Permutation representations of loops

Abstract

The paper proposes a permutation representation concept for loops. A permutation representation of a loop includes a Markov chain for each element of the loop. If the loop is associative, then the concept specializes to the usual notion of a permutation representation of a group, the transition matrices of the Markov chains becoming permutation matrices in this case. The class of permutation representations of a given loop is closed under disjoint unions and direct products, each representation decomposing into a disjoint union of irreducible representations. In contrast with the group case, where regular actions abound as summands in large direct powers of a faithful representation, it is shown that a loop need not be recoverable to within isomorphism from a faithful permutation representation. The paper concludes with an application of loop permutation representations to the investigation of Lagrangian properties of a loop.