Schur quasisymmetric functions and Macdonald polynomials (08rit138)
Recently Haglund-Mason-van Willigenburg discovered certain linear combinations of nonsymmetric Macdonald polynomials that are quasisymmetric functions, called Schur quasisymmetric functions. These functions not only form a new basis for Qsym, but also naturally refine Schur functions. This raises the exciting question of what properties of Schur functions extend to these Schur quasisymmetric functions? Desirable properties would include combinatorial descriptions of the product of two such functions: the existence of Pieri rules and Littlewood-Richardson rules that refine the classical rules; a Jacobi-Trudi formula for expressing these functions as a product of more \"fundamental\" ones; a combinatorial description of the dual basis in NC.
During the BIRS workshop \"Applications of Macdonald Polynomials\" members of our group discovered a potential Pieri rule for multiplying a simple Schur function and a Schur quasisymmetric function and proved the rule for a special case. From this initial breakthrough we have been able to precisely conjecture a Pieri rule, and thus progress towards a Littlewood-Richardson rule for multiplying any Schur function and a Schur quasisymmetric function.
Therefore, our aims for the week include
1) Proving our Pieri rule for all cases, and precisely conjecturing and proving a Littlewood-Richardson rule using the tools employed in our Pieri rule.
2) Extending the above to the analogous product of a Schur function and a nonsymmetric Schur function; and an analogous product of a Schur function and a Demazure character. In this latter case connections with the Schubert polynomials of algebraic geometry already seem to be emerging.
3) Investigating whether Schur quasisymmetric functions can be extended to Hall-Littlewood quasisymmetric functions, where Hall-Littlewood polynomials are one parameter refinements of Macdonald polynomials.
As can be seen from the subject overview, the resolution to these questions is at the cutting edge of algebraic combinatorics, and will impact a number of fields. The timing of the workshop is ideally placed between the closely related MSRI program and Fall AMS special session on combinatorial representation theory, giving us a forum to collect ideas in the former, and to disseminate them at the latter.
Our justification for desiring to research at BIRS is that since the BIRS workshop \"Applications of Macdonald Polynomials\" members of our group have met pairwise (Luoto-van Willigenburg Jul 07, Nov 07; Haglund-Mason Jan 08, Feb 08; Haglund-Luoto Apr 08). However, we have not had the opportunity to meet as a group and we firmly believe that the BIRS Research in Teams program will be the opportunity we seek. This opportunity will be invaluable in resolving important conjectures first discovered a BIRS.