Schur polynomial ($s_{\lambda }$)

For an integer partition $\lambda$ with at most $N$ parts, padded with zeros, the Schur polynomial in $N$ variables isBoth determinants change sign when two variables are exchanged. Every factor of the denominator's Vandermonde determinant divides the numerator, so the quotient is a symmetric polynomial, including at repeated variable values. The bialternant formula and Jacobi–Trudi identity give equivalent descriptions; the latter writes it as $\det [h_{{\lambda }_i-i+j}]$ in complete homogeneous symmetric functions.^[ref]

Equivalently, $s_{\lambda }$ sums the weight monomials of all Semistandard Young tableaux of shape $\lambda$, with entries in $\{1,\dots ,N\}$, weakly increasing along rows and strictly increasing down columns. For example,The two monomials correspond to the two possible fillings of this three-cell shape. A single-row shape gives $s_{(r)}=h_r$, while a single-column shape gives an elementary symmetric polynomial.

Adding a zero variable leaves these expressions unchanged, defining a Schur function in the ring of symmetric functions. In each fixed degree, the Schur functions form a basis indexed by the integer partitions of that degree. The Frobenius characteristic map identifies this basis with the irreducible characters of symmetric groups, connecting tableau enumeration to representation theory.

For an integer partition $\lambda$, $s_{\lambda }(1^m)={\prod }_{(i,j)\in \lambda }(m+j-i)/h_{ij}$, where $h_{ij}$ is the cell's hook length. It counts Semistandard Young tableaux of shape $\lambda$ whose entries belong to $\{1,\dots ,m\}$.

Let $a_{\gamma }=\det [x_j^{{\gamma }_i}]$ and $\delta =(N-1,\dots ,0)$. Multiply the Jacobi–Trudi identity matrix by $B_{jk}=(-1{)}^{N-j}{e}_{N-j}(X\setminus x_k)$. The generating-function identity $H(t){\prod }_{i\ne k}(1-x_{i}t)=(1-x_{k}t{)}^{-1}$ makes the product matrix $[x_k^{{\lambda }_i+N-i}]$. Setting $\lambda =0$ shows $\det B=a_{\delta }$, proving the formula as a polynomial identity.

Put $b_i={\lambda }_i+N-i$. In the bialternant formula, substitute $x_j=e^{t{y}_j}$ with distinct $y_j$ and compare the first nonzero determinant coefficients as $t\to 0$. They arise from the distinct degrees $0,\dots ,N-1$ in the exponential expansions. The common factor in the $y_j$ and the factorials cancel, leaving the product of differences of the $b_i$ divided by the differences of $N-i$. This proves the displayed integral specialization without dividing by a vanishing Vandermonde value.

Sum the weight monomials of all Semistandard Young tableaux on the skew Young diagram $\lambda /\mu$, with weak rows and strict columns. A Bender-Knuth involution interchanges the counts of adjacent letters without changing the shape, so the sum is a symmetric function. Taking $\mu$ empty gives the Schur function.

The Jacobi–Trudi identity expresses a Schur polynomial asIn particular, $s_{(a,b)}=h_a{h}_b-h_{a+1}{h}_{b-1}$.

Here $[a]$ is the trivial character of $S_a$, $[0]$ is the induction-product unit, $[a]=0$ for $a<0$, and multiplication is induction of external tensor products. The Frobenius characteristic map turns the expression into the Jacobi–Trudi identity. A sign-reversing involution on intersecting lattice-path systems leaves precisely the nonintersecting systems corresponding to Semistandard Young tableaux, proving the determinant identity.