#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, \(\{ S_\lambda \}_\lambda\) is a basis for \(\setQ[\xvec],\) which is a bit unusual compared to other families of Schur polynomials.
In [BG20], the main result is that when \(\lambda \vdash d,\) \[\frac{\partial^d}{\partial x_1^d} S_\lambda(x) = f^\lambda.\]
Bibliography
- [BG20]Ommolbanin Behzad and Letterio Gatto. Bosonic and fermionic representations of endomorphisms of exterior algebras. arXiv:2009.00479, 2020.
.bib
@article{BehzadGatto2020x, Author = {Ommolbanin Behzad and Letterio Gatto}, Title = {Bosonic and Fermionic Representations of Endomorphisms of Exterior Algebras}, Year = {2020}, Eprint = {2009.00479}, url = {https://arxiv.org/abs/2009.00479}, journal = {arXiv e-prints} }