### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:**
**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu
**Nathan Fox**, Rutgers University, fox {at} math [dot] rutgers [dot] edu)

Title: Searching for Tableaux Statistics and Schur Expansions

Speaker: **Emily Sergel**, Universtiy of Pennsylvania

Date: Thursday, November 17, 2016 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

This talk covers joint work with Jim Haglund. It will discuss our search for a new family of inversion statistics on standard Young tableaux. Our motivation is the problem of expanding certain LLT polynomials in terms of the Schur basis. The Schur expansions of certain symmetric functions encode the decompositions of some Sn-modules into irreducible representations. Many symmetric functions appearing in the study of Macdonald polynomials and the module of Diagonal Harmonics can be written as positive sums of LLT polynomials. Hence, finding combinatorial Schur expansions for all LLT polynomials would solve many open problems. For example, it would finally give a combinatorial proof of the Macdonald positivity conjecture. The recent proof of the Shuffle Conjecture gives new tools for studying the special case of unicellular LLT polynomials. I will show how computer experimentation is guiding our (ongoing) search for their Schur expansions and the corresponding inversion statistics.

See: http://www.math.rutgers.edu/~nhf12/expmath/