Sequences of Symmetric Polynomials and Combinatorial Properties of Tableaux

Abstract

In 1977 G. P. Thomas showed that the sequence of Schur polynomials associated to a partitionλcan be comfortably generated from the sequence of variablesx=(x1, x2, x3, …) by the application of mixed Baxter/multiplication operators, which in turn can be easily computed from the set SYT(λ) of standard Young tableaux of shapeλ. We generalize this construction, thereby making possible the explicit and effective computation of the Hall–Littlewood, Jack, and Macdonald polynomials used in representation theory, combinatorics, multivariate statistics, and quantum algebra. These generalized formulas have a pleasing recursive structure with respect to the Young lattice and they can easily be specialized to yield “skew” forms in all cases and “super” forms in the Schur case. We introduce and investigate: (1) the “descent polynomial of a partitionλ,” which arises naturally in the enumeration of semistandard Young tableaux of shapeλ; (2) the Boolean latticeG(ζ) associated to anyζ∈SYT(λ), which is fundamental for the “weighted” generalization of Thomas' approach to Schur polynomials; and (3) an action of the symmetric groups on semistandard Young tableaux, which is connected with Knuth's combinatorial proof of the symmetry of Schur functions. Moreover, we argue that a generalization of Thomas' approach is a natural starting point in search of “universal weighted symmetric functions.”