Flagged Schur functions, Schubert polynomials, and symmetrizing operators

Abstract

Flagged Schur functions are generalizations of Schur functions. They appear in the work of Lascoux and Schutzenberger [2] in their study of Schubert polynomials. Gessel [ 1 ] has shown that flagged Schur functions can be expressed both as a determinant in the complete homogeneous symmetric functions and in terms of column-strict tableaux just as can ordinary Schur functions (Jacobi-Trudi identity). For each row of these tableaux there is an upper bound (flag) on the entries. The Schubert polynomials are obtained by applying certain symmetrizing operators to a monomial. In Section 1 we study the effect of applying these symmetrizing operators to flagged Schur functions. Although it is trivial to do this for the determinantal expression, we show, by direct means, how to apply the symmetrizing operators to the tableau expression (without the use of determinants). This produces another proof of Gessel’s result and hence a new inductive proof of the Jacobi-Trudi identity. Each Schubert polynomial is determined by some permutation. Lascoux and Schutzenberger [Z] state a result which enables one to identify those permutations whose Schubert polynomial is a flagged Schur function. In Section 2 we present an explicit expression for the shape and flags (row bounds) in terms of the permutation. We do this by applying the symmetrizing operators to flagged Schur functions. We also show that any flagged Schur function can be obtained by applying a sequence of symmetrizing operators to some monomial. In Section 4 we consider row (column) flagged skew Schur functions. Here, for each row (column) of the skew tableaux there is an upper and lower bound on the entries. In fact the above cited work of Gessel actually