Noncommutative Differentials and Yang-Mills on Permutation Groups Sn

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 ℰ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter11_189_1.tif"/> 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 ℰ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter11_189_2.tif"/> . We also construct Λ _N and ℰ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter11_189_3.tif"/> 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.