Good Reason for Partitions Indexing Symmetric Functions?

Question

I'm mostly unfamiliar with the study of symmetric functions. However, it's my understanding that:

• We are interested in, as a basic object, the vector spaces $\Lambda_n$ of symmetric polynomials in $n$ variables.
• These spaces have many canonical bases which are indexed by partitions, like the elementary symmetric polynomials and the Schur functions.

My question is: Does this deserve to be true? I mean, is there some deeper reason that so many of the bases currently recognized as important/interesting are organized by this particular combinatorial object, or is it a historical accident, or something else?

Answer 1

Ultimately this happens because the conjugacy classes of the symmetric group $S_n$ correspond to partitions of $n$, and the irreducible representations of a finite group correspond to the conjugacy classes. $S_n$ acts naturally on the ring of symmetric polynomials by permuting variables, and the Schur polynomials are the characters of the irreducible representations of $S_n$ (which is why they are indexed by partitions.) In this sense, the Schur polynomials are arguably the most natural basis for the symmetric polynomials.

There are many books on the representation theory of the symmetric group; working through one is worth the investment.