Noncommutative Differentials and Yang-Mills on Permutation Groups S_N

Abstract

We 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.