Séminaire Lotharingien de Combinatoire, B39a(1997), 28pp. SCHUBERT FUNCTIONS AND THE NUMBER OF REDUCED WORDS OF PERMUTATIONS

It is well known that a Schur function is the ‘limit’ of a sequence of Schur polynomials in an increasing number of variables, and that Schubert polynomials generalize Schur polynomials. We show that the set of Schubert polynomials can be organized into sequences, whose ‘limits’ we call Schubert functions. A graded version of these Schubert functions can be computed effectively by the application of mixed shift/multiplication operators to the sequence of variables x = (x1, x2, x3, . . . ). This generalizes the Baxter operator approach to graded Schur functions of G.P. Thomas, and allows the easy introduction of skew Schubert polynomials and functions. Since the computation of these operator formulas relies basically on the knowledge of the set of reduced words of permutations, it seems natural that in turn the number of reduced words of a permutation can be determined with the help of Schubert functions: we describe new algebraic formulas and a combinatorial procedure, which allow the effective determination of the number of reduced words for an arbitrary permutation in terms of Schubert polynomials. Let Sn denote the symmetric group on the ‘letters’ {1, . . . , n} and Z[x1, . . . , xn] the Z-algebra of symmetric polynomials in n variables. There are several well known Z-bases of this algebra (cf. [M1, Sa]), which are indexed by the partitions λ ≡ λ1 . . . λs (λ1 ≥ . . . ≥ λs ≥ 1) with length l(λ) := s ≤ n. The most important of these bases are the Schur polynomials s (n) λ (x) := sλ(x1, . . . , xn), which can be defined alternatively by determinant formulas or combinatorially with the help of semistandard Young tableaux. The Schur polynomials are cumulative in the following sense: if Z[x1, . . . , xn] is extended to Z[x1, . . . , xn, xn+1]n+1 , then (setting s (n) λ (x) := 0 for λ with l(λ) > n) one has ∀λ : s λ (x1, . . . , xn) = s (n+1) λ (x1, . . . , xn, 0) . (0.1)