### 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:
Drew Sills, Rutgers University, asills {at} math [dot] rutgers [dot] edu
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Nonsymmetric Schur functions and an analogue of the RSK algorithm

Speaker: Sarah Mason, University of Pennsylvania

Date: Thursday, February 9, 2006 5:00pm

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

Abstract:

The Schur functions, $s_{\mu}$, form a basis for the ring of symmetric functions. Macdonald polynomials are symmetric functions $P_{\mu}(x;q,t)$ in variables $x=x_1,x_2,...$, with coefficients which are rational functions of two parameters $q$ and $t$. The Schur functions are obtained from Macdonald polynomials by setting $q=t=0$. Recently Haglund, Haiman, and Loehr derived a combinatorial formula for nonsymmetric Macdonald polynomials, which gives a new decomposition of the Macdonald polynomial into nonsymmetric components and provides a combinatorial description of the nonsymmetric Schur functions, $NS_{\lambda}$}. Letting $q=t=0$ in this identity implies $s_{\mu}(x) = \sum_{\lambda}NS_{\lambda}(x)$, where the sum is over all rearrangements $\lambda$ of the partition $\mu$. We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-standard skyline fillings to give a combinatorial proof of the formula. The bijection involves an analogue of the Robinson-Schensted-Knuth Algorithm. We also provide a non-recursive combinatorial interpretation of the standard bases of Lascoux and Schkutzenberger.