The Schubert bialgebra

We introduce a Schubert bialgebra $\mathcal{A}=\bigoplus_{n\geq 0}\mathbb{Z}[x_1,\ldots,x_n]$, in which polynomials in different numbers of variables multiply to zero and the coproduct splits the variables into two consecutive blocks, and study its graded dual $\mathcal{D}$, whose product is concate