  Over the past years, Polar codes have arisen as a highly effective class of
linear codes, equipped with a decoding algorithm of low computational
complexity. This family of codes share a common algebraic formalism with the
well-known Reed-Muller codes, which involves monomial evaluations. As useful
algebraic codes, more specifically known as decreasing monomial codes, a lot of
decoding work has been done on Reed-Muller codes based on their rich code
automorphisms. In 2021, a new permutation group decoder, referred to as the
automorphism ensemble (AE) decoder, was introduced. This decoder can be applied
to Polar codes and has been shown to produce similar decoding effects. However,
identifying the right set of code automorphisms that enhance decoding
performance for Polar codes remains a challenging task. This paper aims to
characterize the full automorphism group of Polar codes. We will prove a
reduction theorem that effectively reduces the problem of determining the full
automorphism group of arbitrary random Polar codes to that of a specified class
of Polar codes. Besides, we give exact classification of the full automorphism
groups of families of Polar codes that are constructed using the Reed-Muller
codes.
