# 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.