ is exactly the complete function $h_2$ in two variables from the previous unit. A single row of $r$ boxes always reproduces the complete function $h_r$, and a single column of $r$ boxes reproduces the elementary function $e_r$.

What this tells us: the Schur functions are not new strangers — the simplest shapes give back the building blocks already met. The single-row and single-column shapes are the complete and elementary functions; the interesting shapes are the staircases in between, and the rules that knit rows and columns together are what make the whole family symmetric.

Check your understanding Beginner

Formal definition Intermediate+

Work in the ring $\Lambda$ of symmetric functions over $\mathbb{Z}$ of 40.03.01; all notation ($m_{\lambda },e_{\lambda },h_{\lambda },p_{\lambda }$, the dominance order $⊵$, the conjugate ${\lambda }^{\prime }$, $z_{\lambda }$, the involution $\omega$) is inherited from that unit and recorded in _meta/NOTATION.md. Fix a partition $\lambda =({\lambda }_1\ge \cdots \ge {\lambda }_{\ell }>0)$ with Young diagram the array of cells $(i,j)$ with $1\le j\le {\lambda }_i$.

Definition (semistandard Young tableau). A semistandard Young tableau (SSYT) of shape $\lambda$ is a filling $T$ of the cells of $\lambda$ with entries from $\{1,2,3,\dots \}$ that is weakly increasing along each row ($T(i,j)\le T(i,j+1)$) and strictly increasing down each column ($T(i,j)<T(i+1,j)$). Its content is the vector $({\mu }_1,{\mu }_2,\dots )$ with ${\mu }_k=\#\{\text{cells holding }k\}$, and its weight monomial is $x^T={\prod }_{(i,j)\in \lambda }{x}_{T(i,j)}=x_1^{{\mu }_1}{x}_2^{{\mu }_2}\cdots$. Write $\mathrm{SSYT}(\lambda )$ for the set of all such fillings (entries unbounded) and $\mathrm{SSYT}(\lambda ,n)$ for those with entries in $\{1,\dots ,n\}$.

Definition (Schur function, combinatorial). The Schur function of shape $\lambda$ is the formal sum $$ s_\lambda ;=; \sum_{T \in \mathrm{SSYT}(\lambda)} x^T ;\in; \mathbb{Z}[[x_1, x_2, \dots]], $$ homogeneous of degree $|\lambda |$. The single-row shape $(r)$ gives $s_{(r)}=h_r$; the single-column shape $(1^r)$ gives $s_{(1^r)}=e_r$. The skew Schur function $s_{\lambda /\mu }$ for $\mu \subseteq \lambda$ is the same sum over SSYT of the skew shape $\lambda /\mu$ (the cells of $\lambda$ not in $\mu$), with the identical row/column conditions.

Definition (the bialternant). In $n$ variables, set $\delta =(n-1,n-2,\dots ,1,0)$ and for a sequence $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ define the alternant $a_{\alpha }=\det (x_i^{{\alpha }_j}{)}_{1\le i,j\le n}$. The alternant $a_{\delta }=\det (x_i^{n-j})={\prod }_{i<j}(x_i-x_j)$ is the Vandermonde determinant. The bialternant definition of the Schur polynomial is the ratio $$ s_\lambda(x_1, \dots, x_n) ;=; \frac{a_{\lambda + \delta}}{a_\delta} ;=; \frac{\det!\big(x_i^{\lambda_j + n - j}\big)}{\det!\big(x_i^{n - j}\big)}, $$ defined for partitions with at most $n$ parts. Each $a_{\lambda +\delta }$ is alternating (a transposition of two variables changes its sign), hence divisible by $a_{\delta }$, so the quotient is a genuine polynomial. The Key theorem proves it equals the SSYT sum restricted to entries $\le n$.

Kostka expansion. Collecting SSYT by content gives $s_{\lambda }={\sum }_{\mu }{K}_{\lambda \mu }{m}_{\mu }$, where the Kostka number $K_{\lambda \mu }=\#\{T\in \mathrm{SSYT}(\lambda ):\mathrm{content}(T)=\mu \}$ counts SSYT of shape $\lambda$ and content $\mu$. One has $K_{\lambda \lambda }=1$ (the unique filling of row $i$ entirely by $i$) and $K_{\lambda \mu }=0$ unless $\lambda ⊵\mu$ in dominance order, so the transition $\{s_{\lambda }\}\to \{m_{\mu }\}$ is unitriangular and $\{s_{\lambda }:\lambda ⊢d\}$ is a $\mathbb{Z}$-basis of ${\Lambda }^d$ ^{[Macdonald §3]}.

Counterexamples to common slips Intermediate+

• "The SSYT rules are symmetric in rows and columns." They are not: rows are weak ($\le$), columns are strict ($<$). This asymmetry is exactly what makes a single row give $h_r$ (repetition allowed) and a single column give $e_r$ (repetition forbidden). The symmetry that does hold is of the output $s_{\lambda }$ under permuting variables, not of the rules.

• "$s_{\lambda }(x_1,\dots ,x_n)$ is nonzero for every partition $\lambda$." In $n$ variables, $s_{\lambda }=0$ as soon as $\lambda$ has more than $n$ parts: a column of height exceeding $n$ has no strictly increasing filling from $\{1,\dots ,n\}$, and correspondingly $\lambda +\delta$ then has a repeated entry, killing the alternant.

• "Jacobi-Trudi needs $\lambda$ and the matrix to be square of size $\ell (\lambda )$." The determinant $\det (h_{{\lambda }_i-i+j})$ may be taken over any $N\times N$ block with $N\ge \ell (\lambda )$; padding $\lambda$ by trailing zeros adds rows/columns that are ${\delta }_{ij}$ on the diagonal ($h_0=1$) and leave the determinant unchanged. The convention $h_k=0$ for $k<0$ is essential.

Key theorem with proof Intermediate+

The signature theorem is that the three definitions of $s_{\lambda }$ coincide: the combinatorial SSYT sum is symmetric, equals the bialternant, and equals the Jacobi-Trudi determinant in the complete functions.

Theorem (symmetry, bialternant, and Jacobi-Trudi). For every partition $\lambda$ with $\ell (\lambda )\le n$, $$ \sum_{T \in \mathrm{SSYT}(\lambda, n)} x^T ;=; \frac{a_{\lambda+\delta}}{a_\delta} ;=; \det!\big(h_{\lambda_i - i + j}\big){1 \le i, j \le n}, $$ *with the convention $h_0=1$, $h_k=0$ for $k<0$. In particular $s\lambda$isasymmetricpolynomial,anditsstable($n \to \infty$)limitisthesymmetricfunction$s_\lambda \in \Lambda$.Dually,$s_\lambda = \det(e_{\lambda'_i - i + j})$.* ^{[Stanley §7.16]}

Proof. Three identifications.

(i) Symmetry of the SSYT sum. It suffices to show $s_{\lambda }$ is invariant under each adjacent transposition $s_k=(k,k+1)$ of variables, i.e. that swapping the exponents of $x_k$ and $x_{k+1}$ permutes SSYT among themselves. The Bender-Knuth involution ${\beta }_k$ does this. In an SSYT $T$, restrict attention to entries equal to $k$ or $k+1$. In each column at most one such entry occurs that is "free" — a $k$ with no $k+1$ directly below in the same column-pair, or vice versa — while $k$-over-$(k+1)$ vertical pairs are "fixed". In each row the free $k$s and free $(k+1)$s form a block $k^a(k+1{)}^b$; replace it by $k^b(k+1{)}^a$. This swap preserves the row-weak and column-strict conditions (the fixed pairs guard the columns) and exchanges the multiplicities of $k$ and $k+1$, so ${\beta }_k$ is an involution on $\mathrm{SSYT}(\lambda )$ with $x^{{\beta }_k(T)}=s_k\cdot x^T$. Hence $s_k$ fixes the sum, and as the $s_k$ generate $S_n$, $s_{\lambda }$ is symmetric.

(ii) Bialternant equals the SSYT sum. Both sides are symmetric polynomials of degree $|\lambda |$; compare them after multiplying by $a_{\delta }$. Expand $a_{\lambda +\delta }={\sum }_{w\in S_n}\mathrm{sgn}(w)x^{w(\lambda +\delta )}$ and $a_{\delta }={\sum }_w\mathrm{sgn}(w)x^{w\delta }$. A Lindström-type sign-reversing involution on pairs (lattice point configurations / column-strict fillings carrying a sign) cancels all cross terms except those assembling a column-strict tableau: writing $a_{\lambda +\delta }=a_{\delta }\cdot \left({\sum }_T{x}^T\right)$ amounts to the Gessel-Viennot model below read in the variables $x_i$. Concretely, the quotient $a_{\lambda +\delta }/a_{\delta }$ is the Weyl character of ${\mathrm{GL}}_n$ at highest weight $\lambda$, and the Weyl character is the trace over a basis indexed by SSYT (Gelfand-Tsetlin patterns), giving the same monomial sum.

(iii) Jacobi-Trudi by nonintersecting paths. Use the Lindström-Gessel-Viennot lemma of 40.01.04: in an acyclic weighted digraph, the signed sum over $N$-tuples of paths between sources $A_i$ and sinks $B_j$ equals $\det (M)$ with $M_{ij}$ the path-generating function $A_i\to B_j$, and only the nonintersecting tuples survive. Take the planar lattice ${\mathbb{Z}}^2$ with east and north steps, sources $A_i=(1-i,0)$ and sinks $B_j=({\lambda }_j+N-j,N-1)$, weighting each east step at height $r$ by $x_r$. The generating function $A_i\to B_j$ is then $h_{{\lambda }_j-j+i}$ (a single row of a tableau), so the LGV determinant is $\det (h_{{\lambda }_j-j+i})=\det (h_{{\lambda }_i-i+j}{)}^{\mathsf{T}}$. A nonintersecting tuple of such paths is precisely an SSYT of shape $\lambda$ (the $i$-th path traces the cells holding entries $\ge i$, the heights of its east steps reading off the columns), so the determinant equals ${\sum }_T{x}^T=s_{\lambda }$. Applying $\omega$ (which sends $h_k\mapsto e_k$ and, as shown in Advanced results, $s_{\lambda }\mapsto s_{{\lambda }^{\prime }}$) to the row form yields the dual column form $s_{\lambda }=\det (e_{{\lambda }_i^{\prime }-i+j})$. $\square$

Bridge. This theorem is the foundational reason the Schur functions are inevitable: it shows that the combinatorial, the determinantal, and the representation-theoretic descriptions are one object, so any property provable in one language transfers to the others. It builds toward the Pieri and Littlewood-Richardson rules of Advanced results, where the SSYT picture controls the structure constants, and it appears again in the Hall inner product of 40.03.03, under which these same $s_{\lambda }$ are orthonormal. The bialternant side is exactly the Weyl character formula, which is dual to the Jacobi-Trudi side through the Vandermonde $a_{\delta }$; this is the central insight that the antisymmetrisation ${\sum }_w\mathrm{sgn}(w)$ in $a_{\lambda +\delta }$ and the sign-reversing involution of the LGV lemma are the same cancellation seen twice. Putting these together, the SSYT definition supplies positivity (a sum of monomials), the bialternant supplies symmetry and the basis property, and Jacobi-Trudi supplies the algebra inside $\Lambda =\mathbb{Z}[h_1,h_2,\dots ]$ — three faces of one function.

Exercises Intermediate+

Advanced results Master

The Schur basis carries the multiplicative and inner-product structure of $\Lambda$, and the determinantal formulas extend to skew shapes, specialisations, and the Cauchy kernel.

Theorem 1 (Pieri, dual Pieri, and Littlewood-Richardson). The complete and elementary functions act on the Schur basis by adding strips: $$ s_\lambda, h_r = \sum_{\mu} s_\mu \ \ (\mu/\lambda \text{ a horizontal } r\text{-strip}), \qquad s_\lambda, e_r = \sum_{\mu} s_\mu \ \ (\mu/\lambda \text{ a vertical } r\text{-strip}). $$ These are the $\nu =(r)$ and $\nu =(1^r)$ cases of the general structure constants $s_{\lambda }{s}_{\mu }={\sum }_{\nu }{c}_{\lambda \mu }^{\nu }{s}_{\nu }$, where $c_{\lambda \mu }^{\nu }$ is the Littlewood-Richardson coefficient counting LR skew tableaux of shape $\nu /\lambda$ and content $\mu$; the $c_{\lambda \mu }^{\nu }$ are non-negative integers, developed in the co-produced 40.03.05. The two Pieri rules are conjugate to each other under $\omega$, since $\omega (h_r)=e_r$ and $\omega (s_{\lambda })=s_{{\lambda }^{\prime }}$ ^{[Macdonald §5]}.

Theorem 2 (skew Jacobi-Trudi and Giambelli). For $\mu \subseteq \lambda$ the skew Schur function satisfies the determinantal formulas $$ s_{\lambda/\mu} = \det!\big(h_{\lambda_i - \mu_j - i + j}\big){1 \le i,j \le \ell(\lambda)} = \det!\big(e{\lambda'i - \mu'j - i + j}\big), $$ again by the LGV lemma with sources displaced by $\mu$. The expansion $s{\lambda/\mu} = \sum\nu c^\lambda_{\mu\nu} s_\nu$hasthesameLittlewood-Richardsoncoefficients,soskewSchurfunctionsareSchur-positive.The\ast \ast Giambelli(hook)determinant\ast \ast writesany$s_\lambda$inFrobeniushookcoordinates$\lambda = (a_1, \dots, a_d \mid b_1, \dots, b_d)$as$s_\lambda = \det\big(s_{(a_i \mid b_j)}\big)$, a determinant of single-hook Schur functions — the Plücker/Schubert-calculus form.

Theorem 3 (Cauchy identity and orthonormality). The Schur functions diagonalise the Cauchy kernel: $$ \prod_{i,j} \frac{1}{1 - x_i y_j} = \sum_{\lambda} s_\lambda(x), s_\lambda(y), \qquad \prod_{i,j}(1 + x_i y_j) = \sum_\lambda s_\lambda(x), s_{\lambda'}(y). $$ Under the Hall inner product of the co-produced 40.03.03 — defined by $⟨h_{\lambda },m_{\mu }⟩={\delta }_{\lambda \mu }$, equivalently $⟨p_{\lambda },p_{\mu }⟩=z_{\lambda }{\delta }_{\lambda \mu }$ — the Cauchy identity is equivalent to $⟨s_{\lambda },s_{\mu }⟩={\delta }_{\lambda \mu }$: the Schur functions are an orthonormal $\mathbb{Z}$-basis, the unique (up to sign) basis that is both orthonormal and unitriangular against $\{m_{\mu }\}$. The involution $\omega$ is then an isometry with $\omega (s_{\lambda })=s_{{\lambda }^{\prime }}$.

Theorem 4 (principal specialisation, hook-content and hook-length). Setting $x_i=q^{i-1}$ for $i=1,\dots ,n$ and the rest $0$ gives the principal specialisation $$ s_\lambda(1, q, q^2, \dots, q^{n-1}) = q^{,n(\lambda)} \prod_{(i,j) \in \lambda} \frac{1 - q^{,n + c(i,j)}}{1 - q^{,h(i,j)}}, $$ the hook-content formula, where $c(i,j)=j-i$ is the content of a cell, $h(i,j)$ its hook length, and $n(\lambda )={\sum }_i(i-1){\lambda }_i$. Letting $q\to 1$ recovers $s_{\lambda }(1^n)={\prod }_{(i,j)}\frac{n+c(i,j)}{h(i,j)}$, the number of SSYT of shape $\lambda$ with entries $\le n$, i.e. the dimension of the irreducible ${\mathrm{GL}}_n$-representation; and the standard-tableau count $f^{\lambda }=\frac{|\lambda |!}{{\prod }_{(i,j)}h(i,j)}$ is the hook-length formula for the symmetric-group irreducible, cross-referenced to 07.05.02. The coefficient of the lowest power, $q^{n(\lambda )}$, is the charge/cocharge statistic that recurs in the Kostka-Foulkes refinement.

Synthesis. Putting these together, the Schur basis is the fixed point at which every structure on $\Lambda$ becomes diagonal. The combinatorial definition makes each $s_{\lambda }$ manifestly Schur-positive and the Kostka unitriangularity makes them a $\mathbb{Z}$-basis; the Cauchy identity of Theorem 3 is exactly the statement that this basis is orthonormal for the Hall inner product of 40.03.03, so the foundational reason the $s_{\lambda }$ dominate the subject is that they are the self-dual basis, and $\omega$ — conjugation of diagrams in 40.03.01 — is precisely the isometry $s_{\lambda }\mapsto s_{{\lambda }^{\prime }}$. The central insight is that the antisymmetrisation of the bialternant, the sign-reversing cancellation of the LGV proof of Jacobi-Trudi, and the orthogonality of characters are one phenomenon: the Weyl character formula is dual to the Jacobi-Trudi determinant through the Vandermonde, and this is exactly why the Pieri rules of Theorem 1 (multiplication by $h_r$ and $e_r$) are conjugate under $\omega$ and why the Littlewood-Richardson coefficients are symmetric. The bridge is the Frobenius characteristic of the co-produced 40.03.06: it carries the irreducible $S_n$-characters to the $s_{\lambda }$, so the hook-length formula of Theorem 4 — counting standard tableaux — generalises into the principal specialisation, and the entire determinantal calculus developed here reappears as the character theory of 07.05.02 and the Murnaghan-Nakayama rule of 07.05.10. The same diagram conjugation that defines $\omega$ in 40.03.01 is the duality threading Pieri, Cauchy, and the hook-content formula.

Full proof set Master

Proposition 1 (the SSYT sum is symmetric). For every partition $\lambda$, $s_{\lambda }={\sum }_{T\in \mathrm{SSYT}(\lambda )}{x}^T$ is a symmetric function.

Proof. Fix $k\ge 1$; we produce an involution ${\beta }_k$ on $\mathrm{SSYT}(\lambda )$ exchanging the multiplicities of $k$ and $k+1$ while fixing all others, which gives invariance under the adjacent transposition $s_k$ and hence (as the $s_k$ generate the symmetric group) full symmetry. In $T$, call a cell holding $k$ or $k+1$ paired if it lies directly above a $k+1$ (for a $k$) or directly below a $k$ (for a $k+1$) in the same column; paired cells form vertical $\left(\genfrac{}{}{0ex}{}{k}{k+1}\right)$ dominoes and are left untouched. The remaining free $k$s and $(k+1)$s lie in horizontal runs within rows: in a given row the free entries equal to $k$ or $k+1$ form a contiguous block $k^a(k+1{)}^b$ (any intervening other value would separate them into independent runs, handled identically). Replace each such run by $k^b(k+1{)}^a$. Row-weak monotonicity is preserved because the run was the only place $k,k+1$ were adjacent and free; column-strictness is preserved because the paired dominoes that could conflict were excluded from the free set. The map is an involution ($a\leftrightarrow b$ twice returns the original) and replaces ${\mu }_k,{\mu }_{k+1}$ by ${\mu }_{k+1},{\mu }_k$, so $x^{{\beta }_{k}T}=s_k\cdot x^T$. Summing, $s_k(s_{\lambda })=s_{\lambda }$. $\square$

Proposition 2 (Jacobi-Trudi via Lindström-Gessel-Viennot). With $h_0=1$, $h_m=0$ for $m<0$, and $N\ge \ell (\lambda )$, $s_{\lambda }=\det (h_{{\lambda }_i-i+j}{)}_{1\le i,j\le N}$.

Proof. Consider the directed lattice on ${\mathbb{Z}}^2$ with unit east and north steps; weight an east step from $(\cdot ,r)$ to $(\cdot +1,r)$ by $x_{r+1}$, so a path's weight is $\prod x$ over its east steps. Place sources $A_i=(1-i,0)$ and sinks $B_j=({\lambda }_j-j+1,N-1)$ for $1\le i,j\le N$ (padding $\lambda$ with zeros to length $N$). The generating function of all paths $A_i\to B_j$ is the complete homogeneous function $h_{({\lambda }_j-j+1)-(1-i)}=h_{{\lambda }_j-j+i}$, since a monotone path making $m$ east steps across heights records a weakly increasing length-$m$ word, whose weight sum is $h_m$. The Lindström-Gessel-Viennot lemma 40.01.04 gives $$ \sum_{w \in S_N} \mathrm{sgn}(w) \prod_{i} (\text{paths } A_i \to B_{w(i)}) = \det\big(h_{\lambda_j - j + i}\big) = \sum_{(P_1,\dots,P_N) \text{ nonintersecting}} \mathrm{sgn}(\sigma), x^{P_1}\cdots x^{P_N}, $$ and for this source/sink placement only the identity permutation admits nonintersecting tuples, all with sign $+1$. A nonintersecting tuple is in content-preserving bijection with an SSYT of shape $\lambda$: path $P_i$ contributes the east-step heights as the entries of the cells in the $i$-th row read against the columns, the nonintersection condition translating exactly into the column-strict, row-weak conditions. Therefore $\det (h_{{\lambda }_j-j+i})={\sum }_T{x}^T=s_{\lambda }$, and transposing the matrix (a determinant is invariant under transpose) gives the stated $\det (h_{{\lambda }_i-i+j})$. $\square$

Proposition 3 (bialternant equals Jacobi-Trudi). In $n$ variables, $a_{\lambda +\delta }/a_{\delta }=\det (h_{{\lambda }_i-i+j}{)}_{1\le i,j\le n}$.

Proof. Use the generating-function identity ${\sum }_m{h}_m{t}^m={\prod }_{i=1}^n(1-x_{i}t{)}^{-1}=1/H_n(t{)}^{-1}$, where here $H_n(t{)}^{-1}={\prod }_i(1-x_{i}t)$. In matrix form, the complete functions are the entries of the inverse of the Vandermonde-style matrix built from the $e$'s; concretely, the Jacobi-Trudi determinant arises by Cramer's rule. Form the $n\times n$ matrices $X=(x_i^{n-j})$ (so $\det X=a_{\delta }$) and $X^{(\lambda )}=(x_i^{{\lambda }_j+n-j})$ (so $\det X^{(\lambda )}=a_{\lambda +\delta }$). Expanding ${\prod }_k(1-x_{i}t{)}^{-1}$ shows that the matrix $H=(h_{i-j}{)}_{1\le i,j}$ (lower-triangular-unipotent in the appropriate truncation) satisfies $X^{(\lambda )}=X\cdot M_{\lambda }$ in each graded piece, where $M_{\lambda }$ is the matrix with $(M_{\lambda }{)}_{jk}=h_{{\lambda }_k-k+j}$ shifted by the staircase; taking determinants, $a_{\lambda +\delta }=a_{\delta }\cdot \det (h_{{\lambda }_i-i+j})$. Equivalently, both sides are symmetric, agree in the stable limit by Proposition 2, and the finite-variable identity follows by setting $x_{n+1}=\cdots =0$, under which $h_m$ specialises correctly and a column of $\lambda$ exceeding $n$ kills both sides. Hence the ratio equals the determinant. $\square$

Proposition 4 (Pieri rule). $s_{\lambda }{h}_r=\sum _{\mu :\mu /\lambda \text{ horizontal }r\text{-strip}}{s}_{\mu }$.

Proof. Pair the Jacobi-Trudi expression with the Hall inner product of 40.03.03, under which $\{s_{\mu }\}$ is orthonormal, so the coefficient of $s_{\mu }$ in $s_{\lambda }{h}_r$ is $⟨s_{\lambda }{h}_r,s_{\mu }⟩=⟨s_{\lambda },h_r^{\perp }{s}_{\mu }⟩$. The adjoint $h_r^{\perp }$ of multiplication by $h_r$ acts as the skewing operator $s_{\mu }\mapsto s_{\mu /(r)}={\sum }_{\nu }(\text{coeff})s_{\nu }$; by the dual-basis property $⟨s_{\lambda },s_{\mu /(r)}⟩$ is the multiplicity of $s_{\lambda }$ in $s_{\mu /(r)}$, which equals $1$ if $\mu /\lambda$ is a horizontal $r$-strip and $0$ otherwise. Combinatorially: $s_{\lambda }{h}_r={\sum }_{T,T_0}{x}^T{x}^{T_0}$ over SSYT $T$ of shape $\lambda$ and single rows $T_0$ of length $r$; the RSK-free insertion of $T_0$'s weakly increasing row onto $\lambda$ produces exactly the SSYT of skew shapes $\mu /\lambda$ that are horizontal strips, each once, since two inserted cells in one column would force a strict-column violation. Grouping by $\mu$ gives ${\sum }_{\mu }{s}_{\mu }$ over horizontal $r$-strips. $\square$

Proposition 5 (orthonormality from Cauchy). Under $⟨h_{\lambda },m_{\mu }⟩={\delta }_{\lambda \mu }$, one has $⟨s_{\lambda },s_{\mu }⟩={\delta }_{\lambda \mu }$.

Proof. The Cauchy kernel factors two ways: ${\prod }_{i,j}(1-x_i{y}_j{)}^{-1}={\sum }_{\lambda }{h}_{\lambda }(x)m_{\lambda }(y)={\sum }_{\lambda }{m}_{\lambda }(x)h_{\lambda }(y)$, exhibiting $\{h_{\lambda }\}$ and $\{m_{\mu }\}$ as dual bases, which is the definition of the inner product. Two graded bases $\{u_{\lambda }\},\{v_{\lambda }\}$ are dual ($⟨u_{\lambda },v_{\mu }⟩={\delta }_{\lambda \mu }$) if and only if ${\sum }_{\lambda }{u}_{\lambda }(x)v_{\lambda }(y)={\prod }_{i,j}(1-x_i{y}_j{)}^{-1}$. Write $s_{\lambda }={\sum }_{\mu }{K}_{\lambda \mu }{m}_{\mu }$ and the dual expansion $s_{\lambda }={\sum }_{\mu }{K}^{-1}\cdots$; substituting the Jacobi-Trudi/Kostka relations, ${\sum }_{\lambda }{s}_{\lambda }(x)s_{\lambda }(y)={\sum }_{\lambda }({\sum }_{\mu }{K}_{\lambda \mu }{m}_{\mu }(x))({\sum }_{\nu }{K}_{\lambda \nu }{m}_{\nu }(y))$. The Kostka matrix $K$ is unitriangular, and the orthogonality ${\sum }_{\lambda }{K}_{\lambda \mu }{K}_{\lambda \nu }$ against the $h$-$m$ duality collapses the double sum to ${\sum }_{\mu }{h}_{\mu }(x)m_{\mu }(y)$ precisely when $\{s_{\lambda }\}$ is self-dual. Thus ${\sum }_{\lambda }{s}_{\lambda }(x)s_{\lambda }(y)={\prod }_{i,j}(1-x_i{y}_j{)}^{-1}$ is equivalent to $⟨s_{\lambda },s_{\mu }⟩={\delta }_{\lambda \mu }$; the Cauchy identity (proved by RSK in the co-produced 40.03.04) supplies the left equality, giving orthonormality. $\square$

Connections Master

• The Schur functions are built directly on the ring and bases of 40.03.01: the Kostka unitriangularity $s_{\lambda }=m_{\lambda }+{\sum }_{\mu ⊲\lambda }{K}_{\lambda \mu }{m}_{\mu }$ uses the dominance order and monomial basis defined there, the Jacobi-Trudi determinants live in $\Lambda =\mathbb{Z}[h_1,h_2,\dots ]$, and the diagram conjugation that gives the involution $\omega$ of 40.03.01 acts here as $\omega (s_{\lambda })=s_{{\lambda }^{\prime }}$, interchanging the row Jacobi-Trudi formula with its dual column form. The single-row and single-column Schur functions recover the complete $h_r$ and elementary $e_r$ generators of that unit.

• The Hall inner product making $\{s_{\lambda }\}$ orthonormal, and the Cauchy identity $\prod (1-x_i{y}_j{)}^{-1}={\sum }_{\lambda }{s}_{\lambda }(x)s_{\lambda }(y)$ that is equivalent to it, are the subject of the co-produced 40.03.03; the RSK correspondence that proves the Cauchy identity bijectively is the co-produced 40.03.04; and the Littlewood-Richardson coefficients $c_{\lambda \mu }^{\nu }$ generalising the Pieri rules proved here are the co-produced 40.03.05. The Frobenius characteristic of the co-produced 40.03.06 carries the Schur functions to the irreducible characters of the symmetric groups, turning every Schur identity into a character identity.

• The bialternant $s_{\lambda }=a_{\lambda +\delta }/a_{\delta }$ is the Weyl character formula for ${\mathrm{GL}}_n$, so this unit is the combinatorial face of the representation theory in 07.05.02 (Young diagrams and the irreducibles) and 07.05.04 (Schur-Weyl duality, where $s_{\lambda }$ is simultaneously a ${\mathrm{GL}}_n$ and an $S_n$ object); the hook-length formula $f^{\lambda }=|\lambda |!/\prod h(i,j)$ of Theorem 4 is the dimension of the Specht module of 07.05.03, and the power-sum expansion of $s_{\lambda }$ is the Murnaghan-Nakayama rule of 07.05.10. The Jacobi-Trudi proof reuses the Lindström-Gessel-Viennot lemma of 40.01.04, the same nonintersecting-path determinant that computes transfer matrices and $P$-partition generating functions there.

Historical & philosophical context Master

The bialternant quotient $a_{\lambda +\delta }/a_{\delta }$ that defines the Schur function appears first in Cauchy's 1815 memoir on alternating functions ^{[Cauchy 1815]}, where the ratio of an alternant by the Vandermonde is studied as the basic symmetric object built from a determinant; the determinant of alternants and the manipulation that yields what is now Jacobi-Trudi were developed by Jacobi in his 1841 paper De functionibus alternantibus in Crelle's Journal ^{[Jacobi 1841]}, from which the modern name of the identity descends. The functions acquired their combinatorial and representation-theoretic meaning in Issai Schur's 1901 Berlin dissertation, which identified $a_{\lambda +\delta }/a_{\delta }$ with the irreducible polynomial character of ${\mathrm{GL}}_n(\mathbb{C})$ of highest weight $\lambda$; the name Schur function is due to the later literature.

The combinatorial definition by semistandard tableaux, the symmetry by what are now called the Bender-Knuth involutions, and the systematic Pieri and Littlewood-Richardson calculus belong to the tableau tradition running from Alfred Young's substitutional analysis (1900-1930s) through Dudley Littlewood and Archibald Richardson's 1934 paper stating the multiplication rule, to its rigorous proofs decades later by Schützenberger, Thomas, and others. The nonintersecting-lattice-path proof of the determinantal formulas is due to Lindström (1973) and, in the combinatorial form used here, to Gessel and Viennot (1985). The bialternant-to-tableau equivalence is the combinatorial shadow of the Weyl character formula proved by Hermann Weyl (1925-26) for compact Lie groups, of which the ${\mathrm{GL}}_n$ case is the Schur polynomial.

Bibliography Master

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  author    = {Stanley, Richard P.},
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}

@book{macdonald1995symmetric,
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}

@book{fulton1997young,
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}

@article{jacobi1841functionibus,
  author  = {Jacobi, Carl Gustav Jacob},
  title   = {De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum},
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  volume  = {22},
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  year    = {1841}
}

@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},
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}

@phdthesis{schur1901,
  author  = {Schur, Issai},
  title   = {\"{U}ber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen},
  school  = {Universit\"{a}t Berlin},
  year    = {1901}
}

@article{littlewoodrichardson1934,
  author  = {Littlewood, Dudley E. and Richardson, Archibald R.},
  title   = {Group characters and algebra},
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}

@article{gesselviennot1985,
  author  = {Gessel, Ira and Viennot, G\'{e}rard},
  title   = {Binomial determinants, paths, and hook length formulae},
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}