A Littlewood-Richardson Rule for 2-Grassmannian Permutations

Giving  a  manifestly positive  formula  for  the structure  constants
arising  from  the  multiplication  of two  Schubert  classes  in  the
cohomology  of a  flag manifold  is  a long-standing  open problem  in
algebraic combinatorics.   This has  only seen limited  progress. Most
results have restrictions on both classes that are multiplied, and the
only formula with a restriction on  only one is the Pieri-type formula
that is 30 years old.

In  this  talk,  I  will  describe  formulas,  in  both  ordinary  and
equivariant  cohomology,   for  the   coefficients  that   arise  when
multiplying a Schubert class by one pulled back from a Grassmannian of
2-planes.   This  uses  geometric   constructions  including  a  novel
description of certain positroid varieties.  The cohomological formula
identifies  each   coefficient  as   particular  Littlewood-Richardson
coefficient on a Grassmannian of k-planes, for some k, not necessarily
equal to 2.

This is joint work with Changzheng Li and Mingzhi Yang.