#Symplectic \(Q\) functions
The symplectic \(Q\) functions are the type \(C\) version of the Schur’s Q functions.
The symplectic \(Q\) functions \(\schurSpQ_{\lambda}(\xvec)\) are indexed by strict partitions, and may be defined as the \(t=-1\) specialization of the symplectic Hall–Littlewood polynomial, multiplied by a power of 2.
Let \(W_n\) be the \(\symS_n \ltimes (\setZ/2\setZ)^n,\) acting on variables by permuting indices and inverting. Let \(\lambda\) be a partition of length \(\leq n.\) The symplectic Hall–Littlewood polynomial is defined as \[\hallLittlewoodPsp_\lambda(\xvec;t) \coloneqq \frac{1}{v_\lambda^{(n)}(t)} \sum_{w \in W_n} w\left( \prod_{i=1}^n x_i^{\lambda_i} \prod_{i=1^n} \frac{1-tx_i^{-2}}{1-x_i^{-2}} \prod_{1 \leq i\lt j \leq n} \frac{1-tx_i^{-1}x_j}{1-x_i^{-1}x_j} \frac{1-tx_i^{-1}x^{-1}_j}{1-x_i^{-1}x^{-1}_j} \right).\] The normalizing constant \(v_\lambda^{(n)}(t)\) is defined as \[v_\lambda^{(n)}(t) \coloneqq \prod_{j=1}^{m_0} \frac{1-t^{2j}}{1-t} \cdot \prod_{k \geq 1} \prod_{j=1}^m \frac{1-t^{j}}{1-t}\] where \(m_k \coloneqq |\{i \in [n] : \lambda_i = k \}|.\)
Finally, we set \(\schurSpQ_{\lambda}(\xvec) \coloneqq 2^{\length(\lambda)}\hallLittlewoodPsp_\lambda(\xvec;-1).\)
Several properties for symplectic \(Q\) functions are proved in [Oka20]. Moreover, the author also introduces the skew factorial universal symplectic \(Q\) functions.
S. Yanagida introduces intermediate symplectic \(Q\)-functions, a Laurent-polynomial family between Schur \(Q\)-functions and S. Okada’s symplectic \(Q\)-functions [Yan22]. They are \(Q\)-function analogues of Proctor’s intermediate symplectic characters, and admit both a tableau-sum formula and a Józefiak–Pragacz-type Pfaffian formula.
#Tableau formula
In [Oka20], S. Okada proves a conjecture by King–Hamel [Conj. 3.1, KH07], which presents \(\schurSpQ_{\lambda}(\xvec)\) as a sum over shifted tableaux.
#Pieri rule
A Pieri rule for computing the coefficients in the expansion \[\schurSpQ_{\mu}(\xvec) \schurSpQ_{(r)}(\xvec) = \sum_{\mu} f^{\lambda}_{\mu,(r)} \schurSpQ_{\lambda}(\xvec)\] is also given in [Oka20] (with a combinatorial formula).
Conjecture (See [Conj. 6.1, Oka20]).
The multiplicative structure constants \(f^{\lambda}_{\mu,\nu}\) (indexed by strict partitions) defined via \[\schurSpQ_{\mu}(\xvec) \schurSpQ_{\nu}(\xvec) = \sum_{\mu} f^{\lambda}_{\mu,\nu} \schurSpQ_{\lambda}(\xvec)\] are non-negative integers.
Bibliography
- [KH07]Ronald King and A. Hamel. Combinatorial realisation of Hall–Littlewood polynomials at $t = -1$. 19th International conference on formal power series and algebraic combinatorics:1–12, 2007.
.bib
@inproceedings{KingHamel2007, author = {King, Ronald and Hamel, A.}, year = {2007}, pages = {1--12}, url = {http://igm.univ-mlv.fr/~fpsac/FPSAC07/SITE07/PDF-Proceedings/Posters/75.pdf}, title = {Combinatorial realisation of {H}all--{L}ittlewood polynomials at $t = -1$}, booktitle = {19th {I}nternational Conference on Formal Power Series and Algebraic Combinatorics}, venue = {Tianjin}, year = {2007} } - [Oka20]Soichi Okada. Symplectic Q-functions. arXiv:2007.04034, 2020.
.bib
@article{Okada2020x, Author = {Soichi Okada}, Title = {Symplectic {Q}-functions}, Year = {2020}, Eprint = {2007.04034}, url = {https://arxiv.org/abs/2007.04034}, journal = {arXiv e-prints} } - [Yan22]Shintarou Yanagida. Intermediate symplectic $Q$-functions. arXiv:2207.03354, 2022.
.bib
@article{Yanagida2022x, author = {Shintarou Yanagida}, title = {Intermediate symplectic {$Q$}-functions}, year = {2022}, eprint = {2207.03354}, url = {https://arxiv.org/abs/2207.03354}, journal = {arXiv e-prints} }