Spin-Preserving Knuth Correspondences for Ribbon Tableaux

The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in N, and standard Young tableaux by semistandard ones. For r ∈ N>0, the Robinson-Schensted correspondence can be trivially extended, using the r-quotient map, to one between r-coloured permutations and pairs of standard r-ribbon tableaux built on a fixed r-core (the Stanton-White correspondence). Viewing r-coloured permutations as matrices with entries in N (the non-zero entries being unit vectors), this correspondence can also be generalised to arbitrary matrices with entries in N and pairs of semistandard r-ribbon tableaux built on a fixed r-core; the generalisation is derived from the RSK correspondence, again using the r-quotient map. Shimozono and White recently defined a more interesting generalisation of the Robinson-Schensted correspondence to r-coloured permutations and standard r-ribbon tableaux; unlike the Stanton-White correspondence, it respects the spin statistic on standard r-ribbon tableaux, relating it directly to the colours of the r-coloured permutation. We define a construction establishing a bijective correspondence between general matrices with entries in N and pairs of semistandard r-ribbon tableaux built on a fixed r-core, which respects the spin statistic on those tableaux in a similar manner, relating it directly to the matrix entries. We also define a similar generalisation of the asymmetric RSK correspondence, in which case the matrix entries are taken from {0, 1}r. More surprising than the existence of such a correspondence is the fact that these Knuth correspondences are not derived from Schensted correspondences by means of standardisation. That method does not work for general r-ribbon tableaux, since for r ≥ 3, no r-ribbon Schensted insertion can preserve standardisations of horizontal strips. Instead, we use the analysis of Knuth correspondences by Fomin to focus on the correspondence at the level of a single matrix entry and one pair of ribbon strips, which we call a shape datum. We define such a shape datum by a nontrivial generalisation of the idea underlying the Shimozono-White correspondence, which takes the form of an algorithm traversing the edge sequences of the shapes the electronic journal of combinatorics 12 (2005), #R10 1