The Weyl Character formula tells us how to write the character of a representation as a linear combination of integral weights. Since characters are invariant under the action of the Weyl group, $W$, we can write a character as a linear combination of $W$-symmetrized dominant integral weights. It is know that the representation ring of a Lie algebra is isomorphic to $\mathbb{C}\[P\]^W$ as a vector space, where $P$ is the weight lattice of some Cartan subalgebra.
So we have two bases for the same vector space: the $W$-symmetrized dominant integral weights and the the character basis. The Weyl character formula tells us how to go from the former to the latter. My question is: is there much known about the matrix of going from the latter to the former? I've gone through a few low rank examples, and many of the coefficients are coming out to be zero. Does anyone know of a reference for this question in general?
Addendum: Jim makes a good point. The Weyl character formula isn't really needed. Perhaps we should just say that the matrix from the weight basis to the character basis is precisely the matrix of weight multiplicities. From this point of view it is clear that the matrix will be "upper triangular" (since weight multiplicities are zero above the highest weight). Thus the inverse should also be upper triangular. So my modified question is there any way to interpret the coefficients of the inverse matrix as counting anything interesting? As the matrix is upper triangular, we can certainly give recursive formulas for the coefficients. Does anyone have any other insight?
Best Answer
For type A, a combinatorial interpretation of the entries of the inverse matrix was given by O. Egecioglu and J.B. Remmel, A combinatorial interpretation of the inverse Kostka matrix, Linear Multilinear Algebra 26 (1990) 59-84. The formula involves a lot of cancellation which suggests that the entries are relatively small. Another interpretation of the entries was given by H. Duan, On the inverse Kostka matrix, J. Combinatorial Theory (A) 103 (2003), 363-376.