Bijective Proofs of Character Evaluations Using the Trace Forest of Jeu de Taquin

Abstract

International audienceIrreducible characters of the symmetric group are of special interest in combina-torics. They can be expressed either combinatorially using ribbon tableaux, or algebraically using contents. In this paper, these two expressions are related in a combinatorial way. We first introduce a fine structure in the famous jeu de taquin called " trace forest " , with the help of which we are able to count certain types of ribbon tableaux, leading to a simple bijective proof of a character evaluation formula in terms of contents that dates back to Frobenius (1901). Inspired by this proof, we give an inductive scheme that provides combinatorial proofs of more complicated character formulae in terms of contents