A new Littlewood-Richardson rule for Schur 𝑃-functions

Abstract

A new description of shifted Littlewood-Richardson coefficients is given in terms of semistandard decomposition tableaux which were recently introduced by L. Serrano. We also show that the set of semistandard decomposition tableaux is invariant under the action of Lascoux-Schützenberger involution, providing a combinatorial proof of the symmetry of Schur P P -functions. We find counterexamples to the conjecture made by L. Serrano on skew Schur P P -functions, proving the falsity of the conjecture. Many combinatorial properties of semistandard decomposition tableaux are also shown.