"Non UBC"@en .
"DSpace"@en .
"Pun, Ying Anna"@en .
"2019-07-23T08:37:41Z"@en .
"2019-01-23T10:40"@en .
"Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase $(n-1, ..., 1,0)$ (of which there are Catalan many) and a composition weight of length $n$. They include the Schur functions ,the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture.\n\nA key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are $k$-Schur functions and proved that graded $k$-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded $k$-Schur functions and resolved the Schur positivity and $k$-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are $k$-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems.\n\nThis is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers."@en .
"https://circle.library.ubc.ca/rest/handle/2429/71070?expand=metadata"@en .
"38.0"@en .
"video/mp4"@en .
""@en .
"Author affiliation: Drexel University"@en .
"Banff (Alta.)"@en .
"10.14288/1.0379942"@en .
"eng"@en .
"Unreviewed"@en .
"Vancouver : University of British Columbia Library"@en .
"Banff International Research Station for Mathematical Innovation and Discovery"@en .
"Attribution-NonCommercial-NoDerivatives 4.0 International"@en .
"http://creativecommons.org/licenses/by-nc-nd/4.0/"@en .
"Postdoctoral"@en .
"BIRS Workshop Lecture Videos (Banff, Alta)"@en .
"Mathematics"@en .
"Group theory and generalizations"@en .
"Combinatorics"@en .
"Catalan Functions and $k$-Schur functions"@en .
"Moving Image"@en .
"http://hdl.handle.net/2429/71070"@en .