Affine Quasisymmetric Functions

Desired Properties of $\tilde{L}_c$

• The affine Schur functions expand positively into AF quasisymmetric functions $\tilde{L}_c$.
• $\tilde{L}_c$ form a basis of the $QSymm^{(n)} = (M_c: c \in Comp^{n})$

Examples

$n=4$

$\tilde{s}_{421} = -2 s_{111111} + s_{211111} = 4*m[1, 1, 1, 1, 1, 1, 1] + m[2, 1, 1, 1, 1, 1]$ (r = [3,2,0,1,2,3,0], w = [-5, 2, 5, 8]);