Monodromy in the Osculating Schubert Calculus

An  important special  case of  Schubert calculus  for the  Grassmannian
concerns flags osculating the rational normal curve, which is equivalent
to the  Bethe Ansatz  in the  Gaudin model for  gl_n.  The  most natural
family of  these problems are  indexed by partitions \lambda.   Liao and
Rybnikov  recently studied  a subgroup  of the  monodromy group  for the
Bethe Ansatz  equivalent to the action  of the cactus group  on standard
Young tableau of shape \lambda.  When \lambda is a hook or is symmetric,
they  showed that  it  was  not 2-transitive,  but  was otherwise  giant
(contains the alternating group).

I will describe  this background and then give  some geometric arguments
which  refine their  work on  hooks and  symmetric partitions,  and then
[resent  some computational  evidence that  Harris's principle  holds in
that the monodromy is as large as  possible. This is based on joint work
with Leonid Rybnikov (UdeM).