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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 Young tableaux, triangles, hives or honeycombs have
been considered. Littlewood-Richardson coefficients hide several symmetries
and there are several Young tableau bijections for these
symmetries whose relationship is sometimes obscure.
I. Pak and E. Vallejo have called fundamental
symmetry to any Young
tableau bijection exhibiting the commutativity, and they have
conjectured that four of them
are equivalent. Three are based on standard algorithms
on Young tableau theory. V. I. Danilov and G. A. Koshevoy
have proved the conjecture for those three. We show
that the fourth one is equivalent to the one based on the
switching-tableau operation.
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