Graded tensor products, Schur positivity and mock theta functions
Final Report Abstract
This project investigates further the interplay between the fields of representation theory, combinatorics and number theory. The focus is on graded tensor products and their independence from the choice of parameters, the Schur positivity conjecture, the existence of excellent filtrations, the connection of graded multiplicities to mock-theta functions, the family of truncated representations and their realization as graded tensor products. Among the main achievements of the project are: • A presentation for a class of graded tensor products and the independence of this class from the choice of parameters, • Existence of excellent filtrations and a Pieri-type character decomposition, • Formulas for the graded decompositions of truncated representations in terms of zeros and poles of rational functions, • Recursion formulas and identities for the numerical multiplicities in excellent filtrations, • Proof of the Schur positivity conjecture for all rank 2 Lie algebras and the types A3 and A4 , and the parametrization of the Schur expansion as integer points in convex polytopes.
Publications
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Simplified presentations and embeddings of Demazure modules
Deniz Kus & R. Venkatesh
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Quantum Affine Algebras, Graded Limits and Flags. Journal of the Indian Institute of Science, 102(3), 1001-1031.
Brito, Matheus; Chari, Vyjayanthi; Kus, Deniz & Venkatesh, R.
