Macdonald-RSK local move integrability


macdonald_polynomials macdonald_processes

Given an usual-RSK-type algorithm, let us write down the Local move integrability equation for the Macdonald processes.

Let (c.f. Explicit formula of Macdonald polynomials)

\[ g(n, k) = {(q t^k)_n \over (t^{k + 1})_n}. \]

Also given \(\Lambda\), let \(\alpha\) and \(\beta\) be the "weight / type / content" of the array \(s(\Lambda)\):

\begin{align} \alpha_1 &= t_{\Lambda'_1, 1} \\ \alpha_j &= \sum_{k = 1 : (j \wedge \Lambda'_j)} t_{\Lambda'_j - k + 1, j - k + 1} - \sum_{k = 1 : ((j - 1) \wedge \Lambda'_j)} t_{\Lambda'_j - k + 1, j - k}, \qquad j = 2 : \Lambda_1 \\ \beta_1 &= t_{1, \Lambda_1} \\ \beta_i &= \sum_{k = 1 : (i \wedge \Lambda_i)} t_{i - k + 1, \Lambda_i - k + 1} - \sum_{k = 1 : ((i - 1) \wedge \Lambda_i)} t_{i - k, \Lambda_i - k + 1} \qquad i = 2 : \Lambda'_1 \end{align}

In this setting, we have

Theorem. The local move integrability equation for Macdonald-RSK is thus

\[ \sum_{s'} {(t)_{s'_{n k}} \over (q)_{s'_{n k}}} {M_\Theta(s') \over M_\Lambda(s)} \prob(\rho_{n, k} s' = s) = 1. \qquad (1) \]

Proof. Plugging (1) in Local move integrability and simplify. \(\square\)