 Schur polynomial

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.

The Schur polynomial is a specific type of symmetric polynomial that plays a significant role in algebraic combinatorics, representation theory, and geometry. It is associated with a given partition of integers and is used in the study of symmetric functions.