On the Descent Algebra of Type $D$

Here we give a combinatorial interpretation of Solomon’s rule for multiplication in the descent algebra of Weyl groups of type $D$, $\mathrm{\Sigma }{D}_n$. From here we show that $\mathrm{\Sigma }{D}_n$ is a homomorphic image of the descent algebra of the hyperoctahedral group, $\mathrm{\Sigma }{B}_{n-2}$.