The Schubert bialgebra

We introduce a Schubert bialgebra $\mathcal{A}={⨁}_{n\ge 0}\mathbb{Z}[x_1,\dots ,x_n]$, 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 concatenation of weak compositions. The Schubert polynomials ${\mathfrak{S}}_u^n$ form a basis of $\mathcal{A}$; we give an explicit formula for the dual Schubert basis ${\mathrm{\Xi }}_u^n$ of $\mathcal{D}$ via a transition formula to an elementary symmetric basis, and show that the Schubert structure constants $c_{u,v}^w$ are the coefficients of the coproduct of $\mathcal{D}$ in the dual Schubert basis. The product of $\mathcal{D}$ is then lifted to an associative product $\bigsqcup$ on the free abelian group $\mathcal{B}\mathcal{R}\mathcal{C}$ of bounded RC graphs, built from a transition map $\mathcal{Z}$ on RC graphs that realizes the Lascoux-Schützenberger transition formula and generalizes Little’s bumping algorithm. Quotienting $\mathcal{B}\mathcal{R}\mathcal{C}$ by ideals coming from Demazure crystal equivalence, Nadeau-Tewari forest equivalence, and quasi-Yamanouchi equivalence yields manifestly positive Littlewood-Richardson rules for the dual Schubert, dual key, dual forest, and dual slide bases of $\mathcal{D}$. Read on $\mathcal{A}$, these are branching rules: enumerative formulas for splitting a Schubert, key, forest, or slide polynomial across two blocks of variables, and in the Schubert case they compute the Levi branching coefficients of Ressayre-Richmond for ${\mathrm{G}\mathrm{L}}_p\times {\mathrm{G}\mathrm{L}}_q\hookrightarrow {\mathrm{G}\mathrm{L}}_{p+q}$.