  We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk
formula in the torus-equivariant $K$-group of the formal power series model of
semi-infinite flag manifolds, both of which are described explicitly in terms
of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum
Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas
give an explicit description of the structure constants for the Pontryagin
product in the torus-equivariant $K$-group of affine Grassmannians and that for
the quantum multiplication of the torus-equivariant (small) quantum $K$-group
of finite-dimensional flag manifolds. Our proof of these formulas is based on
standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, which is
established in our previous work, and also uses a string property of
Demazure-like subsets of the crystal basis of a level-zero extremal weight
module over a quantum affine algebra.
