Alladi Ramakrishnan Hall

Matrix Schubert Varieties

K N Raghavan

IMSc

Consider the polynomial ring in n variables with integer coefficients. 
Consider its quotient by the ideal generated by symmetric polynomials
without constant term.    This quotient ring is identified with the integral
cohomology ring of the variety of full flags in a vector space of dimension
n.

To each permutation of n letters there is associated a polynomial in n
variables with integer coefficients,  called the Schubert polynomial.  
These polynomials (as we run over all permutations of n letters) form an
integral module basis for the quotient ring described above.   In fact,
 they represent cohomology classes of Schubert subvarieties in the variety
of full flags.

To each permutation of n letters there is associated an affine variety in
n-squared dimensional affine space,  called matrix Schubert variety. 
Knutson and Miller (Annals, 2005) interpret Schubert polynomials (and other
related ones,  namely,  the double Schubert, the Grothendieck, and double
Grothendieck polynomials) in terms of the geometry, algebra, and
combinatorics naturally associated to corresponding matrix Schubert
varieties. Our aim is to review the main ideas of their paper entitled "A
Groebner geometry of Schubert polynomials".