Seminar - Thomas McConville

Title: Determinantal formulas with major indices

Abstract: Krattenthaler and Thibon discovered a beautiful formula for the determinant of the matrix indexed by permutations whose entries are q^maj( u*v^{-1} ), where “maj” is the major index. Previous proofs of this identity have applied the theory of nonsymmetric functions or the representation theory of the Tits algebra to determine the eigenvalues of the matrix. I will present a new, more elementary proof of the determinantal formula. Then I will explain how we used this method to prove several conjectures by Krattenthaler for variations of the major index over signed permutations and colored permutations. This is based on joint work with Donald Robertson and Clifford Smyth.