Macdonald-RSK local move integrability

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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

• \(\mu_\Lambda(s) := M_\Lambda(s) \prod_{ij} {(a_i b_j)_\infty \over (a_i b_j t)_\infty} a^\alpha b^\beta\),
  • where \(M_\Lambda(s) := \prod_i g(s_{ii}, i - 1)^{-1} {\prod_{ijk} g(s_{ij} - s_{i - k, j - k}, k - 1) g(s_{ij} - s_{i - k, j - k}, k) \over \prod_{ij\ell} g(s_{ij} - s_{i - \ell, j - \ell + 1}, \ell - 1) \prod_{ij\ell}g(s_{ij} - s_{i - \ell + 1, j - \ell + 1}, \ell - 1)}\).
  • When \(\Lambda = (n^k)\) is a rectangular Young diagram, summing over non-diagonal terms \(\{s_{i, j}: n - i \neq k - j\}\) results in the Macdonald measure for the diagonal terms \(\lambda = (s_{n, k}, s_{n - 1, k - 1}, ..., s_{(n - k)^+ + 1, (k - n)^+ + 1})\): \[ \sum_{s_{i, j}: n - i \neq k - i} \mu_\Lambda(s) = P_\lambda(a) Q_\lambda(b) \prod_{ij} {(a_j b_i)_\infty \over (a_j b_i t)_\infty}. \]
• \(w_{n, k} \sim qt\text{Geom}(a_n b_k)\), the qt-deformed geometric distribution.
• \(\rho_{n, k}\) are unknown / to be found.

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. \(\square\)