It’s time to set up the Schubert stratification of , 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 -theory rings of the Grassmannian. And, as a result, intersection theory on has a complete, concrete description in terms of the combinatorics of the Schubert decomposition (i.e. partitions and the Littlewood-Richardson rule).
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.
Goals for this post:
- coordinate systems
- line and vector bundles and divisors
- the functor of points perspective
- the homogeneous coordinate ring is a UFD
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.
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).
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.
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.