The Combinatorics of A2-webs

The nonelliptic A2-webs with k “+”s on the top boundary and 3n − 2k “−”s on the bottom boundary combinatorially model the space Homsl3(V,V) of sl3-module maps on tensor powers of the natural 3-dimensional sl3-module V, and they have connections with the combinatorics of Springer varieties. Petersen, Pylyavskyy, and Rhodes showed that the set of such A2-webs and the set of semistandard tableaux of shape (3n) and type {12, . . . , k, k+1, . . . ,3n−k} have the same cardinalities. In this work, we use the m-diagrams introduced by Tymoczko and the Robinson-Schensted correspondence to construct an explicit bijection, different from the one given by Russell, between these two sets. In establishing our result, we show that the pair of standard tableaux constructed using the notion of path depth is the same as the pair constructed from applying the Robinson-Schensted correspondence to a 3 2 1-avoiding permutation. We also obtain a bijection between such pairs of standard tableaux and Westbury’s A2 flow diagrams. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2011-0012398). †This research was supported by the Basic Science Research Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0022003). The hospitality of the Mathematics Department at the University of Wisconsin-Madison while this research was done is gratefully acknowledged. the electronic journal of combinatorics 21(2) (2014), #P2.25 1