Iwahori-Hecke algebras and beyond

For the irreducible representations of the Hecke algebras, the minimal elements in each conjugacy class play an important role. In this talk, we try to review the minimal length elements and characterize in a more efficient way to find the maximal elements in each conjugacy class, which in turn, means the minimal elements in each conjugacy class by a given automorphism of the Weyl Group, treating type A mainly