Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson–Schensted–Knuth-type correspondence for quasi-ribbon tableaux
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- 作者：Alan J. Cain ; António Malheiro
- 关键词：Hypoplactic ; Quasi ; ribbon tableau ; Robinson–Schensted–Knuth correspondence ; Kashiwara operator ; Crystal graph
- 刊名：Journal of Algebraic Combinatorics
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：45
- 期：2
- 页码：475-524
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Combinatorics; Convex and Discrete Geometry; Order, Lattices, Ordered Algebraic Structures; Computer Science, general; Group Theory and Generalizations;
- 出版者：Springer US
- ISSN：1572-9192
- 卷排序：45
文摘
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–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 for 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.