#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 OPAC YouTube lectures on this topic. The induced character basis was previously studied by D. Speyer and S. Assaf [AS19] under the name stable Specht polynomials.
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 \((\cdot)_i\) denotes a 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 \[\chSym_\lambda \coloneqq \sum_{\gamma} \chi^{\lambda}(\gamma) \frac{ \hat{\powerSum}_\gamma }{z_\gamma} \qquad \indCharBasis_\lambda \coloneqq \sum_{\gamma} \chi^{\lambda}(\gamma) \frac{ \bar{\powerSum}_\gamma }{z_\gamma} \quad \indTrivBasis_\lambda \coloneqq \sum_{\gamma} \langle \completeH_\gamma, \powerSum_\gamma \rangle \frac{ \bar{\powerSum}_\gamma }{z_\gamma}\]
The irreducible character symmetric functions \(\{\chSym_\lambda\}\) were introduced by R. Orellana and M. Zabrocki already in [OZ16]. They are non-homogeneous symmetric functions and can alternatively be defined as follows.
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. M. Marciniak studies the quadratic coefficients of the Goulden–Rattan character polynomials [Mar21]. In particular, he proves the Goulden–Rattan positivity conjecture for the coefficient of the quadratic term \(C_2^2,\) using bijections involving maps on surfaces. C. Gaetz and L. Pierson study character polynomials coming from moments of permutation-pattern counts [GP22]. For identity patterns, they verify several cases of a positivity conjecture by showing that the corresponding coefficient polynomials are real-rooted with all roots below the pattern length.
#Properties
The multiplicative structure constants are given by the reduced Kronecker coefficients, \[\chSym_\lambda \chSym_\mu = \sum_{\nu} \bar{g}^{\nu}_{\lambda,\mu} \chSym_\nu.\] 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.
R. Orellana and M. Zabrocki prove a Murnaghan–Nakayama type identity.
The restriction problem is closely related to character polynomials. S. P. Narayanan, D. Paul, A. Prasad, and S. Srivastava study this connection in [NPPS21]. In a companion paper, they construct a polynomial induction functor adjoint to restriction from polynomial representations of general linear groups to representations of the corresponding Weyl groups [NPPS21]. For background on eventual character polynomials and representation stability, see the work of T. Church, J. S. Ellenberg, and B. Farb on FI-modules [CEF15].
Bibliography
- [AS19]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):1015–1029, October 2019.
.bib
@article{AssafSpeyer2019, doi = {10.1090/proc/14815}, url2 = {https://doi.org/10.1090/proc/14815}, year = {2019}, month = oct, publisher = {American Mathematical Society ({AMS})}, volume = {148}, number = {3}, pages = {1015--1029}, author = {Sami H. Assaf and David E. Speyer}, title = {Specht modules decompose as alternating sums of restrictions of {S}chur modules}, journal = {Proceedings of the American Mathematical Society} } - [CEF15]Thomas Church, Jordan S. Ellenberg and Benson Farb. FI-modules and stability for representations of symmetric groups. Duke Mathematical Journal, 164(9):1833–1910, 2015.
.bib
@article{ChurchEllenbergFarb2015, author = {Thomas Church and Jordan S. Ellenberg and Benson Farb}, title = {{FI}-modules and stability for representations of symmetric groups}, journal = {Duke Mathematical Journal}, volume = {164}, number = {9}, pages = {1833--1910}, year = {2015}, doi = {10.1215/00127094-3120274} } - [GP22]Christian Gaetz and Laura Pierson. Positivity of permutation pattern character polynomials. arXiv:2204.10633, 2022.
.bib
@article{GaetzPierson2022x, author = {Christian Gaetz and Laura Pierson}, title = {Positivity of permutation pattern character polynomials}, year = {2022}, eprint = {2204.10633}, url = {https://arxiv.org/abs/2204.10633}, journal = {arXiv e-prints}, journalref = {Advances in Applied Mathematics, Volume 147, June 2023}, doi = {10.1016/j.aam.2023.102507} } - [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.
.bib
@article{GarsiaGoupil2009, doi = {10.37236/85}, url2 = {https://doi.org/10.37236/85}, year = {2009}, month = jul, publisher = {The Electronic Journal of Combinatorics}, volume = {16}, number = {2}, author = {A. M. Garsia and A. Goupil}, title = {Character Polynomials, their $q$-Analogs and the {K}ronecker Product}, journal = {The Electronic Journal of Combinatorics} } - [Mar21]Mikołaj Marciniak. Quadratic coefficients of Goulden-Rattan character polynomials. arXiv:2104.13512, 2021.
.bib
@article{Marciniak2021GouldenRattan, author = {Miko{\l}aj Marciniak}, title = {Quadratic coefficients of {G}oulden-{R}attan character polynomials}, year = {2021}, eprint = {2104.13512}, url = {https://arxiv.org/abs/2104.13512}, journal = {arXiv e-prints} } - [NPPS21]Sridhar P. Narayanan, Digjoy Paul, Amritanshu Prasad and Shraddha Srivastava. Character Polynomials and the Restriction Problem. Algebraic Combinatorics, 4(4):703–722, 2021.
.bib
@article{NarayananPaulPrasadSrivastava2021character, author = {Sridhar P. Narayanan and Digjoy Paul and Amritanshu Prasad and Shraddha Srivastava}, title = {Character {P}olynomials and the {R}estriction {P}roblem}, journal = {Algebraic Combinatorics}, volume = {4}, number = {4}, pages = {703--722}, year = {2021}, doi = {10.5802/alco.176} } - [NPPS21]Sridhar P. Narayanan, Digjoy Paul, Amritanshu Prasad and Shraddha Srivastava. Polynomial induction and the restriction problem. Indian Journal of Pure and Applied Mathematics, 52(3):643–651, 2021.
.bib
@article{NarayananPaulPrasadSrivastava2021polynomial, author = {Sridhar P. Narayanan and Digjoy Paul and Amritanshu Prasad and Shraddha Srivastava}, title = {Polynomial induction and the restriction problem}, journal = {Indian Journal of Pure and Applied Mathematics}, volume = {52}, number = {3}, pages = {643--651}, year = {2021}, doi = {10.1007/s13226-021-00185-7} } - [OZ16]Rosa Orellana and Mike Zabrocki. Symmetric group characters as symmetric functions. arXiv:1605.06672, 2016.
.bib
@article{OrellanaZabrocki2016, Author = {Rosa Orellana and Mike Zabrocki}, Title = {Symmetric group characters as symmetric functions}, Year = {2016}, Eprint = {1605.06672}, url = {https://arxiv.org/abs/1605.06672}, journal = {arXiv e-prints} } - [OZ18]Rosa Orellana and Mike Zabrocki. The Hopf structure of symmetric group characters as symmetric functions. arXiv:1901.00378, 2018.
.bib
@article{OrellanaZabrocki2018, Author = {Rosa Orellana and Mike Zabrocki}, Title = {The {H}opf structure of symmetric group characters as symmetric functions}, Year = {2018}, Eprint = {1901.00378}, url = {https://arxiv.org/abs/1901.00378}, journal = {arXiv e-prints} }