Brian Jurgelewicz

Hall-Littlewood polynomials and the classical Hall algebra

The last two weeks we have been learning about MacDonald polynomials. This week we take a step back and consider Hall-Littlewood polynomials, which have only one "parameter", instead of two. They form a basis for a certain one-parameter deformation of the ring of symmetric polynomials. Specializing at t = 0 gives the Schur polynomials, and t = 1 gives the monomial symmetric polynomials. The ring of symmetric polynomials is isomorphic to the classical Hall algebra. We will get our hands dirty by calculating a few products in the classical Hall algebra. Such calculations are equivalent to multiplications rules for HL-polynomials.