Abstract: In this survey talk, we will discuss several combinatorial, geometric and algebraic problems
from a representation theoretic viewpoint.
The problems include:
(1) Counting the number of solutions of the equation $x^k=g$ in classical Weyl groups.
(2) Constructing multiplicity-free modules (Gelfand models and Gelfand pairs).
(3) Counting permutations by cycle type and descent statistics.
(4) Finding cyclic symmetries of permutation statistics (the cyclic descent extension)..
(5) Studying flips of polygon triangulations as group actions.
The key tools, used to solve such problems, are higher Lie characters.
We will define them and explain their role in the solutions.
Time allowing, we will present an asymptotic solution to an 83 years old open problem of Thrall:
Decompose the higher Lie characters into irreducibles.
Based on joint works with Ron Adin, Pal Hegedus and Natalia Tsilevich.