International conference
Combinatorial Methods in Physics and Knot Theory

Gleb KOSHEVOY (CEMI and Poncelet laboratory, Moscow): Combinatorics of Young tableaux and octahedral recurrence

The octahedron recursion is a rule for propagation a function given at 5 vertices of a unitary octahedron to the 6th vertex, $f(x+1,y,z+1)=\max (f(x,y,z+1)+f(x+1,y,z), f(x,y-1,z)+f(x+1,y+1,z+1))-f(x,y,z))$, the ultradiscretezation of the Hirota equation. Recently, using this propagation rule for a function with special (discretely concave) values at two adjoint facets of the tetrahedron, Knutson, Tao and Woodward, established associativity of the Hives category, that is indeed an interesting bijection between two sets of special pairs of Young tableaux (we call them Standard pairs). Our main result is a construction of the modified RSK using the octahedron recursion on a prism with boundary values at 3 facets. As a consequence we obtain that the KTW bijection coincides with the bijection constructed by the authors for associativity of the array category. This bijection might be also seen as a flip for Thompson group. More intriguing connections occur for the commutativity bijection (R-matrix). We prove that our commutativity bijection in the array category coinsides with the bijection due to Henriques and Kamnitzer for Hives category and the first and second fundamental symmetries due to Pak and Vallejo.