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Schubert Calculus II: Schubert Stratification

It’s time to set up the Schubert stratification of G(k,n), a decomposition of the whole variety into locally closed pieces (Schubert cells), each of which is isomorphic to an affine space.

These cells are indexed by partitions and, by general theory, give rise to an additive basis of the Chow, cohomology, and K-theory rings of the Grassmannian. And, as a result, intersection theory on G(k,n) has a complete, concrete description in terms of the combinatorics of the Schubert decomposition (i.e. partitions and the Littlewood-Richardson rule).

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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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