Hall-Littlewood polynomial, root system, affine Weyl group, alcove walk, gallery, Schwer's formula
This paper proves a Hall-Littlewood polynomial formula in a paper by Ram [Ram06] using a theorem by Schwer [Sch06]. We review materials relating to root systems, affine Weyl groups and ane Hecke algebras that are required to study alcoves, galleries and the Hall-Littlewood polynomials. In order to prove the Hall-Littlewood polynomial formula, we formulate in a special case Schwer's formula in Theorem 5.5 [Sch06] computing right multiplication of the alcove basis by standard basis elements. We show that Ram's formula for Hall-Littlewood polynomials in terms of positively folded alcove walks coincides with the formulation of Schwer's formula in terms of positively folded galleries.
Wai Ling Yee
Master of Science
Mathematics and Statistics
Major Research Paper