Lower bound on the dimensions or irreducible representations of symmetric groups and on the exponents of varieties of Lie algebras

Two main results of this paper are singled out. The first one relates to the representation theory of symmetric groups. The second one deals with varieties of Lie algebras over a field of characteristic zero. The first result can be presented as follows: given a symmetric group of sufficiently large degree $n$, every irreducible representation of it with Young diagram fitting into a square with side $n/k$ is of dimension at least $k^n$. The second result states that there are no varieties of Lie algebras over a field of characteristic zero with lower exponent strictly less than two. At the same time, examples of varieties with exponent two are presented.