Multiplicities of points on Schubert varieties in the Grassmannian-II

Abstract: 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.

Mathematics

Audience: advanced learners

---

IIT Bombay Virtual Commutative Algebra Seminar

Series comments: Note: To get meeting ID,fill the Google form on the web-site of the series sites.google.com/view/virtual-comm-algebra-seminar/home

---