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Báo cáo toán học: "Spin-preserving Knuth correspondences for ribbon tableaux"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Spin-preserving Knuth correspondences for ribbon tableaux. | Spin-preserving Knuth correspondences for ribbon tableaux Marc A. A. van Leeuwen Universite de Poitiers Departement de Mathematiques UFR Sciences SP2MI Teleport 2 BP 30179 86962 Futuroscope Chasseneuil Cedex France Marc.van-Leeuwen@math.univ-poitiers.fr http www-math.univ-poitiers.fr maavl Submitted Dec 1 2003 Accepted Jan 3 2005 Published Feb 14 2005 Mathematics Subject Classifications 05E10 Abstract 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 2 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 Nr the non-zero entries being unit vectors this correspondence can also be generalised to arbitrary matrices with entries in 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 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 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. More surprising than the existence of such a correspondence