Semistandard tableaux for Demazure characters (key polynomials) and their atoms

The Schur function indexed by a partition λ with at most n parts is the sum of the weight monomials for the Young tableaux of shape λ. Let π be an npermutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by π and λ. (These polynomials are the characters of the Demazure modules for GL(n).) The “atom” indexed by π is the sum of weight monomials of the tableaux whose right keys are the “key” tableau for π. Schur functions and key polynomials can be decomposed into sums of atoms. We also describe the tableaux that contribute to an atom, the tableaux that have a left key equal to a given key, and the tableaux that have a left key bounded below by a given key.