Symmetric functions are mathematical expressions that remain unchanged when the variables are permuted. For example, if you have a function of variables x and y, swapping these variables does not alter the function's value. This property makes symmetric functions particularly useful in various areas of mathematics, including algebra and combinatorics. There are different types of symmetric functions, such as elementary symmetric functions and power sum symmetric functions. Elementary symmetric functions are formed by taking products of the variables, while power sum symmetric functions involve summing powers of the variables. These functions play a crucial role in polynomial theory and representation theory.