Keys associated with the symmetric group S4 and matrix realizations of pairs of tableaux

Abstract

A variant of the dual RSK correspondence [10, 12] gives a bijection between classes of skew-tableaux and tableau-pairs of conjugate shapes. The problem of a matrix realization, over a local principal ideal domain with prime p, of the pair (T ,K(σ)) with K(σ) a key associated with the permutation σ ∈ St, and T a skew-tableau with the same evaluation as K(σ), is addressed. If T corresponds by this variant of the dual RSK to the tableau-pair (P,Q) of conjugate shapes, there exists a matrix realization for (T ,K(σ)), σ ∈ St, only if P = K(σ) [2, 4, 5, 6]. This necessary condition has also been proved to be sufficient [7], by exhibiting an explicit matrix realization, in the case the frank word σQ is a union of row words whose lengths define the conjugate shape of Q. Here, we extend the matrix realization given in [7] to any tableau-pair (K(σ),Q) of conjugate shapes, with σ ∈ S4. This is carried out by stretching the frank words with shape (2, 1, 1, 2) which are not the union of one row of length four with one of length two, those associated to 431421 ≡ K(ǫ, (1, 0, 1, 0)), with ǫ ∈ {1423, 1432, 4123, 4132}, to row words of length six associated with the key 654321 in S6.