Geodesic
An explicit bijection between semistandard tableaux and non-elliptic $sl _{3}$ webs
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 4, pp. 851-862.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The $sl _{3}$ spider is a diagrammatic category used to study the representation theory of the quantum group $U _{q }(sl _{3})$. The morphisms in this category are generated by a basis of non-elliptic webs. M. Khovanov and G. Kuperberg [Pac. J. Math. 188, No. 1, 129--153 (1999; Zbl 0929.17012)] observe that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, J. Tymoczko [J. Algebr. Comb. 35, No. 4, 611--632 (2012; Zbl 1242.05277)] gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and join of webs. This article builds on Tymoczko's bijection to give a simple and explicit algorithm for constructing all non-elliptic $sl _{3}$ webs. As an application, we generalize results of T. K. Petersen et al. [J. Algebr. Comb. 30, No. 1, 19--41 (2009; Zbl 1226.05260)] and Tymoczko [loc. cit.] proving that, for all non-elliptic $sl _{3}$ webs, rotation corresponds to jeu de taquin promotion and join corresponds to shuffling.
Classification : 05E10, 17B15, 17B37
Keywords: spider category, Young tableaux, jeu de taquin 
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Russell, Heather M. An explicit bijection between semistandard tableaux and non-elliptic $sl _{3}$ webs. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 4, pp. 851-862. http://geodesic.mathdoc.fr/item/JAC_2013__38_4_a7/