A Proof of Solomon’s Rule

We put forward a proof of Solomon’s rule, in terms of matrices, for multiplication in the descent algebra of the symmetric group. Our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions, {(i, i + 1)} i=1 . Let W be a Coxeter group with generating set, S, of fundamental reflections. If J is any subset of S, let WJ be the subgroup generated by J . Let XJ be the unique set of minimal length left coset representatives of WJ . Note that X J = {x |x ∈ XJ} is the unique set of minimal length representatives for the right cosets of WJ . Let l(y) denote the length of y in W . Solomon [9] then gives us the following theorem: Theorem 1 For every subset K of S, let