Factorial Schur functions and the YangBaxter equation
Abstract
Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statisticalmechanical system (six vertex model). The proof is based on the YangBaxter equation. There is a deformation parameter $t$ which may be specialized in different ways. If $t=1$, then we recover the expression of the factorial Schur function as a ratio of alternating polynomials. If $t=0$, we recover the description as a sum over tableaux. If $t=\infty$ we recover a description of Lascoux that was previously considered by McNamara. We also are able to prove using the YangBaxter equation the asymptotic symmetry of the factorial Schur functions in the shift parameters. Finally, we give a proof using our methods of the dual Cauchy identity for factorial Schur functions. Thus using our methods we are able to give thematic proofs of many of the properties of factorial Schur functions.
 Publication:

arXiv eprints
 Pub Date:
 August 2011
 arXiv:
 arXiv:1108.3087
 Bibcode:
 2011arXiv1108.3087B
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics;
 05E05 (Primary) 16T35;
 16T30 (Secondary)