Littlewood-Richardson fillings and their symmetries

Abstract Considering the classical definition of the Littlewood-Richardson rule and its 2-dimensional representation by means of rectangular tableaux, we exhibit 24 symmetries of this rule when considering dualization, conjugation and their composition. Extending the Littlewood-Richardson rule to sequences of nonnegative real numbers, six of these symmetries may be generalized. Our point is to stress the role of different Littlewood-Richardson fillings, opposite (or increasing) [1, 2, 12] and column, [1, 2, 7] in guiding these symmetries. The main result is a bijection in the set of Littlewood-Richardson rectangular tableaux which transforms LittlewoodRichardson fillings of type [a, b, c] into [b, a, c]. This bijection is based on the projection of Littlewood-Richardson tableaux of order r into Littlewood-Richardson tableaux of order r − 1, for each r ∈ N.