## 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

$0 \to \mathcal{S} \to \mathbb{C}^n \to \mathcal{Q} \to 0$

on the Grassmannian $G(k,n)$. 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 $X$ a smooth projective (integral) variety and $\mathcal{E}$ a locally-free sheaf on $X$,

$H^i(X,\mathcal{E}) \cong H^{n-i}(X, \mathcal{E}^* \otimes \omega_X)^*$,

where $\omega_X$ is the canonical bundle and $n = \dim X$. This isomorphism is almost canonical: it depends on the choice of an isomorphism

$t: H^n(X,\omega_X) \to k$,

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, $\mathbb{S}^\lambda(V)$, where $\lambda$ is a partition and $V$ is a vector space. These will turn out to be the complete set of irreducible polynomial representations of $GL_n$ (for all $n$). The main facts to strive for are:

• Every irreducible representation of $GL_n$ is a unique Schur functor. Conversely, every Schur functor is irreducible.
• The character of $\mathbb{S}^\lambda(V)$ is the Schur polynomial $s_\lambda$.
• The dimension of $\mathbb{S}^\lambda(V)$ is the number of SSYTs of shape $\lambda$ and entries from $1, \ldots, n$ (where $n = \dim(V)$.) 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 $Rep(\text{GL}) \to \lambda_n$ 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 $\Lambda$ are involved in the representation theory of the symmetric group $S_n$, and the general linear group $GL_n$, 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 $GL_n$, 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 $\Lambda$ and its interaction with the $\omega$-involution. As a side benefit, we’ll get the “dual” Jacobi-Trudi and Pieri rules, with $e$ and $h$ swapped.