Character Rings and Hecke Algebras

In this project, we are going to study some results concerning the character rings and Heche Algebras in Sk and GL(k, F ). In next section, we will introduce some notation system in Sk and GL(k, F ) first, and with that, we will deduce some interesting and important results in this field, such as the results about the length function on W and the double B − B cosets, where W is the Weyl group of GL(k, F ). In Section 3, we will show a basic result concerning the WI-WJ double cosets in W and the PI-PJ double cosets in GL(k, F ). After this a beautiful connection between Rk and Rk(q) follows, where Rk is the character ring of Sk and Rk(q) is the character ring of GL(k,Fq). In Section 4, we move to the Hecke algebras H and Hk(q), where H is the convolution ring of B-bi-invariant functions on GL(k,Fq). Based on the properties of H, we will present the more general definition of Hk(q), with q any complex number and as we will see, if q is prime power, then H and Hk(q) coincide, which reaches the end of this project. Statement: All of the materials in this project come from Bump’s book, [1]; concretely, it occupies the whole Chapter 48, main part of Chapter 30, half of Chapter 28 and 21, and the main theorem in Chapter 37. Except the Jacobi-Trudi Identity, we deduce all of the results from the beginning to the end. Besides, all of these chapters in Bump’s are based on the concept root system, however, in this specific case, we do not need its general definition and I successfully avoid the discussion about it throughout this project.