A t-Pieri rule for Hall-Littlewood P-functions and QS(t)-functions (12frg164)
James Haglund (University of Pennsylvania)
Stephanie van Willigenburg (University of British Columbia)
In their article published in the Journal of Combinatorial Theory Series A (JCTA) , Haglund, Luoto, Mason and van Willigenburg (HLMV) introduced a new basis for Qsym they call Quasisymmetric Schur functions (QS functions). In this and a subsequent article by Bessenrodt, Luoto, and van Willigenburg published in Advances in Marthematics titled ``Skew quasisymmetric Schur functions and noncommutative Schur functions", many of the wonderful properties of Schur functions are shown to have refinements involving QS functions. One example of this is the Pieri rule for the expansion in the Schur basis of the product of a Schur function corresponding to a partition with a one-row Schur function. In the JCTA article, HMLV show that the product of a one-row Schur function, with a QS function corresponding to a composition, in terms of the QS basis, has a nice combinatorial description. At a follow-up Research in Teams meeting at BIRS, HMLV proved a significant generalization of this result, by describing the coefficients in the expansion of an arbitrary Schur function with a QS function. The coefficients are describable in terms of tableaux, and to prove their result HLMV developed extensions of many of the results on the combinatorics of tableaux in Fulton's book ``Young Tableaux" to the setting of Mason's ``skyline tableaux", which occur in the definition of QS functions. They published this result in Transactions of the AMS. In their JCTA article HLMV also show that some of the combinatorial structure of the QS functions can be extended to include an extra parameter t. In particular they show that these the QS(t)-functions refine the well-known Hall-Littlewood symmetric function basis in the same way that QS functions refine the Schur basis, i.e. that the sum, over all compositions whose parts rearrange the parts of a given partition, of the QS(t)-function corresponding to the composition, equals the Hall-Littlewood P-polynomial corresponding to the partition. They do not develop the theory of the QS(t)-functions any further though.
Recently Haglund has noted empirically using Maple that the expansion of the product of a Schur function and a Hall-Littlewood P-function , when expanded in terms of the P- basis, has coefficients which are polynomials in t with nonnegative integral coefficients. Since a given P-function reduces to a Schur function when t=0, the constant terms in these coefficients are given by the famous Littlewood-Richardson rule for the expansion of the product of two Schur functions in terms of the Schur basis. In the special case when we multiply by a one-row Schur function, Meesue Yoo has refined this conjecture and obtained an elegant combinatorial description of these coefficients, called a t-Pieri rule. When t=0 this reduces to the classical Pieri rule. Another special case of interest is when the P-polynomial is the constant 1, in which case we are asking for the coefficients when a Schur function is expanded in terms of the P-basis. By a celebrated theorem of Lascoux and Sch"utzenberger, these coefficients are the t-Kostka polynomials, which have nonnegative integral coefficients describable in terms of an intricate combinatorial statistic on tableaux called charge. Yoo also has an elegant conjecture for the coefficients in the expansion of a one-row Schur function and a QS(t)-function in the QS(t)-basis, which reflects their (conjectural) nonnegativity. In addition to understanding the combinatorics behind the nonnegativity of these coefficients, one could also hope to obtain a geometric interpretation, which is known in certain important special cases. The investigations of Haglund and Yoo are partly motivated discussions between Yoo, Mason, van Willigenburg and others at the BIRS 2011 Algebraic Combinatorixx workshop.
Goals for the week:
As a first step towards understanding the conjecture of Haglund that the coefficients in the expansion of the product of a Schur function and a P-function in terms of the P-basis have nonnegative coefficients, we would like to spend a week at BIRS trying to prove Yoo's t-Pieri rule conjecture for the special case when the Schur function is a one-row, and also trying to prove her refinement of this conjecture involving the coefficients of the QS(t)-functions in the expansion of the product of a one-row Schur function and a QS(t) function.
If we are successful in proving these conjectures, we would then extend the study to the case of an arbitrary Schur function. This work would build directly on the 2008 HLMV BIRS Research in Teams program ``Schur quasisymmetric functions and Macdonald polynomials", the recent 2011 BIRS 5-day workshop ``Algebraic Combinatorixx", and to a lesser extent on the 2007 BIRS 5-day workshop ``Applications of Macdonald Polynomials" organized by Haglund, F. Bergeron, and J. Remmel, and attended by Mason, van Willigenburg, and Yoo.