2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61561We study noncommutative differential structures on the group of permutations $S_N$, defined by conjugacy classes. The 2-cycles class defines an exterior algebra $Λ_N$ which is a super analogue of the Fomin-Kirillov algebra $\CE_N$ for Schubert calculus on the cohomology of the $GL_N$ flag variety. Noncommutative de Rahm cohomology and moduli of flat connections are computed for $N<6$. We find that flat connections of submaximal cardinality form a natural representation associated to each conjugacy class, often irreducible, and are analogues of the Dunkl elements in $\CE_N$. We also construct $Λ_N$ and $\CE_N$ as braided groups in the category of $S_N$-crossed modules, giving a new approach to the latter that makes sense for all flag varieties.Final version to appear Marcel Dekker Lect. Notes Pure Appl. Maths; improved intro and moved some technical material to an appendixQuantum AlgebraAlgebraic Geometry58B32, 58B34, 14N15text