2003/12/01 by Marc A. A. Van Leeuwen, Marc van Leeuwen, Van Leeuwen, Marc A. A.
Mathematics · #05E10 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications #math.CO #msc:05E10
paper · pdf · doi:10.48550/arxiv.math/0312020
arxiv created 2003/12/01 · openalex publication_date 2003/12/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in \bf N, and standard Young tableaux by semistandard ones. For r>0, the Robinson-Schensted correspondence can be trivially extended, using the r-quotient map, to one between coloured permutations and pairs of standard r-ribbon tableaux built on a fixed r-core (the Stanton-White correspondence). This correspondence can also be generalised to arbitrary matrices with entries in \bf Nr 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 coloured permutations and standard r-ribbon tableaux, one that (unlike the Stanton-White correspondence) respects the spin statistic (total height of ribbons) on standard r-ribbon tableaux, relating it directly to the colours of the coloured permutation. We define a construction establishing a bijective correspondence between general matrices with entries in \bf Nr 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.