A key polynomial 
is a type-A Demazure character indexed by a weak composition 
of nonnegative integers.
Let 
interchange 
and 
and define the isobaric divided difference operator by
 |
The initial condition is 
for weakly decreasing 
, and the recursion is 
whenever 
. The braid relations make this
independent of the chosen recursion. Reversing a partition
padded with zeros gives the corresponding Schur polynomial.
Replacing 
by 
gives Demazure atoms (Mason 2009).
