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

## TL;DR

This paper presents two explicit formulae for computing exterior power operations on higher K-groups of schemes, aiding in understanding their algebraic structure and confirming their consistency with existing definitions.

## Contribution

It introduces two new formulae for exterior power operations on higher K-groups, specifically for external products and n-cubes, advancing computational methods in algebraic K-theory.

## Key findings

- Formula for exterior powers of external products
- Formula for exterior powers of n-cubes
- Evidence that operations agree with Hiller's definitions

## Abstract

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.