We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem”. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a ...

We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of “specia...

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 for Schubert calculus on the cohomology of the GL N flag variety. Noncom-mutative de Rahm cohomology and moduli of flat connections are computed for N < 6. We find that flat...

Let G be a reductive group over an algebraically closed field whose characteristic is not a bad prime for G. Let w be an elliptic element of the Weyl group which has minimum length in its conjugacy class. We show that there exists a unique unipotent class X in G such that the following holds: if V is the variety of pairs (g,B) where g ∈ X and B is a Borel subgroup such that B, gBg−1 are in rela...

For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G. Although numerically equal, there is no general formula to map the conjugacy classes of a group to its irreducible representations. For the case of the symmetric group Sn, however, there is a remarkably simple correspondence; we will see that each irreducible...