438 research outputs found
Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted--Knuth-type correspondence for quasi-ribbon tableaux
Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure
given by identifying words labelling vertices that appear in the same position
of isomorphic components of the crystal. In the particular case of the crystal
graph for the -analogue of the special linear Lie algebra
, this monoid is the celebrated plactic monoid, whose
elements can be identified with Young tableaux. The crystal graph and the
so-called Kashiwara operators interact beautifully with the combinatorics of
Young tableaux and with the Robinson--Schensted--Knuth correspondence and so
provide powerful combinatorial tools to work with them. This paper constructs
an analogous `quasi-crystal' structure for the hypoplactic monoid, whose
elements can be identified with quasi-ribbon tableaux and whose connection with
the theory of quasi-symmetric functions echoes the connection of the plactic
monoid with the theory of symmetric functions. This quasi-crystal structure and
the associated quasi-Kashiwara operators are shown to interact just as neatly
with the combinatorics of quasi-ribbon tableaux and with the hypoplactic
version of the Robinson--Schensted--Knuth correspondence. A study is then made
of the interaction of the crystal graph of the plactic monoid and the
quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal
structure is applied to prove some new results about the hypoplactic monoid.Comment: 55 pages. Minor revision to fix typos, add references, and discuss an
open questio
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