2023-09-26

## 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. https://arxiv.org/abs/1008.2501

### 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

## References

- [Mou23] Vassilis Dionyssis Moustakas. Descent representations and colored quasisymmetric functions. arXiv e-prints, 2023.