A Monoid for the Grassmannian Bruhat Order

Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are related to the enumeration of chains in a new partial order on S∞, the Grassmannian Bruhat order. Here we present a monoid M related to this order. We develop a notion of reduced sequences for M and show that M is analogous to the nil-Coxeter monoid for the weak order on S∞.