Abstract
We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the \emphshifted plactic monoid. It can be defined in two different ways: via the \emphshifted Knuth relations, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.
Highlights
Knuth (6): the plactic classes [u] and [v] of two words u and v uniquely determine the plactic class [uv] of their concatenation. This gives the set of all plactic classes the structure of a plactic monoid P = P(X)
Cλμ,ν is equal to the number of pairs (Tμ, Tν) of plactic classes of shapes μ and ν such that TμTν = Tλ
We prove that the plactic Schur P -functions span the ring they generate, and this ring is canonically isomorphic to the ring spanned/generated by the ordinary Schur P -functions
Summary
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. We introduce a shifted analog of the plactic monoid of Lascoux and Schutzenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P -function; a shifted counterpart of the LascouxSchutzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more. [...] pour affirmer la necessited’installer le monoıde plaxique parmi les structures remarquables
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