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The shifted plactic monoid

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arxiv 0811.2057 v2 pith:3ABK5AOX submitted 2008-11-13 math.CO

classification math.CO
keywords shiftedplacticmonoidschuremphutzenbergeranalogapplications
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abstract

We introduce a shifted analog of the plactic monoid of Lascoux and Sch\"utzenberger, the \emph{shifted plactic monoid}. It can be defined in two different ways: via the \emph{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 Lascoux-Sch\"utzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

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