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Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted-type correspondence for quasi-ribbon tableaux | | Alan J. Cain
; António Malheiro
; | | Date: |
24 Jan 2016 | | Abstract: | 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 $q$-analogue of the special linear Lie algebra
$mathfrak{sl}_{n}$, 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 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 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. | | Source: | arXiv, 1601.6390 | | Services: | Forum | Review | PDF | Favorites | |
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