Applications of the Brauer Complex: Card Shuffling, Permutation Statistics, and Dynamical Systems

✍ Scribed by Jason Fulman

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 187 KB
• Volume: 243
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.8814

✦ Synopsis

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.