Some New Symmetric Function Tools and their Applications A. Garsia ∗Department of Mathematics University of California, La Jolla, CA 92093
[email protected] J. Haglund †Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104-6395
[email protected] M. Romero ‡Department of Mathematics University of California, La Jolla, CA 92093
[email protected] June 30, 2018 MR Subject Classifications: Primary: 05A15; Secondary: 05E05, 05A19 Keywords: Symmetric functions, modified Macdonald polynomials, Nabla operator, Delta operators, Frobenius characteristics, Sn modules, Narayana numbers Abstract Symmetric Function Theory is a powerful computational device with applications to several branches of mathematics. The Frobenius map and its extensions provides a bridge translating Representation Theory problems to Symmetric Function problems and ultimately to combinatorial problems. Our main contributions here are certain new symmetric func- tion tools for proving a variety of identities, some of which have already had significant applications. One of the areas which has been nearly untouched by present research is the construction of bigraded Sn modules whose Frobenius characteristics has been conjectured from both sides. The flagrant example being the Delta Conjecture whose symmetric function side was conjectured to be Schur positive since the early 90’s and there are various unproved recent ways to construct the combinatorial side. One of the most surprising applications of our tools reveals that the only conjectured bigraded Sn modules are remarkably nested by Sn the restriction ↓Sn−1 . 1 Introduction Frobenius constructed a map Fn from class functions of Sn to degree n homogeneous symmetric poly- λ nomials and proved the identity Fnχ = sλ[X], for all λ ⊢ n.