Math 595 The Grassmannian : Preliminaries

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Math 595 “The Grassmannian”: Preliminaries

Alexander Yong

University of Illinois at Urbana-Champaign

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Outline

I Overview of the course

II Definition(s) of the Grassmannian

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Overview

This course concerns the GrassmannianGrk(Cn): the parameter

space of all k-dimensional planes inCn_.

Link to Wikipedia entry on Hermann Grassmann (1809–1877).

I will emphasize work of recent research interest related to:

◮ combinatorics;

◮ algebraic geometry;

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Topics I

want

to cover (time permitting...)

◮ Preliminaries

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Topics I

want

to cover (time permitting...)

◮ Preliminaries

◮ Matroid theory: GGMS theory, the matroid strata,Tx. ◮ Total positivity: planar directed networks, matroid

decomposition of Grk(Rn₎TNN

◮ Cluster algebras: the cluster structure for Grk(Cn), the

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Topics I

want

to cover (time permitting...)

◮ Preliminaries

Laurent phenomenon

◮ Singularity theory: the singular locus, measures of

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Topics I

want

to cover (time permitting...)

◮ Preliminaries

Laurent phenomenon

singularities, resolutions, Kazhdan-Lusztig theory

◮ Schubert calculus: cohomologies ofGrk(Cn), symmetric

polynomials, the Littlewood-Richardson rule

◮ Goresky-Kottwitz-Macpherson theory: The moment map and

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Topics I

want

to cover (time permitting...)

◮ Preliminaries

Laurent phenomenon

singularities, resolutions, Kazhdan-Lusztig theory

◮ Schubert calculus: cohomologies ofGrk(Cn), symmetric

polynomials, the Littlewood-Richardson rule

◮ Goresky-Kottwitz-Macpherson theory: The moment map and

polytope, the GKM conditions and equivariant cohomology

◮ Degeneration theory: the toric, Hodge-Gr¨obner degenerations ◮ Tropical geometry: the tropical Grassmannian

◮ Reality: real enumerative geometry, transversality, the

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Philosophy of this course

Many of these concepts are available much more widely (and at least for generalized flag manifolds). However:

◮ Combinatorial theorems for the Grassmannian have crystalized

to a state of substantial beauty and simplicity not available in general.

◮ Desire to seek relationships among topics in the “prototypical

case”

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Preliminary definitions

Definition: Grk(Cn) ={V is ak-dimensional subspace ofCn}. Suppose V =hv1~, . . . ,vk~i. Thinking of these as row vectors, we have the association

V ∈Grk(Cn)_↔M ∈Mat⋆ k,n,

where Mat⋆

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Preliminary definitions

where Mat⋆

k,n is the space of all full rankk×n matrices. In class exercise: Is this correspondence one-to-one? If not, do

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Preliminary definitions

where Mat⋆

k,n is the space of all full rankk×n matrices. In class exercise: Is this correspondence one-to-one? If not, do

you make it so? Are all k×k subdeterminants nonzero?

Answer: “No” to the first question, however the correspondence

V ∈Grk(Cn)_↔M ∈GLk\Mat⋆ k,n.

Also, “no” to the second, e.g.,

0 0 1 0 0 0 0 1

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Pl¨ucker coordinates I

For V ∈Grk(Cn) with basishv1~, . . . ,vk~ilet [vi(j)] denote the associated matrix     v1(1) v1(2) v1(3) . . . v1(n) v2(1) v2(2) v2(3) . . . v2(n) . . . . vk(1) vk(2) vk(3) . . . vk(n)     ∈Mat⋆ k,n.

Definition: For a Schubert symbol J, i.e., a sequence 1≤j1 <j2< . . . <jk ≤n, let

PJ =k×k minor of [vi(j)]using columns J. be the Pl¨ucker coordinateassociated to J.

Let [· · ·:PJ :· · ·]denote the list of all n_k

Pl¨ucker coordinates listed lexicographically.

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Pl¨ucker coordinates II

In class exercise: Compute the Pl¨ucker coordinates when

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Pl¨ucker coordinates II

k =2,n=4 andV =h~e2+2~e3+3~e4,~e1+_~e3i.

Solution: The corresponding matrix is

0 1 2 3 1 0 1 0

.

The Pl¨ucker coordinates are[−1: −2: −3:1:0: −3].

In class exercise: Can all Pl¨ucker coordinates be zero?

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Pl¨ucker coordinates II

k =2,n=4 andV =h~e2+2~e3+3~e4,~e1+_~e3i.

Solution: The corresponding matrix is

0 1 2 3 1 0 1 0

.

The Pl¨ucker coordinates are[−1: −2: −3:1:0: −3].

In class exercise: Can all Pl¨ucker coordinates be zero?

What happens if we choose a different basis for V?

Solution: “No” to the first question, since [vi(j)] is rankk. Change of basis corresponds to left multiplication of [vi(j)] by

g ∈GLk. Hence the Pl¨ucker coordinates differ by rescaling by det(g).

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The Grassmannian as a projective algebraic variety I

Summary: We have shown that

V ∈Grk(Cn₎_←→_[_{· · ·}_:_PJ _:_{· · ·}_]_∈P( n k)−1_,

where

Definition: PN

is (complex) projective space: all nonzero N+1

tuples [x0 :x1:. . .:xN]modulo [x0:x1 :. . .:xN]∼[x0′ :x ′ 1:· · ·:x ′ N] if xi =αx′

i for alli and someα∈C⋆.

In fact: Grk(Cn)⊆P( n

k)−1 as a projective algebraic variety (it’s cut

out by homogeneous polynomial equations in the Pl¨ucker

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The Grassmannian as a projective algebraic variety II

In class exercise: Explain why we can extend the definition of PJ

to any sequence PJ by setting

P(· · ·:j :j :· · ·) =0

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The Grassmannian as a projective algebraic variety II

In class exercise: Explain why we can extend the definition of PJ

to any sequence PJ by setting

P(· · ·:j :j :· · ·) =0

P(· · ·:i :j :· · ·) = −P(· · ·:j :i :· · ·).

Solution: These are immediate from elementary properties of determinants.

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The Grassmannian as a projective algebraic variety III

Theorem: The set of points[· · ·:PJ :· · ·]∈P( n

k)−1 are exactly

those that satisfy the Pl¨ucker relations:

k+1 X t=1 (−1)tp(j1,j2, . . . ,jk−1,jt′)·p(j ′ 1, . . . ,^j ′ t, . . . ,j ′ k+1) where {jt}k t=1 and{j ′ t} k+1

t=1 is anysequence in [n] :={1,2, . . . ,n} and

^_j′

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The Grassmannian as a projective algebraic variety III

Theorem: The set of points[· · ·:PJ :· · ·]∈P( n

k)−1 are exactly

those that satisfy the Pl¨ucker relations:

k+1 X t=1 (−1)tp(j1,j2, . . . ,jk−1,jt′)·p(j ′ 1, . . . ,^j ′ t, . . . ,j ′ k+1) where {jt}k t=1 and{j ′ t} k+1

t=1 is anysequence in [n] :={1,2, . . . ,n} and

^_j′

t means “omit” jt′.

Proof: See [Kleiman-Laksov; pg 1064–1065].

Definition-Theorem: ThePl¨ucker ideal Ik,n is the ideal of C[· · ·:PJ :· · ·]generated by the Pl¨ucker relations. It is the homogeneous coordinate ring of Grk(Cn)⊆P(

n k)−1.

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The Grassmannian as an algebraic variety IV

In class exercise: Determine the Pl¨ucker ideal that cuts out

Gr2(C4

) in P5

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The Grassmannian as an algebraic variety IV

Gr2(C4

) in P5

.

Solution: It’s cut out by a single equation:

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The Grassmannian as an algebraic variety IV

Gr2(C4

) in P5

.

Solution: It’s cut out by a single equation:

P(12)P(34) −P(13)P(24) +P(14)P(23) =0.

Definition: This model of Grk(Cn)as a subvariety of P( n k)−1 is

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Decomposing

Gr

_k

₍

C

n

₎

_I

Linear algebra fact: By Gaussian (row) elimination M ∈Mat⋆ k,n

can be put into some unique J-row echelon form:

    ⋆ 1 0 0 0 0 0 0 0 ⋆ 0 ⋆ ⋆ 1 0 0 0 0 ⋆ 0 ⋆ ⋆ 0 ⋆ 1 0 0 ⋆ 0 ⋆ ⋆ 0 ⋆ 0 ⋆ 1    

Definition: TheSchubert cell is

X◦

J ={V ∈Grk(C

n

) :Mat(V) reduces to J-row echelon form.} ∼=C#⋆′s

Here Mat(V)∈Mat⋆

k,n is the matrix associated toV. Definition: TheSchubert cell decomposition is

Grk(Cn) =a

J XJ◦.

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Decomposing

Gr

_k

₍

C

n

₎

_II

Work on any (or all) of the following, depending on taste:

In class exercise A: Write down the echelon forms forGr2(C4).

In class exercise B: To each Schubert symbolJ we can associate a lattice path from(0,0)to(k,n−k) and thus a partitionλ(J)(let

the path be the SE border). What is dim(X◦

J) in terms of λ(J)? In class exercise C: Prove that

X◦

J ={V ∈Grk(C

n

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Decomposing

Gr

_k

₍

C

n

₎

_III

Solution A: 1 0 0 0 0 1 0 0 , 1 0 0 0 0 ⋆ 1 0 , 1 0 0 0 0 ⋆ ⋆ 1 , ⋆ 1 0 0 ⋆ 0 1 0 , ⋆ 1 0 0 ⋆ 0 ⋆ 1 , ⋆ ⋆ 1 0 ⋆ ⋆ 0 1 .

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Decomposing

Gr

_k

₍

C

n

₎

_III

Solution A: 1 0 0 0 0 1 0 0 , 1 0 0 0 0 ⋆ 1 0 , 1 0 0 0 0 ⋆ ⋆ 1 , ⋆ 1 0 0 ⋆ 0 1 0 , ⋆ 1 0 0 ⋆ 0 ⋆ 1 , ⋆ ⋆ 1 0 ⋆ ⋆ 0 1 . Solution B: dim(X◦ J) =|λ|.

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Decomposing

Gr

_k

₍

C

n

₎

_III

Solution A: 1 0 0 0 0 1 0 0 , 1 0 0 0 0 ⋆ 1 0 , 1 0 0 0 0 ⋆ ⋆ 1 , ⋆ 1 0 0 ⋆ 0 1 0 , ⋆ 1 0 0 ⋆ 0 ⋆ 1 , ⋆ ⋆ 1 0 ⋆ ⋆ 0 1 . Solution B: dim(X◦ J) =|λ|. Solution C: [Fulton, Section 9.4].

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Group actions I

GLn acts transitively on Cn_{and thus on} _Grk₍_Cn₎_.

The following subgroups of GLn also thus act:

◮ T =invertible diagonal matrices

◮ B =invertible upper triangular matrices

Given Mat(V), the action of g ∈GLn onV is represented by

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Group actions I

GLn acts transitively on Cn_{and thus on} _Grk₍_Cn₎_.

The following subgroups of GLn also thus act:

◮ T =invertible diagonal matrices

◮ B =invertible upper triangular matrices

Given Mat(V), the action of g ∈GLn onV is represented by

replacing row i, i.e,~vi by g ·~vi.

In class exercise A: How doesg =

    1 2 3 4 0 1 2 3 0 0 1 2 0 0 0 1     act onV represented by 1 0 0 0 0 0 1 0 ?

In class exercise B: Show that theV ∈Grk(Cn₎

fixed by the action of T are exactly EJ, theJ-echelon matrix where all⋆=0.

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Group actions II

Solution A: The action sends us to the matrix

1 0 0 0 3 2 1 0

. However, this has echelon form

1 0 0 0 0 2 1 0

. In general, the

action of B gives matrices of the form

1 0 0 0 0 ⋆ 1 0 =X◦ 13. Solution B: Clear.

Solution C: Generalize Solution A, noting the action of B does leftward column operations.

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Schubert (sub)varieties I

In class exercise: Consider X◦

13⊆Gr2(C 4

). What is in the (topological) closure of this cell? (Taking the unique free ⋆to , what vector space(s) occur in the limit?)

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Schubert (sub)varieties I

In class exercise: Consider X◦

13⊆Gr2(C 4

). What is in the (topological) closure of this cell? (Taking the unique free ⋆to , what vector space(s) occur in the limit?)

Solution: We have X13◦ = 1 0 0 0 0 a 1 0 .

Hence a vector space in this cell is spanned by~e1,a~e2+e3. As we take a_{→ ∞} we obtain a new vector space h~e1,~e2i in the limit. In other words X◦ 13=X ◦ 13∪X ◦ 12.

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Schubert (sub)varieties II

Definition: TheSchubert variety XJ is the Zariski (or equivalently, the classical topology induced from the complex

numbers) closure of X◦

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Schubert (sub)varieties II

Definition: TheSchubert variety XJ is the Zariski (or equivalently, the classical topology induced from the complex

numbers) closure of X◦

J.

Proposition: The following facts hold:

(I) XJ =`_I_≤_JX◦

I (I ≤J is the Bruhat order: it≤jt for allt) (II) XJ ={V ∈Grk(Cn) :dim(V ∩ h~e1, . . . ,~eti)≥

#J∩{1,2, . . . ,t}}. (This is the same as the definition ofX◦

J

except “=” is replaced by “≥”.)

(III) XJ is an irreducible subvariety.

(IV) dimXJ =|λ(J)|

Proof: See [Fulton, Section 9.4]; (II) follows once you know (I) and the similar definition of X◦

J. (III) holds becauseXJ is the

closure of a B-orbit. (IV) holds since |λ(J)| is the dimension of the largest cell it contains.

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Schubert (sub)varieties III

In class exercise A: What is the relationship between λ(I) and

λ(J) ifI ≤J?

In class exercise B: (More of a “home exercise”.) Can you describe XJ as a subvariety of Grk(Cn) in its Pl¨ucker embedding?

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Schubert (sub)varieties III

λ(J) ifI ≤J?

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Schubert (sub)varieties III

λ(J) ifI ≤J?

Solution A: It is not hard to argue that λ(I)⊆λ(J).

Solution B: The additional equations needed are the Pl¨ucker coordinates PI =0 whenever I 6≤J. In factPI 6=0 iff I ≤J. This follows from the echelon form. See also [Kleiman-Laksov,

Section 3].

In terms of solution A, if λ=λ(I) then Xλis cut out by Pµwhere µ6⊆λ.

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Schubert (sub)varieties

In class exercise/discussion: Write down the equations cutting

out X24and X23. If you know how, use a computer package such

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Schubert (sub)varieties

In class exercise/discussion: Write down the equations cutting

out X24and X23. If you know how, use a computer package such

as Macaulay 2 to analyze.

Solution: For X24 the defining ideal is the unique Pl¨ucker relation together with P34. This is actually singular at the pointE12 as one

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Summary

The main notions from this section are:

◮ The Grassmannian

◮ Pl¨ucker coordinates, relations, embedding

◮ Schubert cells

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References

M. Brion, Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, 33–85, Trends Math., Birkh¨auser, Basel, 2005.

W. Fulton, Young tableaux, London Mathematical Society, 35,

1999.

S. L. Kleiman and D. Laksov, Schubert calculus,