Hecke Algebras and Orthogonal Polynomials

Research Focus Group

Department of Mathematics

University of California Davis

Both Hecke algebras and orthogonal polynomials are a rich source of combinatorics. Furthermore, the relationship between the two topics is deep and interesting and has led to exciting results in current research. Orthogonal polynomials and special functions have applications to many areas, such as quantum integrable systems, harmonic analysis of simple Lie groups,

The general theme of the RFG is to introduce students to areas of research available in the department, such as orthogonal polynomials, elliptic hypergeometric functions, symmetric functions, Coxeter groups, combinatorics, the representation theory of Hecke algebras, Kazhdan-Lusztig theory, crystal base theory, and applications in physics.

One quarter will be spent studying Hecke algebras, another studying orthogonal polynomials, and another developing the connections between the two topics. While some connections will be pointed out along the way, students and faculty could
easily participate in the second quarter's activities without having been involved in those of the first quarter.

In the reading seminar we expect active participation by students who will take turns to present chapters of the books or articles
of common interest. In addition we expect the students with combinatorial interest to participate in the Bay Area Discrete Math day in October of 2004 and March 2005.

Suggested books and papers include:

Macdonald ``Symmetric functions and orthogonal polynomials,'' QA212.M33
Szegő ``Orthogonal Polynomials,''
( Colloquium publications (American Mathematical Society) ; v. 23) Shields Library QA404.5 .S9
Andrews, Askey, Roy ``Special functions,"
( Encyclopedia of mathematics and its applications ; v. 71) Shields Library QA351 .A74
Koornwinder ``Askey-Wilson polynomials for root systems of type $BC$,''
Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), 189--204, Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992. QA353.H9 H97 1992
Kirillov ``Lectures on affine Hecke algebras and Macdonald's conjectures,''
Kazhdan-Lusztig, ``Representations of Coxeter groups and Hecke algebras.''

Charles Dunkl and Yuan Xu "Orthogonal polynomials of several variables. "
(Encyclopedia of Mathematics and its Applications, 81. Cambridge University Press, Cambridge, 2001. . ISBN 0-521-80043-9 ) QA404.5.D86