Local move integrability


(So far local moves are only seen in RSK, gRSK and \(q\)RSK, but let us try to generalise it. In generality "local move integrability" may not be a good name because \(\rho\) does not have to decompose into mpas that are local, but let us keep it until we find some non-local type.)

(We do not know the local move integrability for either dual-spec algorithms or Warren-type processes so we don't generalise it above the usual-spec RSK-type algorithms.)

The local move integrability is a symmetric treatment to the usual-RSK-type integrabilities, and generalises the usual-RSK-type algorithms from taking matrix input, to taking tableaux input.

In the case of RSK algorithm it is related to the ocotahedron recurrence [{hopkins14}].

For a partition \(\Lambda\) with a growth sequence

\[ \emptyset \nearrow \Lambda(1) \nearrow \Lambda(2) \dots \nearrow \Lambda(N - 1) \nearrow \Lambda \]

We define \(T_\Lambda\) by

\[ T_\Lambda = \rho_{\Lambda / \Lambda(N - 1)} \circ \dots \circ \rho_{\Lambda(2) / \Lambda(1)} \circ \rho_{\Lambda(1) / \emptyset}. \]

We assume this definition does not depend on the growth sequence.

Definition. The local move integrability is

\[ \sum_{w_{n, k}, s'} \prob(w_{n, k}) \mu_\Lambda(s') \prob(\rho_{\Theta / \Lambda} s' = s) = \mu_\Theta(s). \qquad (1) \]

Theorem. Given \(A = (w_{n, k})\), the local move integrability implies

\[ \prob(T_\Lambda A(\Lambda) = s) = \mu_\Lambda(s). \]

Proof. Just apply (1) recursively. \(\square\)

Examples of local move integrability include RSK, gRSK, \(q\)RSK and (hypothetically) Macdonald RSK.