Two Formulae for Exterior power operations on higher $K$-groups

Exterior power operations on the higher $K$-groups of a quasi-compact scheme have recently been constructed by Taelman and the authors by purely algebraic means. In this paper, we prove two formulae that help to compute these operations. The first is a formula for exterior powers of external products. The second is a formula for exterior powers of $n$-cubes, i.e., of acyclic binary multi-complexes supported on $[0,1]^n$. These formulae provide evidence for the expectation that our exterior power operations agree with those defined by Hiller.