Reading Class 290-084: Hecke Algebras and Orthogonal Polynomials

Fall 2004

Meeting:	The reading class meets Fridays 4:10pm in Kerr 693 during the Fall Quarter.
Instructor:	Monica Vazirani, Kerr Hall 579, phone: 752-2218, vazirani@math.ucdavis.edu
CRN:	49762 OOPS , someone told me two numbers were transposed here (as 49672 previously).
Grading:	Students who plan to attend regularly and participate in the discussion can register for one credit. Students who will present a paper in class can register for three credits.
Web:	http://www.math.ucdavis.edu/~vazirani/F04/290.html

Course description

This reading seminar runs in the context of the Research Focus Group on "Hecke Algebras and Orthogonal Polynomials". We will read and discuss topics of mutual interest. Suggested topics are:

review of symmetric function. The monomial, power, and Schur symmetric functions. the scalar product
Schur functions as orthogonal polynomials. Jack polynomials. Macdonald polynomials as orthogonal polynomials.
Macdonald polynomials as simultaneous eigenfunctions of operators and existence proof.
Review of other root systems, in particular type BC, and the Askey-Wilson polynomials
The double affine Hecke algebra.
Macdonald polynomials as simultaneous eigenfunctions of Dunkl-like operators in the double affine Hecke algebra.
More on the double affine Hecke algebra. Related work of Cherednik, Opdam, Gordon, etc.
Other viewpoints of Macdonald polynomials, as in Haiman's work. Connection to diagonal harmonics. A combinatorial description.

Suggested texts/papers are:

Macdonald ``Symmetric functions and orthogonal polynomials,'' QA212.M33
Koornwinder ``Askey-Wilson polynomials for root systems of type $BC$,'' QA353.H9 H97
Hypergeometric functions on domains of positivity, Jack polynomials, and applications
(Tampa, FL, 1991), 189--204, Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992.
Kirillov ``Lectures on affine Hecke algebras and Macdonald's conjectures,''
Charles Dunkl and Yuan Xu "Orthogonal polynomials of several variables." QA404.5.D86
Encyclopedia of Mathematics and its Applications, 81. Cambridge University Press, Cambridge, 2001.

Further suggestions are welcome!

Schedule Fall

October 1: Organizatorial meeting
October 8: Dan Rutherford on the basics of symmetric functions
October 15: Dan continued.
Gus on Jack and Macdonald polynomials as orthogonal polys
October 22: Gus continued
October 29: Gus continued
(notice Saturday, Oct 30 was the bay area discrete math day). Thanks to all who attended.
November 5: Yvonne Lai on root systems
November 12: Monica on another existence proof of Macdonald polys
November 19: James Parmenter on the double affine Hecke algebra
December 3:Lipika Deka on Koornwinder polynomials; type BC
December 10: James continued

Organizatorial meeting, what are people thinking for next quarter??