# The search for Macdonald RSK

robinson_schensted_knuth macdonald_polynomials local_move_integrability directed_last_passage_percolation

We have established in equivalence_m_rsk_md_lm_integrabilities that the Markov-Doob integrability and the local move integrability equations for the usual-spec Macdonald-RSK dynamics are the same.

However, it is not known if there exists a Macdonald-RSK dynamics that satisfy these equations.

In dlpp_dp_rsk, we associated the various RS(K) algorithms with various directed last passage percolation or polymer models, whereas in burke_property, we showed a less complicated integrability result (than the Markov-Doob integrability) called the Burke property.

A $$qt$$-deformation of the DLPP exhibiting a $$qt$$-deformation of the Burke property was also written down in that entry. That specific $$qt$$DLPP dynamics is natural and unique in a certain sense (see Claim 1 there).

A natural question thus arises: is there a dynamics satisfying the following conditions?

1. the first edge of the output GT pattern evolves as the qtDLPP,
2. it is an RSK-type dynamics, namely it is weight-preserving
3. and the Macdonald-RSK dynamics satisfies the usual-spec integrability equation for Macdonald processes,

Claim. The answer is no.

Proof. Consider the integrability equation when $$n = k = 2$$ (so the two paritions in (0) are $$\Theta = (2, 2)$$ and $$\Lambda = (2, 1)$$), namely in the language of local move integrability, when the local move $$\rho_{22}$$ transforms $$\begin{pmatrix} \lambda^1_1(1) & \lambda^1_1(2) \\ \lambda^2_1(1) & w_{22} \end{pmatrix}$$ to $$\begin{pmatrix} \lambda^2_2(2) & \lambda^1_1(2) \\ \lambda^2_1(1) & \lambda^2_2(1) \end{pmatrix}$$, where $$w_{22}$$ is the 22-entry of the input matrix.

We translate the three conditions. The first condition says (recall the $$qt$$DLPP dynamics in discrete_dlpp_dp and note our $$w_{ij}$$ is their $$a_{ij}$$ and c.f. dlpp_dp_rsk)

$\lambda^2_1(2) = w_{22} + \lambda^2_1(1) + \lambda^1_1(2) - \lambda^1_1(1) - X' \qquad (1)$

where $$X' \sim qt$$IHyp$$(\lambda^1_1(2) - \lambda^1_1(1), \lambda^2_1(1) - \lambda^1_1(1))$$ and this random variable determines $$\prob (\rho_{22} s' = s)$$ in (0).

The second condition says

$\lambda^2_1(2) + \lambda^2_2(2) - \lambda^1_1(2) = \lambda^2_1(1) - \lambda^1_1 (1) + w_{22}. \qquad (2)$

Combining this with (1) we have

$X' = \lambda^2_2(2), \qquad (3)$

thus

$\prob(\rho_{22} s' = s) = f_{\lambda^1_1(2) - \lambda^1_1(1), \lambda^2_1(1) - \lambda^1_1(1)} (\lambda^2_2(2)), \qquad (4)$

The third condition states that (0) is true. Denote $$a_1 = \lambda^2_1(2), a_2 = \lambda^2_2(2), b = \lambda^2_1(1), c = \lambda^1_1(2), d = \lambda^1_1(1)$$, $$g_k(n) = g(n, k)$$. By plugging (-1), (4), and note that $$s_{22}' = w_{22} = a_1 + a_2 - b - c + d$$, (0) becomes

\begin{align} &{g_1(a_1) g_0(b - a_2) g_0(a_1 - b) g_0(c - a_2) g_0(a_1 - c) \over g_0(a_1 - a_2) g_1(a_1 - a_2)}\\ &\qquad\times\sum_d g_0(d)^{-1} g_0(a_1 + a_2 - b - c + d)^{-1} \sum_k h_{1 / t} (c - a_2 - d - k) t^{c - a_2 - d - k} g_0(k + b - c)^{-1} g_0(k)^{-1} = 1. \qquad (5) \end{align}

Observe that $$g_0(j) = h_{1 / t} (j) = 0$$ whenever $$j < 0$$ due to the $$(q)_j$$ in the denominators, so any summand in the $$d$$- and $$k$$-sums vanishes when the instantiation of $$k$$ or $$d$$ results in negative values in the brakets in the sums.

Let $$a_1 = b = c = a_2 = 1$$. The only nonvanishing summand is when $$k = d = 0$$, which we plug in formula (5) to obtain the left hand side equals

$g_1(1) = {(qt)_1 \over (t^2)_1} = {1 - qt \over 1 - t^2} \neq 1.$

$$\square$$

## Acknowledgement

Thanks to Christian Krattenthaler who provided a counterexample $$(a_1, b, c, a_2) = (12, 5, 5, 2)$$ to show the falsehood of (5).