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