Sequences of Symmetric Polynomials and Combinatorial Properties of Tableaux

In 1977 G.P. Thomas has shown that the sequence of Schur polynomials associated to a partition can be comfortabely generated from the sequence of variables x = (x1; x2; x3; : : :) by the application of a mixed shift/multiplication operator, which in turn can be easily computed from the set SY T () of standard Young tableaux of shape. We generalise this construction, thereby making possible | for the rst time | the explicit and eeective computation of the Hall-Littlewood, Jack, and Macdonald polynomials used in representation theory, combinatorics, multivariate statistics, and quantum algebra. These generalised formulas have a pleasing recursiv structure with respect to the Young lattice; they can easily be specialised to yield`skew' and`super' forms; and they are a natural starting point for the construction ofùniversal weighted symmetric functions'. Moreover we introduce and investigate: (1) thèdescent polynomial of a partition ', which arises naturally in the enumeration of semistandard Young tableaux of shape ; (2) the Boolean lattice G() associated to any 2 SY T (), which is fundamental for thèweighted' generalisation 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. For a partition`N of a natural number N we denote by s (m) (x) 2 Zx 1 ; : : : ; x m ] the Schur polynomial associated to in the variables x 1 that these Schur polynomials can be deened algebraically by various determinantal formulas or combinatorially by the formula s (m) := X 2SSY T (m) () x ; where x x () := x 1 () 1 x 2 () 2 x 3 () 3 : : : is the monomial associated to the content () of and SSY T (m) () is the set of semistandard Young tableaux of shape with entries in f1; : : :; mg. (The exact deenition of these and other no(ta)tions appearing subsequently has been collected in an Appendix.) The determinantal formulas are very compact and appropriate for many theoretical purposes, but diicult to evaluate: it is even hard to decide, which monomials in a given s (m) (x) occur. To the contrary this is an easy task from the combinatorial deenition, but the latter is clearly not very compact due to the large number of possible SSYT (semistandard Young tableaux). Therefore one seeks …