Arrays and the octahedron recurrence

Details Arrays and the octahedron recurrence Arrays and the octahedron recurrence

2005-04-14

arxiv.org/abs/math/0504299v1

Mathematics

Combinatorics

34 pages, 11 figures

Scientific paper

Recently, in papers by Knutson, Tao and Woodward, Henriques and Kamnitzer, Pak and Vallejo have been constructed several interesting bijections of associativity and commutativity. In the first two papers bijections relate special sets of discretely concave functions (hives) on triangular grids and the octahedron recurrence plays the key role for these bijections. Pak and Vallejo related special sets of Young tableaux and constructions of these bijections based on standard algorithms in this theory, jeu de taquen, Schutzenberger involution, tableaux switching, etc. In this paper we investigate these constructions from the third point of view, combinatorics of arrays, theory worked out recently by the authors. Arrays naturally related as well to functions on the lattice of integers as to Young tableaux. In the tensor category of arrays, the bijections of associativity and commutativity arise naturally. We establish coincidence of our bijections with that defined in the first two papers and in the integer-valued set-up with the bijection in the third paper (that is, in particular, a solution of Conjecture 1 by Pak and Vallejo). In order to relate different approaches and to reveal combinatorics of the octahedron recurrence, we, first, show that the octahedron recurrence agrees with discrete convexity and, second, we construct another bijection using the octahedron recurrence, the functional form of the RSK correspondence.

Affiliated with

Also associated with

No associations

Rating

Arrays and the octahedron recurrence does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Arrays and the octahedron recurrence, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Arrays and the octahedron recurrence will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-146080