Lecture Notes/Videos

YouTube playlist link can be found here.

  • 5th April, Lecture 1: Extended affine Weyl group -Radhika Ganapathy

  • 7th April, Lecture 2: Symmetric functions - Amritanshu Prasad

  • 12th April, Lecture 3: Affine Hecke algebras - Radhika Ganapathy

  • 14th April, Lecture 4: Schur functions & Orthonormal basis-Amritanshu Prasad

  • 19th April, Lecture 5: Double affine braid groups - Radhika Ganapathy

  • 21st April, Lecture 6: Jacobi−Trudi identity - Amritanshu Prasad

  • 26th April, Lecture 7: RSK correspondence - Amritanshu Prasad

  • 28th April, Lecture 8: DAArt-the GLn case - K. N. Raghavan

  • 3rd May, Lecture 9: Hall-Littlewood polynomials I - Arvind Ayyer

  • 5th May, Lecture 10: DAHA-GLn case - K. N. Raghavan

  • 10th May, Lecture 11: Hall-Littlewood polynomials II - Arvind Ayyer

  • 12th May, Lecture 12: The Macdonald polynomials - K. N. Raghavan

  • 17th May, Lecture 13: Jack and Macdonald polynomials - Arvind Ayyer

  • 19th May, Lecture 14: The Macdonald polynomials I - S. Viswanath

  • 26th May, Lecture 15: Combinatorial Interpretation of Symmetric Macdonald polynomials - Arvind Ayyer

  • 31st May, Lecture 16: The Macdonald polynomials II - S Viswanath

  • 2nd June, Lecture 17: Examples: the SL2 case I - R Venkatesh

  • 7th June, Lecture 18: The Macdonald polynomials III - S Viswanath

  • 9th June, Lecture 19: Examples: the SL2 case II - R Venkatesh

  • 14th June, Lecture 20: The Macdonald polynomials IV - S Viswanath

  • 16th June, Lecture 21: Demazure characters - R Venkatesh

  • 21st June, Lecture 22: The (q,t)-Weyl character formula and the (q,t)- Boson-Fermion correspondence - Arun Ram

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.

  • 23rd June, Lecture 23: Specialisations of Macdonald polynomials and Demazure characters - R Venkatesh

  • 28th June, Lecture 24: Five (q,t)-analogues of Kostka numbers - Arun Ram

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.


  • 30th June, Lecture 25: Principal specializations of Macdonald polynomials - Arun Ram

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.