Basic Schubert Calculus, Lecture notes of Vector Analysis

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Basic Schubert Calculus

Sara Billey

University of Washington

ICERM: Introductory Workshop:

Combinatorial Algebraic Geometry

February 1, 2021

Outline

Enumerative Algebraic Geometry and Hilbert’s 15th Problem

Introduction to Flag Manifolds and Schubert Varieties

Schubert Problems in Intersection Theory

Ancient Questions Enumerative Algebraic Geometry

1. How many points are in the intersection of two lines in R

2 ?

Ans: 0 or 1 or ∞.

2. Given 2 circles in the plane, how many common tangents do

they have? Ans: 0,1,2,3,4,∞. Draw pictures!

3. Given 3 circles in the plane, how many circles are tangent to

all 3? Ans: 0,1,2,3,4,5,6,8,∞. The generic solution has 8

circles as drawn below, known as the Circles of Apollonius, ca

200 BC.

https://commons.wikimedia.org/wiki/File:Apollonius8ColorMultiplyV2.svg

Hilbert’s 15th Problem

Mathematical Problems by Professor David Hilbert.

Lecture delivered at the ICM, 1900. (Bull.AMS)

Schubert’s Enumerative Calculus

Hermann C¨asar Hannibal Schubert (22 May 1848 – 20 July 1911)

A German mathematician interested in Enumerative AG.

He wanted a method for finding the typical number of subspaces

meeting other given sequences of vector spaces in a given position.

Classical Problem. How many lines meet 4 given lines in R

3 ?

Ans: 0,1,2,∞.

Modern Problem. How many points, lines, planes,... intersect

a given family of Schubert varieties in a fixed set of dimensions?

(Schubert varieties were named by Bert Kostant, roughly 1960.)

Inspired Consequences

Schubert’s calculus and Hilbert’s 15th problem inspired many

developments in singular homology, cohomoloogy, de Rham

cohomology, Chow cohomology, equivariant cohomology, quantum

cohomology, intersection theory, cobordism, combinatorics,

representation theory and beyond over the past 150 years.

Inspired Consequences

Schubert’s calculus and Hilbert’s 15th problem inspired many

developments in singular homology, cohomoloogy, de Rham

cohomology, Chow cohomology, equivariant cohomology, quantum

cohomology, intersection theory, cobordism, combinatorics,

representation theory and beyond over the past 150 years.

According to Wikipedia (on 1/30/2021), Hilbert’s 15th problem

has been completely solved...

“by Borel, Marlin, Billey-Haiman and Duan-Zhao, et al. ”

Goal. This talk is a Revisionist History of the problem, the

solution, and what it continues to inspire.

The Flag Manifold

Defn. A complete flag F ● = ( F 1 ,... , Fn ) in C

n is a nested

sequence of vector spaces such that dim( Fi ) = i for 1 ≤ i ≤ n. F ● is

determined by an ordered basis ⟨ f 1 , f 2 ,... fn ⟩ where

Fi = span⟨ f 1 ,... , fi ⟩.

Drawn projectively, a flag is a point, on a line, in a plane,...

Go Schubert Team!

The Flag Manifold

Canonical Matrix Form.

F ● =⟨ 6 e 1 + 3 e 2 , 4 e 1 + 2 e 3 , 9 e 1 + e 3 + e 4 , e 2 ⟩

≈⟨ 2 e 1 + e 2 , 2 e 1 + e 3 , 7 e 1 + e 4 , e 1 ⟩

F n (C) ∶= flag manifold over C

n ⊂ (^) ∏

n k = 1 Gr ( n, k ) ⊂ (^) ∏ P

‰ n k

Ž

={complete flags F ●}

≈ B ƒ GLn (C) , B = lower triangular mats.

Flags and Permutations

Example.

F ● = ⟨ 2 e 1 + e 2 , 2 e 1 + e 3 , 7 e 1 + e 4 , e 1 ⟩ ≈

i 1 0 0

i 1 0

i 1

i 1 0 0 0

Note. If a flag is written in canonical form, the positions of the

leading 1’s form a permutation matrix. There are 0’s to the right

and below each leading 1. This permutation determines the

position of the flag F ● with respect to the reference flag

R ● = ⟨ e 1 , e 2 , e 3 , e 4 ⟩.

The Flag Manifold

Defn. Consider two complete flags B ● (black) and R ● (red).

Define pos( B ● , R ●) ∶= w ∈ Sn if dim( Bi ∩ Rj ) = rk NW ( w [ i, j ])

Examples.

R 4

R 3

R 1

R 2 1 1 1 1

2

2 (^2 )

3 4

B B B B

1

2 3 4

1

1

1

1 1 2

2

2 3 3 4

1

1 (^1 )

3 1 2 3 4

1 1 2

1

w = 4321 w = 4312 w = 4132

Schubert Cells

Schubert cell Cw ( R ●). All flags F ● with pos( F ● , R ●) ∶= w.

Cw ( R ●) = { F ● ∈ F n S dim( Fi ∩ Rj ) = rk NW ( w [ i, j ])}

Example. C 4132 ( R ●) =

∶ ∗ ∈ C

Fact. dimC( Cw ) = inv ( w ) = #{( i < j ) ∶ wi > wj } where

{( i < j ) ∶ wi > wj } are the inversions of w.

Fun Facts

Fact 1. The closure relation on Schubert varieties defines a nice

partial order.

Xw = (^)  v ≤ w

Cv = (^)  v ≤ w

Xv

Bruhat order (Ehresmann 1934, Chevalley 1958) is the transitive

closure of

w < wtij ⇐⇒ w ( i ) < w ( j ).

Example. Bruhat order on permutations in S 3.

Bruhat order on S 4.

4 2 3 1

3 1 2 4

4 2 1 3

1 2 3 4

3 4 2 1

1 2 4 3

3 2 1 4

2 1 3 4

2 3 1 4

2 4 3 1 3 2 4 1

2 3 4 1 4 1 2 3

4 1 3 2

1 4 2 3

1 4 3 2

4 3 1 2

3 1 4 2

1 3 4 2

3 4 1 2

2 1 4 3

1 3 2 4

2 4 1 3

4 3 2 1