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 Thomas,
and allows the easy introduction of skew Schubert polynomials and
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.
Received: September 24, 1997; Accepted: January 27, 1998.
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