MATH 7390: Topics in Algebra (spring 2009)

Instructor: Allen Knutson

One could summarize the content of the course I intend as "We'll study Schubert varieties and close relatives thereof, and quiver cycles, using Gröbner bases and algebraic combinatorics. Often their study involves degeneration of the defining equations, and we will develop tools to study degenerations in general." Now in more detail:

Part I. Schubert varieties.

  1. The Grassmannian and its Plücker coordinates. Hodge's degeneration of the Grassmannian to a union of projective spaces, one for each standard Young tableaux. (This trades geometrical complexity for combinatorial complexity.)

Interlude: simplicial complexes and their Stanley-Reisner rings.

  1. Schubert varieties in the Grassmannian and in the full flag manifold.
  2. Matrix Schubert varieties and pipe dreams. Interlude: multigraded Hilbert series and multidegrees.
  3. Schubert polynomials and the transition formula. Stanley symmetric functions and reduced words for permutations.

Part II. Quiver cycles.

  1. Quivers and their representations. Gabriel's theorem characterizing ADE Dynkin diagrams and their indecomposable representations.
  2. Quiver cycles. The Zelevinsky-Lakshmibai-Magyar theorem relating them to Schubert varieties.
  3. Degenerating quiver cycles to unions of product of matrix Schubert varieties. Lacing diagrams.

Part III. Schubert calculus.

  1. More about Schubert polynomials. Descent-cycling.
  2. Puzzles.
  3. Connections to representation theory.

Prerequisites: Primarily, a willingness to follow linear algebra wherever it goes. Enough commutative ring theory to play with primary decomposition of ideals in polynomial rings. Gröbner bases would be very helpful. It would be good to know some algebraic geometry, Lie theory, algebraic combinatorics, and even algebraic topology. But the stuff we'll need we'll develop along the way.