The symmetric functions catalog

An overview of symmetric functions and related topics


Ribbon Schur polynomials

The ribbon Schur polynomials are the skew Schur polynomials indexed by ribbons.

There are $2^{n-1}$ different ribbons $\alpha$ with $n$ boxes, but the corresponding ribbon Schur functions $\schurS_\alpha$ are distinct.

Complete homogeneous expansion

The ribbon Schur function $\schurS_\alpha$ can be expanded as

\[ \sum_{\beta \geq \alpha} (-1)^{\length(\beta)-\length(\alpha)}\completeH_{\beta} \]

where the sum is over coarsenings of $\alpha,$ see [Mou23]. A generalization to colored ribbon Schur functions can also be found in that reference.

Products of ribbon Schur functions