2001/02/14 by Jason Fulman, Fulman, Jason
Chemistry · Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Molecular spectroscopy and chirality #math.CO #math.DS #math.GR
paper · pdf · doi:10.48550/arxiv.math/0102105
One change: we fix a typo in definition of f(m,k,i,d) on page 14
arxiv created 2001/05/09 · arxiv updated 2009/11/30
By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type A are found. This leads to new models of card shuffling and has interesting combinatorial and number theoretic consequences. An analysis of type C gives another solution to a problem of Rogers in dynamical systems: the enumeration of unimodal permutations by cycle structure. The proof uses the factorization theory of palindromic polynomials over finite fields. Contact is made with symmetric function theory.