The symmetric functions catalog

An overview of symmetric functions and related topics


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

\begin{equation*} \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). \end{equation*}

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.

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.