  In this paper, we propose an analysis of the automorphism group of polar
codes, with the scope of designing codes tailored for automorphism ensemble
(AE) decoding. We prove the equivalence between the notion of decreasing
monomial codes and the universal partial order (UPO) framework for the
description of polar codes. Then, we analyze the algebraic properties of the
affine automorphisms group of polar codes, providing a novel description of its
structure and proposing a classification of automorphisms providing the same
results under permutation decoding. Finally, we propose a method to list all
the automorphisms that may lead to different candidates under AE decoding; by
introducing the concept of redundant automorphisms, we find the maximum number
of permutations providing possibly different codeword candidates under AE-SC,
proposing a method to list all of them. A numerical analysis of the error
correction performance of AE algorithm for the decoding of polar codes
concludes the paper.
