The symmetric functions catalog

An overview of symmetric functions and related topics


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.