Local move integrability
local_moves
(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}].
- Let \((s_{ij})_{i, j \ge 1} \in K^{\pint_{>0} \times \pint_{>0}}\) be an infinite array on \(K\) (think of \(K\) as real or integers).
- Let \(\rho_{n, k}\) be a map on \(s\) such that \(\rho_{n, k}\) only alters the diagonal terms \((s_{n - i, k - i})_{i = 0 : n \wedge k - 1}\).
- Let \((w_{n, k})_{n, k \ge 1}\) be independent random variables with probability mass function \((p_{n, k})\), and we informally write \(\prob(w_{n, k}) = p_{n, k}(w_{n, k})\).
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.
- Let for any partition \(\Lambda\) let \(\mu_\Lambda(s)\) be a measure on \(K^\Lambda\).
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.
References
- [hopkins14] RSK via local transformations, , 2014.