2023-07-01

## Character symmetric functions

In [OZ18], the authors define the irreducible character basis, $\{\chSym_\lambda\},$ the induced character basis, $\{\indCharBasis_\lambda\},$ and the induced trivial character basis, $\{\indTrivBasis_\lambda\}.$ See also the \hyperref{https://www.youtube.com/watch?v=4fDmCz9gWac}{OPAC Youtube lectures on this topic}. The induced character basis was previously studied by D. Speyer and S. Assaf [AS19c] under the name stable Specht polynomials.

In order to define these three families, we need some additional notation. First, let

\[ \bar{\powerSum}_{i^r} \coloneqq i^r \left( \frac{1}{i} \sum_{d \mid i } \mu(i/d) \powerSum_d \right)_i \text{ and } \hat{\powerSum}_{i^r} \coloneqq \sum_{k=0}^r (-1)^{r-k} \binom{r}{k} \bar{\powerSum}_{i^k}, \]where we use the notation of falling factorial in the first expression. With these definitions, we can define $\bar{\powerSum}_\gamma$ and $\hat{\powerSum}_\gamma,$ for partitions $\gamma.$ Finally, we set

\[ \chSym_\lambda \coloneqq \sum_{\gamma} \chi^{\lambda}(\gamma) \frac{ \hat{\powerSum}_\gamma }{z_\gamma} \qquad \indCharBasis_\lambda \coloneqq \sum_{\gamma} \chi^{\lambda}(\gamma) \frac{ \bar{\powerSum}_\gamma }{z_\gamma} \quad \indTrivBasis_\lambda \coloneqq \sum_{\gamma} \langle \completeH_\gamma, \powerSum_\gamma \rangle \frac{ \bar{\powerSum}_\gamma }{z_\gamma} \quad \]The irreducible character symmetric functions $\{\chSym_\lambda\}$ were introduced by Orellana and Zabrocki already in [OZ16]. They are non-homogeneous symmetric functions, and may alternatively be defined as follow.

Let $\lambda$ be a fixed partition, and $n \geq |\lambda|+\lambda_1.$ Then for all partitions $\gamma \vdash n,$

\[ \chSym_\lambda(\zeta_{1},\dotsc,\zeta_{n}) = \chi^{(n-|\lambda|,\lambda)}(\gamma) \]where $\zeta_{1},\dotsc,\zeta_{n}$ are the eigenvalues of a permutation matrix with cycle structure $\gamma.$ This property uniquely defines the $\chSym_\lambda.$

This definition makes the connection with character polynomials evident, see the paper [GG09] for more background.

### Properties

The multiplicative structure constants are given by
the *reduced* Kronecker coefficients,

Moreover, this property plus $\schurS_{1^r} = \chSym_{1^r}+\chSym_{1^{r-1}}$ uniquely defines the character symmetric functions.

The $\{\chSym_\mu\}$ are the unique set of solutions to the system of equations

\[ \schurS_\lambda = \sum_{\mu : |\mu| \leq |\lambda|} A_{\lambda,(n-|\mu|,\mu)} \chSym_\mu \]for all $n$ sufficiently large. Here, $A_{\lambda,\mu} = \langle \schurS_\lambda, \schurS_\mu[1 + \completeH_1 + \completeH_2 + \dotsb \rangle,$ where we use plethystic notation. It is an open problem to combinatorially describe the $A_{\lambda,\mu}.$

For a brief overview, see this OPAC blog post.

Orellana and Zabrocki prove a Murnaghanâ€“Nakayama type identity.

See also https://arxiv.org/abs/2004.03928 and https://arxiv.org/pdf/2001.04112.pdf. Also related to https://www.math.uchicago.edu/~farb/papers/FImod.pdf

## References

- [AS19c] Sami H. Assaf and David E. Speyer. Specht modules decompose as alternating sums of restrictions of Schur modules. Proceedings of the American Mathematical Society, 148(3):1015â€“1029, October 2019.
- [GG09] A. M. Garsia and A. Goupil. Character polynomials, their $q$-analogs and the Kronecker product. The Electronic Journal of Combinatorics, 16(2), July 2009.
- [OZ16] Rosa Orellana and Mike Zabrocki. Symmetric group characters as symmetric functions. arXiv e-prints, 2016.
- [OZ18] Rosa Orellana and Mike Zabrocki. The Hopf structure of symmetric group characters as symmetric functions. arXiv e-prints, 2018.