  This paper studies the properties of Demazure atoms and characters using
linear operators and also tableaux-combinatorics. It proves the atom-positivity
property of the product of a dominating monomial and an atom, which was an open
problem. Furthermore, it provides a combinatorial proof to the key-positivity
property of the product of a dominating monomial and a key using skyline
fillings, an algebraic proof to the key-positivity property of the product of a
Schur function and a key using linear operator and verifies the first open case
for the conjecture of key-positivity of the product of two keys using linear
operators and polytopes.
