Definition of Macdonald polynomial $P_\lambda^{\mathfrak{g}}$ associated to a Lie algebra $\mathfrak{g}$ (unlike $P_\lambda$)

Question

I want to find the definition for the Macdonald polynomial associated to a Lie algebra $\mathfrak{c}_n$, i.e. $P_\lambda^{\mathfrak{c}_n}(x,t,q)$. This appears in the physics paper https://arxiv.org/pdf/2005.12282.pdf, equation 2.9 and the explanation in the paragraphs below, citing Macdonald's book $\textit{Symmetric functions and orthogonal polynomials}$. My understanding is that for each partition $\lambda$ (i.e. Young diagram), there is a corresponding Macdonald polynomials $P_\lambda (x,q,t)$. The literature seems to indicate that this is implicitly the Macdonald polynomials of type $A_n$, i.e. $P_\lambda^{\mathfrak{a}_n}(x,q,t)$.

Ultimately, I want to know how to compute these $P_\lambda^{\mathfrak{c}_n}$, where I know how to compute $P_\lambda$ (or $P_\lambda^{\mathfrak{a}_n}$ if I am not wrong). In short, what is $P_\lambda^{\mathfrak{c}_n}$?

Answer 1

In Macdonald's book, as you indicated, he is implicitly working with type $A_{n}$ throughout. The key for Macdonald polynomials of type $C_{n}$ as opposed to $A_{n}$ lies in understanding Equation (9.10) on p.372, which is written specifically for type $A_{n}$. More generally

\begin{equation} \langle f,g \rangle_{q,t}^{W} = \frac{1}{|W|}\int_{T}f(x)\overline{g(x)}\Delta(q,t) dz. \end{equation}

Here the density function also needs to change from the standard Macdonald density to the generalized Macdonald density. You will find this as Equation (3.3) in Macdonald's paper Orthogonal polynomials associated with root systems (2000). So, the more general Macdonald polynomials $P_{\lambda}^{W}(x;q,t)$ correspond to a modified version of Theorem (4.7) on p. 322 of Macdonald's book. They are the unique family of $W$-symmetric functions such that

\begin{equation} P_{\lambda}^{W}(q,t) = m_{\lambda}^{W} + \sum_{\mu < \lambda}u_{\lambda\mu}(q,t)m_{\mu}^{W}, \end{equation}

such that

\begin{equation} \langle P_{\lambda}(q,t), P_{\mu}(q,t) \rangle_{q,t}^{W} = 0, \text{ whenever } \lambda \neq \mu. \end{equation}

Here the $m^{W}$ are generalizations of the monomial symmetric functions you are familiar with from $A_{n}$. They can be defined by

\begin{equation} m_{\lambda}^{W} = \sum_{\alpha \in \mathop{Orb}(\lambda,W)} x^{\alpha}. \end{equation}

For $A_{n}$ these are simply permutations of monomials. For $C_{n}$ these are instead signed permutations as $W$ is instead the hyperoctahedral group. For example

\begin{equation} m_{(1)}(x_{1},x_{2},x_{3}) = x_{1} + x_{1}^{-1} + x_{2} + x_{2}^{-1} + x_{3} + x_{3}^{-1}. \end{equation}