Lecture Notes/Videos

Abstract: One of the favorite formulas for the Schur function is as a quotient of two determinants. I will explain the corresponding formula for Macdonald polynomials. The boson-fermion correspondence is the correspondence between symmetric functions and skew symmetric functions which is given by multiplying by the Vandermonde determinant.

I will explain the (q,t) version of this correspondence and how the Weyl character formula fits into this correspondence.

Abstract: The Kostka numbers arise as the coefficients in the monomial expansion of Schur functions.

In the Macdonald polynomial case, I know 5 different analogues of these numbers coming from

• The monomial expansion of the P_\lambda

• The expansion of e_\mu in terms of the P_\lambda

• The expansion of g_\mu in terms of the P_\lambda

• The expansion of P_\mu(q,qt) in terms of the P_\lambda(q,t)

• The expansion of J_\mu in terms of the big Schurs S_\lambda

In the first three cases, I know formulas for these as weighted sums of column strict tableaux.

I do not understand the relationships between these 5 different analogues of Kostka numbers.

Abstract: The principal specialization of a Schur function (or Weyl character)

has a striking factorization as a product, and specializing the parameter gives

the Weyl dimension formula. In type GL_n this gives the hook formula for the

number of column strict tableaux. The nonspecialized formula is often called the

quantum dimension. Amazingly the Macdonald polynomials have similar formulas

for their principal specializations, which might be thought of as elliptic Weyl dimension

formulas.