On the structure constants of the descent algebra of the hyperoctahedral group

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson-Schensted-Knuth correspondence and some of its generalisations. This provides a new link between these results and the theory of quasisymmetric functions and allows to derive more advanced formulas involving Kronecker coefficients. Furthermore, using the theory of type B quasisymmetric functions introduced by Chow, we can extend our method to the hyperoctahedral group and derive some new formulas.