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Description: |
Schur functions constitute a distinguished basis for the ring of the symmetric functions. The rule of Littlewood-Richardson gives the decomposition of a product of two Schur functions in the basis of the same functions and the coefficients are the so called Littlewood-Richardson coefficients.
Different combinatorial interpretations of these coefficients in terms of certain Young tableaux and triangular grids like
Berenstein-Zelevinsky triangles or hives, have been considered. The relationship between these constructions is now understood. On the other hand, as Littlewood-Richardson coefficients hide several symmetries, there are several
bijections, on Young tableaux, of these
symmetries and the relationship between them is sometimes obscure.
In the case of bijections of the commutativity of Littlewood-Richardson coefficients, $c^{\lambda}_{\mu,\nu}= c^{\lambda}_{\nu,\mu}$, I. Pak and E.
Vallejo have conjectured that four of them are equivalent. V. I. Danilov and G. A. Koshevoy have proved this conjecture for three of them. In this talk, the fourth one will be presented in the language of Young tableaux as well as its translation to triangular grids defining Gelfand-Tsetlin patterns with restrictions on the boundary.
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