The symmetric functions catalog

An overview of symmetric functions and related topics

2021-07-17

Schur-S polynomials

Let us first introduce $S_j(\xvec)$ via the expansion

\[ \sum_{j \geq 0} S_j z^j = \exp\left( \sum_{j \geq 1} x_j z^j\right). \]

Then define

\[ S_\lambda(\xvec) \coloneqq \det[ S_{\lambda_j-j+i} ]. \]

Note that this is the image of $\schurS_\lambda$ under the algebra homomorphism $\completeH_j \mapsto S_j.$

As an example,

\[ S_{32}(\xvec)= \frac{x_1^5}{24}+\frac{1}{6} x_2 x_1^3-\frac{1}{2} x_3 x_1^2+\frac{1}{2} x_2^2 x_1-x_4 x_1+x_2 x_3. \]

After this map, we have that $\{ S_\lambda \}_\lambda$ is a basis for $\setQ[\xvec],$ which is a bit unusual compared to other families of Schur polynomials.

In https://arxiv.org/pdf/2009.00479.pdf the main result is that when $\lambda \vdash d,$

\[ \frac{\partial^d}{\partial x_1^d} S_\lambda(x) = f^\lambda. \]