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John R. Stembridge works on algebraic combinatorics, representation theory, Coxeter groups and symmetric functions. His Michigan profile also identifies computational methods in algebra as a focus.^[ref]

With Richard P. Stanley, he studied immanants of matrices associated with the Jacobi–Trudi identity. An immanant replaces the sign character in a determinant by another symmetric group character. Their paper proves special cases of stronger positivity conjectures and connects the calculation with cycle structure of permutations having restricted positions.^[ref]

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.