The search for Macdonald RSK

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robinson_schensted_knuth macdonald_polynomials local_move_integrability directed_last_passage_percolation

• Recall the local move integrability equation (1) in macdonald_rsk_local_move_integrability, which we refer as (0).
• Recall the formula for \({M_\Theta (s') \over M_\Lambda(s)}\) in formula (2) of equivalence_m_rsk_md_lm_integrabilities, which we refer as (-1).
• Recall the \(qt\)-infhypergeometric distribution and its pmf \(f_{u, v}\) in (3.2) of burke_property.
• Recall \(g(n, k)\) in macdonald_polynomials_formula.
• Recall \(h_t(n)\) in burke_property.

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

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

\(\square\)

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