Combinatorial and geometric aspects of invariant subspaces ...justus/conf/lubbock2014.pdf · Combinatorial and geometric aspects of invariant subspaces of linear operators Justyna

Combinatorial and geometric aspects of

invariant subspaces of linear operators

Justyna Kosakowska,Nicolaus Copernicus University, Torun, Poland

A report on a joint project with Markus Schmidmeier, FAU

AMS - Special Session on Linear Operators in Representation Theoryand in Applications,

Texas Tech University, Lubbock

April, 2014

Justyna Kosakowska, Nicolaus Copernicus University, Torun, PolandCombinatorial and geometric aspects of invariant subspaces of linear operators

Classical case

Fix a positive integer n ∈ N

K – algebraically closed field

Mn = Mn(K ) – vector space of square n × n matrices

Consider Mn as an affine variety (with Zariski topology)

GLn = GL(K ) = {A ∈Mn ; det A 6= 0} – general linear group

A,B ∈Mn are conjugated if there exists T ∈ GLn such thatB = T · A · T−1 (notation: A ∼ B)

M0n subset of Mn consisting of nilpotent matrices

GLn acts on Mn (or on M0n): T ∗ A := T · A · T−1

Orbits of this action: OA = {B ∈Mn ; A ∼ B}


Classical case

A ∈M0n is conjugated to exactly one matrix of the form

Jα =

Jα1 0 . . . 0

0 Jα2 . . . 0...

.... . .

...0 0 . . . Jαm

, Jαi =

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 10 0 0 . . . 0

where α = (α1 ≥ α2 ≥ . . . ≥ αm).Partition of n: α = (α1, . . . , αm) such thatα1 ≥ α2 ≥ . . . ≥ αm and n = α1 + . . . + αm = |α|

Pn – the set of all partitions of nThere is a bijection: { orbits in M0

n } ←→ Pn

OA = OJα 7→ (Jα1 , . . . , Jαm) 7→ α = (α1, . . . , αm)

So we can write Oα instead of OA.


Classical case

Want to understand the following partial order:A ≤deg B if and only if OB ⊆ OA

OA – the closure of OA in the Zariski topology in Mn

Nα =⊕s

i=1 K [T ]/(Tαi ) — nilpotent linear operator ass. withα = (α1, . . . , αm)

Nα – nilpotent K [T ]-module

Nα ∼= Nβ if and only if Oα = Oβ

Example (α – the conjugate partition of α)

α = (5, 3, 3, 3, 2) α = (5, 5, 4, 1, 1)


Dom-order and box-order

Definition

On the set Pn we definie partial orders.

1 γ ≤dom γ if and only if, for any j :γ1 + · · ·+ γj ≤ γ1 + · · ·+ γj

2 ≤box – partial order generated by a sequence of moves oftype (going up with a box):

x≤box

x

Lemma

γ ≤dom γ if and only if γ ≤box γ


Classical results

Theorem (Gerstenhaber, Hesselink)

For α, β ∈ Pn:

Oβ ⊆ Oα ⇐⇒ dimKerB s ≥ dimKerAs for all s ∈ N

It is known:

dimKerAs =s∑

i=1

αi

Corollary

Nα ≤deg Nβ ⇐⇒ Oβ ⊆ Oα ⇐⇒ α ≤dom β ⇐⇒ α ≤box β


The main aim

The main aim. Get similar results for triples (Nα,Nβ, f ),where f : Nα → Nβ is a monomorphism of K [T ]-modules

Fix partitions α, β, γ. Consider triples (Nα,Nβ, f ) withCoker f ∼= Nγ.

Hβα = HomK (Nα,Nβ) = M|α|,|β|(K ) – affine variety (Zariski

topology)

Vβα,γ ⊂ Hβ

α — subset consisting of all monomorphisms

f : Nα → Nβ

with Coker f ∼= Nγ


Group action

Group action: G = AutK [T ]Nα × AutK [T ]Nβ acts on Vβα,γ:

(g , h) · f = h ◦ f ◦ g−1

Nαf−−−−−−−−→ Nβyg

yh

Nαh◦f ◦g−1

−−−−−−−−→ Nβ

For triples X = (Nα,Nβ, f ) and Y = (Nα,Nβ, g) weinvestigate the partial order

X ≤deg Y ⇐⇒ OY ⊆ OX


Combinatorial tools

Definition (Littlewood-Richardson tableaux)

An LR-tableau of type (α, β, γ) is a skew diagram of shapeβ\γ with α1 entries 1 , α2 entries 2 , etc. The entries areweakly increasing in each row, strictly increasing in eachcolumn, and satisfy the lattice permutation property (for eachc ≥ 0, ` ≥ 2 there are at least as many entries `− 1 on theright hand side of the c-th column as there are entries `).

Example

For α = (2, 2, 1, 1), β = (5, 4, 3, 3, 2, 1), γ = (4, 3, 2, 2, 1):

Γ1 :

11

1 12

2

Γ2 :

12

1 11

2

α = (4, 2)


LR-tableaux and LR-sequences

An LR-tableau Γ is given as a sequence of partitions

Γ = [γ(0), . . . , γ(s)]

where γ(i) denotes the region in the Young diagram β whichcontains the entries , 1 , . . ., i . If Γ has shape (α, β, γ),then γ = γ(0), β = γ(s), and αi = |γ(i) \ γ(i−1)| fori = 1, . . . , s.

Γ :

4

32

1

Γ = [(62), (621), (631), (641), (741)]


LR-tableaux and extensions

It is known:There exists a short exact sequence of nilpotent K [T ]-modules

η : 0 −→ Nαf−→ Nβ −→ Nγ −→ 0

if and only if there exists an LR-tableau Γ of type (α, β, γ).

Γ ”controls” Nβ/Ti f (Nα) for all i .

Therefore: Vβα,γ 6= ∅ if and only if there exists an LR-tableau

of type (α, β, γ).

Vβα,γ =

•⋃VΓ

Is Γ enough to ”control” orbits of action of G on Vβα,γ?

In general, answer is NO.


Categorification

Sβα,γ — the category consisting of all systems

X = (Nα,Nβ, f )

where f : Nα → Nβ is a monomorphism and Coker f ∼= Nγ;

Morphisms are defined in a natural way;

The G -orbits in Vβα,γ are in 1− 1-correspondence with the

equivalence classes of objects in Sβα,γ.

In general the problem is difficult (wild). We need additionalassumptions.


Klein tableaux

If α1 ≤ 2 (equivalently, if all entries of LR-tableau are ≤ 2)LR-tableaux ”control” orbits of G in Vβ

α,γ

Definition (Klein tableau)

An LR-tableau such that each entry equal to 2 carries asubscript, subject to the conditions:

1 If a symbol 2r occurs in the m-th row in the tableau, then1 ≤ r ≤ m − 1.

2 If 2r occurs in the m-th row and the entry above 2r is 1,then r = m − 1.

3 The total number of symbols 2r in the tableau cannotexceed the number of entries 1 in row r .


Indecomposable objects in Sβα,γ, α1 ≤ 2

For α = (2, 2, 1, 1), β = (5, 4, 3, 3, 2, 1), γ = (4, 3, 2, 2, 1):

0 −→ Nαf−→ Nβ −→ Nγ −→ 0

Γ :

11

1 12

2

Π :

11

1 122

23

∆ :• • • • •5 4 3 2 1

� ��

Theorem (Beers, Hunter, Walker, 1983)

Let α1 ≤ 2. Each indecomposable object is isomorphic to:

Pm0 : 0→ N(m); (m ≥ 1)

Pm1 : N(1) → N(m); 1→ Tm−1 (m ≥ 1)

Pm2 : N(2) → N(m); 1→ Tm−2 (m ≥ 2)

Bm,r2 : N(2) → N(m,r); 1→ (Tm−2,T r−1) (m − 2 ≥ r ≥ 1)


Combinatorial invariants

Invariants for the indecomposable objectsX : Pm

0 Pm1 Pm

2 Bm,r2

Γ(X ) : ...}

m1

...}

m21

...}

m

m

2

...1

......}

r

Π(X ) : ...}

m1

...}

m 1

...

2r

}m

r=m−1

m

2r

...1

......}

r

r<m−1

∆(X ) : ∅ •m

• •� �m m−1

• • •� �m r


The arc diagram of an object

The Klein tableau for a direct sum M ⊕M ′ has a diagramgiven by the union β ∪ β′ of the partitions representing theambient spaces, and in each row the entries are obtained bylexicographically ordering the entries in the corresponding rowsin the tableaux for M and M ′, with empty boxes coming first.

Sβα,γ 3 X 7→ Π(X ) 7→ ∆(X )

Example:X = B5,3

2 ⊕ B4,22 ⊕ P3

1 ⊕ P11 .

Γ :

11

1 12

2

Π :

11

1 122

23

∆ :• • • • •5 4 3 2 1

� �� Justyna Kosakowska, Nicolaus Copernicus University, Torun, PolandCombinatorial and geometric aspects of invariant subspaces of linear operators

Arc order

If α1 ≤ 2, then there is a bijection between orbits of G in Vβα,γ

and Klein tableaux (equivalently arc diagrams).Sβα,γ 3 X 7−→ ∆(X ) — arc diagram of X

≤arc – partial order defined by:

• • • •� ��

• • • •

� �� (A)

<arc

@@@(C)

>arc

• • • •� � � �

• • •� �

• • •

� �

• • •� �

�� (B)

<arc

@@@(D)

>arc

Definition: X ≤arc Y if and only if ∆(X ) ≤arc ∆(Y )


Main results

Theorem (K-Schmidmeier 2011/12)

K = K, α, β, γ – partitions with α1 ≤ 2, Y ,Z ∈ Sβα,γ1

Y ≤deg Z if and only if Y ≤arc Z .

In Vβα,γ:

2 and there is the unique G-orbit ≤arc-maximal(equivalently ≤deg-maximal),

3 there are cβα,γ G-orbits ≤arc-minimal (equivalently≤deg-minimal).

4 other combinatorial properties ...

cβα,γ – the Littlewood-Richardson coefficient


Example: The deg-order in V4321211,321

Let α = (211), β = (4321), γ = (321). Objects in Sβα,γ (up toiso):

B4,12 ⊕ P3

1 ⊕ P21 • • • •

4 3 2 1

��B4,2

2 ⊕ P11 ⊕ P2

1 • • • •4 3 2 1

� �

B3,12 ⊕ P4

1 ⊕ P21 • • • •

4 3 2 1

� �B4,3

2 ⊕ P21 ⊕ P1

1 • • • •4 3 2 1

� �B3,2

2 ⊕ P41 ⊕ P1

1 • • • •4 3 2 1

� � B2,12 ⊕ P3

1 ⊕ P41 • • • •

4 3 2 1

� �


Example: The deg-order in V4321211,321

∆6 :

• • • •4 3 2 1

��

��

@@I

∆4 :

• • • •4 3 2 1

� � ∆5 :

• • • •4 3 2 1

� �6 6

∆1 :

• • • •4 3 2 1

� � ∆3 :

• • • •4 3 2 1

� ��

@@I

∆2 :

• • • •4 3 2 1

� �Justyna Kosakowska, Nicolaus Copernicus University, Torun, PolandCombinatorial and geometric aspects of invariant subspaces of linear operators

General case

We do not assume that α1 ≤ 2.

LR-tableaux (Klein tableaux) do not control G -orbits in Vβα,γ.

There may be infinitely many G -orbits with the sameLR-tableau (Klein tableaux).

Given an LR-tableau Γ of type (α, β, γ), denote by VΓ ⊆ Vβα,γ

consisting of all f of type Γ.

Given Γ, Γ of type (α, β, γ). Define a preorder (reflexive andantisymmetric):

Γ ≤closure Γ⇐⇒ VΓ ∩ VΓ 6= ∅

≤∗closure – transitive closure of ≤closure


Box-order

Γ, Γ - LR-tableaux of type (α, β, γ)We say Γ <box Γ if Γ is obtained from Γ by exchanging twoentries which are the only entries in their respective column insuch a way that the lower entry is the higher position in Γ.Examples:

12

1

1

<box1

11

2

12

1 13

2

<box

12

1 12

3

≤box – partial order generated by these moves


Dominance-order

Γ, Γ - LR-tableaux of type (α, β, γ)

Definition

Two LR-tableaux Γ = [γ(0), . . . , γ(s)], Γ = [γ(0), . . . , γ(s)]of the same shape are in the dominance order, insymbols Γ ≤dom Γ, if for each i , γ(i) ≤dom γ(i) holds.

Two partitions γ, γ are in the natural partial order, insymbols γ ≤dom γ, if the inequality

γ1 + · · ·+ γj ≤ γ1 + · · ·+ γjholds for each j .


Main results

Theorem (K-Schmidmeier, 2013/14)

Let Γ, Γ be LR-tableaux of type (α, β, γ).

1 Γ ≤box Γ =⇒ Γ ≤∗closure Γ =⇒ Γ ≤dom Γ

2 If β \ γ is vertical and horizontal strip, thenΓ ≤box Γ⇐⇒ Γ ≤∗closure Γ⇐⇒ Γ ≤dom Γ

3 other combinatorial properties ...

The skew diagram β \ γ is said to be a horizontal (resp.

a vertical) strip, if βi ≤ γi + 1 (resp. βi ≤ γi + 1), for all i .

Γ :1

23

1

7→ (1, 3, 2, 1)


Example

β = (6, 5, 4, 3, 2, 1), γ = (5, 4, 3, 2, 1) and α = (3, 2, 1).

322111

231211

231121

213121

121321

211321

213211

232111

123121

321121

321211

312211

312121

123211132121

132211

........

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References

Justyna Kosakowska and Markus Schmidmeier, Operationson arc diagrams and degenerations for invariantsubspaces of linear operators, to appear in Tran. Amer.Math. Soc., arXiv:1202.2813v1 [math.RT],

Justyna Kosakowska and Markus Schmidmeier, Arc diagramvarieties, Contemporary Mathematics series of the AMS, 607,2014, arXiv:1211.5798 [math.RT],

Justyna Kosakowska and Markus Schmidmeier, Varieties ofinvariant subspaces given by Littlewood-Richardsontableaux, Oberwolfach Preprint, OWP 2014-01,


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Combinatorial and geometric aspects of invariant subspaces ...justus/conf/lubbock2014.pdf · Combinatorial and geometric aspects of invariant subspaces of linear operators Justyna