## Addendum to last post: the total Chern classes of tautological bundles on G(k,n)

I realised after finishing the last post that it’s simpler than I had thought to compute the total Chern classes of the tautological bundles appearing in the short exact sequence

on the Grassmannian . I’m keeping it in a separate post, though, both because the last post is already very long and because I haven’t yet defined the Schubert varieties that appear in the answer.

## Schubert Calculus I: Geometry of G(k,n)

**Goals for this post:**

- coordinate systems
- line and vector bundles and divisors
- the functor of points perspective
- the homogeneous coordinate ring is a UFD

## Schubert Calculus Mini-Course

Next week, I am teaching a mini-course, introducing some of my fellow UM grad students to Schubert calculus. It will focus on the Grassmannian. Ultimately, I also hope to have all the notes from the course posted on this blog.

My goal is to help both my combinatorially- and geometrically-oriented friends learn enough of both toolsets to carry out concrete computations in intersection theory.

## On Serre Duality

Serre Duality is the statement, for a smooth projective (integral) variety and a locally-free sheaf on ,

,

where is the canonical bundle and . This isomorphism is almost canonical: it depends on the choice of an isomorphism

,

called a trace map. I’m going to sketch out my understanding of what’s going on with this duality statement and how it comes up (non-rigorously).

## 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.