RSK-type dynamics

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robinson_schensted_knuth

Recall One-one correspondence between Young tableaux and Gelfand-Tsetlin patterns.

The RSK-type dynamics is a kind of weight-preserving dynamics on the space of (semi-standard) Young tableau / Gelfand-Tsetlin patterns.

Like the usual RSK, the basic building block is the insertion of a row to a Young tableau.

Unlike the usual RSK, it is generalised to having random output, namely it maps a deterministic tableau to a random tableau.

Definition. An RSK-type dynamics is a random map \(R\): \((r_{1 : \ell}, (\lambda^k_j)_{1 \le j \le k \ell}) \mapsto (\tilde\lambda^k_j)_{1 \le j \le k \ell}\) satisfying

\[ | \lambda^k | - | \lambda^{k - 1} | + r_k = | \tilde\lambda^k | - | \tilde\lambda^{k - 1} | \qquad (1) \]

Basically it says the following:

For any integer \(k\), the number of \(k\)s in the old tableau plus the number of inserted \(k\)s equals the number of \(k\)s in the new tableau.

If we introduce time index \(n\), namely suppose at time \(n\), \(r = w_{n, 1 : \ell}\) is inserted to \(\lambda^{n - 1, 1 : \ell}\) to produce \(\lambda^{n, 1 : \ell}\), the weight-preserving property (1) can be written in a more symmetric manner

\[ |\lambda^{n - 1, k - 1}| + |\lambda^{n, k}| - |\lambda^{n - 1, k}| - |\lambda^{n, k - 1}| = w_{n, k}. \qquad \text{a.s.} \]

Example. RSK, qRSK and gRSK.