2020-08-31

## Odd Schur functions

The odd Schur functions are introduced in [EKL12EK12]. They constitute a basis for the space of odd symmetric functions.

There are several definitions of the odd Schur functions, $\{\oddSchur_\lambda\},$ using divided difference operators or plactic relations. In [Ell12], it was shown that all the previous definitions coincide, and that we have the following tableau formula.

\[ \oddCompleteH_\mu = \sum_{T \in \SSYT(\lambda,\mu)} \sign(T_\lambda) \sign(T) \oddSchur_\lambda. \]Here, $\sign(T)$ is the sign of the shortest permutation that sorts the reading word of $T$ in an increasing fashion, and $T_\lambda$ is the unique SSYT in $\SSYT(\lambda,\lambda).$

In [Ell12], a Littlewood–Richardson rule is proved for the odd Schur functions.

## References

- [EK12] Alexander P. Ellis and Mikhail Khovanov. The Hopf algebra of odd symmetric functions. Advances in Mathematics, 231(2):965–999, October 2012.
- [EKL12] Alexander P. Ellis, Mikhail Khovanov and Aaron D. Lauda. The odd nilHecke algebra and its diagrammatics. International Mathematics Research Notices, 2014(4):991–1062, November 2012.
- [Ell12] Alexander P. Ellis. The odd Littlewood–Richardson rule. Journal of Algebraic Combinatorics, 37(4):777–799, August 2012.