Description
Date and Time: 2 October 2020, 5:30pm IST/ 12:00GMT / 08:00am EDT
(joining time: 5:15 pm IST - 5:30 pm IST)
Google meet link: meet.google.com/vog-pdxx-fdt
Speaker: K.N. Raghavan, The Institute of Mathematical Sciences
Title: Multiplicities of points on Schubert varieties in the Grassmannian
- Part 2
Given an arbitrary point on a Schubert (sub)variety in a Grassmannian, how
to compute the Hilbert function (and, in particular, the multiplicity) of
the local ring at that point? A solution to this problem based on
"standard monomial theory" was conjectured by Kreiman-Lakshmibai circa
2000 and the conjecture was proved about a year or two later by them and
independently also by Kodiyalam and the speaker. The two talks will be an
exposition of this material aimed at non-experts in the sense that we will
not presume familiarity with Grassmannians (let alone flag varieties) or
Schubert varieties.
There are two steps to the solution. The first translates the problem from
geometry to algebra and in turn to combinatorics. The second is a solution
of the resulting combinatorial problem, which involves establishing a
bijection between two combinatorially defined sets. The two talks will
roughly deal with these two steps respectively.
Three aspects of the combinatorial formulation of the problem (and its
solution) are noteworthy: (A) it shows that the natural determinantal
generators of the tangent cone (at the given point) form a Groebner basis
(in any "anti-diagonal" term order); (B) it leads to an interpretation of
the multiplicity as counting certain non-intersecting lattice paths; and
(C) as was observed by Kreiman some years later, the combinatorial
bijection is a kind of Robinson-Schensted-Knuth correspondence, which he
calls the "bounded RSK".
Determinantal varieties arise as tangent cones at points on Schubert
varieties (in the Grassmannian), and thus one recovers multiplicity
formulas for these obtained earlier by Abhyankar and Herzog-Trung. (The
multiplicity part of the Kreiman-Lakshmibai conjecture was also proved by
Krattenthaler, but by very different methods.)
What about Schubert varieties in other (full or partial) flag varieties
(G/Q with Q being a parabolic subgroup of a reductive algebraic group G)?
The problem remains open in general, even for the case of the full flag
variety GL(n)/B, although there are several papers over the last two
decades by various authors using various methods that solve the problem in
various special cases. Time permitting, we will give some indication of
these results, without however any attempt at comprehensiveness.