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

The Schur function indexed by a partition λλ with at most nn parts is the sum of the weight monomials for the Young tableaux of shape λλ. Let ππ be an nn-permutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by ππ and λλ. (These polynomials are the characters of the Demazure modules for GL(n)GL(n).) The “atom” indexed by ππ is the sum of weight monomials of the tableaux whose right keys are the “key” tableau for ππ. 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.