Some remarks on evolution algebras corresponding to permutations

TL;DR

This paper studies the structure of n-dimensional evolution algebras linked to permutations, focusing on nilpotent and idempotent elements, and decomposes these algebras based on permutation cycles.

Contribution

It provides a detailed description of nilpotent and idempotent elements in evolution algebras associated with permutations and offers a decomposition method based on cycle structures.

Findings

Characterization of absolute nilpotent elements

Identification of idempotent elements in permutation-based evolution algebras

Decomposition of algebras into direct sums based on permutation cycles

Abstract

In the present paper we describe absolute nilpotent and some idempotent elements of an n- dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations.