Description

Date and Time: 29 September 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 1

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 of 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.

(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 1

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 of 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.

Date

Tue, September 29, 2020

Start Time

5:30pm IST

Priority

5-Medium

Access

Public

Created by

DEFAULT ADMINISTRATOR

Updated

Tue, September 29, 2020 10:51am IST