#### 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".

Done