## Schur functors

I’m going to describe the basic ideas of the Schur functors, , where is a partition and is a vector space. These will turn out to be the complete set of irreducible polynomial representations of (for all ). The main facts to strive for are:

- Every irreducible representation of is a unique Schur functor. Conversely, every Schur functor is irreducible.
- The character of is the Schur polynomial .
- The dimension of is the number of SSYTs of shape and entries from (where .) This fact will be explicit: there will be a “tableau basis” for the representation.

As a corollary, we get an improved understanding of the Littlewood-Richardson numbers and the isomorphism between the representation ring and the ring of symmetric polynomials.

## GL-representations, symmetric polynomials, and geometry

One of the many applications of symmetric polynomials is to representation theory, and in this post I want to begin sketching out how.

Symmetric polynomials and the ring are involved in the representation theory of the symmetric group , and the general linear group , in related ways. The precise relationship between the representation theory of these two groups is spelled out in the Schur-Weyl Duality theorem, as well as in explicit constructions of representations of both groups.

I’m mainly interested in the Schur functors, which are representations of , so I’ll be focusing on those.

## Everything you wanted to know about symmetric polynomials, part V

This will be the last post on symmetric polynomials, at least for now. (They’ll continue to come up when I get to representation theory and my true love, algebraic geometry, but only as part of other theories.)

I want to discuss the Hall inner product on the symmetric function ring and its interaction with the -involution. As a side benefit, we’ll get the “dual” Jacobi-Trudi and Pieri rules, with and swapped.

## Everything you wanted to know about symmetric polynomials, part IV

**Alternating and Symmetric Polynomials**

Consider the following recipe for building symmetric polynomials, using alternating polynomials. Consider the Vandermonde determinant

To see the last equality, note that the determinant is zero if for any , so it is divisible by , and all these factors are distinct. By degree-counting (the polynomial is evidently homogeneous of degree ), this is the whole thing, up to a scalar. Finally, to get the scalar, we do some computation (e.g. plugging in convenient values like ).

If, instead of the row , we used some other sequence of polynomials, like , the result would still be alternating in the ‘s, so it would still be divisible by the product above. However, the degree-counting argument might no longer be relevant. (For example, if we use , then the result is the original Vandermonde determinant times .)

Still, we can divide out the Vandermonde determinant, and (surprise!) the result will be a *symmetric* polynomial, since the sign-change is “divided out” as well.