The Combinatorics of Macdonald Polynomial Operators

Several bi-graded character formulas of current interest in algebraic combinatorics and math physics can be expressed as applying a Delta operator, from Macdonald polynomial theory, to an elementary symmetric function.  Haiman has proved the resulting expressions are Schur positive using the HIlbert scheme, but combinatorial models are only known in certain special cases.   In this talk we overview some recent work, joint with Rhoades and Shimozono, which gives combinatorial expressions when one of the two bigrading-parameters is set to \$0\$.