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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.

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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

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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.

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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).

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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.

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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.

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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.

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