Tuesday, 28 November 2000

Day 22: Math 697R

Applicable Algebraic Geometry
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*Discuss Problem Session:  No room yet.

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Equations for the Grassmannian.

* Construct syzygy among the minors z_\alpha of a 2 x 4 matrix
  ---> Get the equation for Grass(2,4).
* Generalize to Grass(2,n).
* Define Young's Lattice & Bruhat order;  Draw for (2,5).
  ---> Observe the form of the equations

* Linearly order variables lexocigraphically, from the end.
   let < be the resulting degree reverse Lexicographic order.

Theorem.  The Pl\"ucker ideal of the Grassmannian has reduced GB
  consisting of quadratic polynomials S(\alpha,\beta)....

Define:  A monomial is standard, if it is sortable, in Bruhat order.

Proof of Theorem is in 3 parts
1) Standard monomials are linearly independent on Grassmannian, that is,
    modulo Pl\"ucker Ideal
2) Construction of Van der Waerden syzygies VdW(\alpha,\beta)
    (State Lemma about them)
3) Gaussian elimination of VdW syzygies obtain S(\alpha,\beta), whose
    initial forms define the monomial ideal of non-standard monomials.
4) Inclusion argument shows that S(\alpha,\beta) are a reduced Groebner
    basis for Pl\"ucker ideal.





Give 3), then 4).

Begin 2), but save 1) for next week.

For 2)
  1) First define VdW(\alpha,\beta)




  2) Then show that it has the order properties




  3) Finally, show that it is a valid relation:
     a) Define Multilinear form, Alternating form
     b) Show that an alternating p+1 form on K^p is identically zero.
     c) Show that VdW is an alternating p+1 form on p-space.