GENERATING FUNCTIONS FOR HECKE OPERATORS

Generating Functions for Hecke Algebra Characters

Certain polynomials in n variables which serve as generating functions for symmetric group characters are sometimes called (Sn) character immanants. We point out a close connection between the identities of Littlewood-Merris-Watkins and Goulden-Jackson, which relate Sn character immanants to the determinant, the permanent and MacMahon’s Master Theorem. From these results we obtain a generalizat...

Hecke Operators on Rational Functions

Contents 1. Introduction 1 2. Some preliminaries 5 3. The point spectrum of U p 7 4. A structure theorem for eigenfunctions 14 5. A decomposition into finite dimensional eigenspaces 21 6. Simultaneous eigenfunctions 23 7. A first application: tensor products of Hecke operatorsand the Riemann zeta function 27 8. A second application: completely multiplicative coefficients 30 9. Appendix: Explici...

Hecke Operators

Probability Generating Functions for Sattolo’s Algorithm

In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. Recently, H. Prodinger analysed two important random variables associated with the algorithm, and found their mean and variance. H. Mahmoud extended Prodinger’s analysis by finding limit laws for the same two random variables.The present article, starting from the ...

Gaussian Hypergeometric Functions and Traces of Hecke Operators

We establish a simple inductive formula for the trace Tr k (Γ0(8), p) of the p-th Hecke operator on the space S k (Γ0(8)) of newforms of level 8 and weight k in terms of the values of 3F2-hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conjecture of Koike relating Tr 6 (Γ0(8), p) to the values 6F5(1)p and 4F3(1)p. Furthermore, we find new congruences...