Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
Basic definitions
Linear free divisors
Groups, Lie algebras
Free discriminants
in prehomogeneous vector spaces
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Mathias Schulze
Joint work with Michel Granger and David Mond
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Warwick, June 3, 2011
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Linear free divisors
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
Definition (Buchweitz-Mond)
A hypersurface D ⊂ Cn =: V is a linear free divisor (LFD) if
Der(log D) has a basis of linear (degree 0) vector fields.
Theorem (Saito’s Criterion)
δ1 , . . . , δn ∈ Der(log D) form a basis iff ∆ = det(δi • xj ) 6= 0
square free. In this case, D = {p ∈ V | ∆(p) = 0}.
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Corollary
Any linear free divisor D ⊂ V is defined by a homogeneous
polynomial ∆ of degree n.
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Groups, Lie algebras
Definition
Mathias Schulze
To a linear free divisor D = {∆ = 0} we associate a
I
I
Free discriminants
in prehomogeneous
vector spaces
Lie algebra (representation)
gD := {A ∈ gln (C) | xAt ∂ t ∈ Der(log D)},
linear algebraic group
GD := {g ∈ GLn (C) | g .D = D}◦ .
Lemma
gD is the Lie algebra of GD .
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Proof.
Taylor expanding ∆ ◦ (I + Aε) = a(ε) · ∆ over C[ε] gives
∆ + ∂(∆)Ax t ε = (a(0) + a0 (0)ε) · ∆.
| {z }
=xAt ∂ t (∆)
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Low dimension
n
∆
1
x
2
xy
Der(log D)
x
x
0
0
y

3
3
xyz
2
(y + xz)z
4
xyzw
4
(y 2 + xz)zw
4
(yz + xw )zw
4
x(y 3 − 3xyz + 3x 2 w )
4
y 2 z 2 − 4xz 3 − 4y 3 w +
18xyzw − 27w 2 x 2
x
0
0

x
 4x
−2y

x
0
0
y

0
0
0
0

x
y

y
 4x
−2y
z
0
0

x
0
0
y

x
0
z
−w

x
y
0
y

0
x
0
0

3x
2y
0
3x

y
2z
0
y

0
0
z

z
−2z 
0

0
0
0
0

z
0
0
w

z
0

−2z
0
0
0
0
w

0
−w
0
w 

z
0 
0
0

z
w
2z
3w 

y
z 
x
y

z
0
2y
z 

3w
0 
2z
3w
0
y
0
y
y
z
gD
reductive?
C
yes
C2
yes
C3
yes
Mathias Schulze
Basic definitions
C ⊕ g2
no
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
4
C
yes
C2 ⊕ g2
no
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
2
C ⊕ g2
no
C ⊕ g3
no
gl2 (C)
yes
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Prehomogeneous vector spaces
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
Which G ⊂ GLn (C) define LFDs? Necessary conditions:
I
G connected, dim G = n.
I
G has open orbit with codim. 1 complement D.
I
D\ Sing D is a union of codim. 1 orbits.
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Theorem (Granger, Mond, S.)
The converse holds above if the connected isotropy groups
at generic points in D are isomorphic to C∗ .
Proof.
OV ⊗ g ∼
= Der(log D), induced by g 3 A 7→ Ax t , can be
checked on V \ Sing D by reflexivity of Der(log D). . . .
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Relative invariants
Mathias Schulze
Let D be a linear free divisor with group G = GD , and
irreducible components Di = {∆i = 0}, i = 1, . . . , N.
Basic definitions
Definition
Structure results
I
f ∈ C(V ) is rel. invariant with character χf ∈
g .f := f ◦ g −1 = χf (g ) · f ,
I
X ∗ (G )
Linear free divisors
Groups, Lie algebras
if
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
∀g ∈ G .
X0∗ (G ) := {χf | f relative invariant}
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Lemma
I
C[∆±1
i ] is the algebra of all relative invariants.
I
X0∗ (G ) is a free abelian group with basis χ∆i .
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Reductive case
Definition
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
A linear free divisor D is called reductive if G = GD is so.
Basic definitions
Lemma
Structure results
A reductive linear free divisor has dim Z (G ) components.
Linear free divisors
Groups, Lie algebras
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Proof.
For p ∈ V \D, Gp is finite, hence N = rk X0∗ (G ) = rk X ∗ (G ).
By reductivity, G = Z · S almost direct product, Z torus, S
semisimple, hence rk X ∗ (G ) = rk X ∗ (Z ) = dim Z .
Theorem (Granger, Mond, S.)
If D is a reductive LFD then the number of irreducible
components of D and V coincide. Moreover, all irreducible
G -modules in V occur with multiplicity one.
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Reductive case (continued)
Corollary
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
All irreducible linear free divisors occur in Saito-Kimura’s list.
Basic definitions
Corollary
Structure results
G /C∗
If
is semisimple, then D is one of the following linear
free divisors (up to a castling transformation):
1. D = {0} ⊂ C.
2. D = {y 2 z 2 −4xz 3 −4y 3 w +18xyzw −27w 2 x 2 = 0} ⊂ C4
with G = GL2 (C) acting naturally on S 3 C2 ∼
= C4 .
3. D ⊂ C12 with G = SL3 (C) × GL2 (C)
4. D ⊂ C40 with G = SL5 (C) × GL4 (C)
Proposition (Granger, Mond, S.)
A linear free divisor D is normal crossing iff G is abelian.
Linear free divisors
Groups, Lie algebras
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Quiver representations
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
Definition
I
quiver = directed graph Q = (Q0 , Q1 , s, t)
I
I
I
I
quiver representation: M = ({Vi }i∈Q0 , {fα }α∈Q1 )
I
I
I
I
I
Q0 set of vertices
Q1 ⊂ Q0 × Q0 set of arrows
s, t : Q1 → Q0 source/target maps
Vi finite dim. vector space, di = dim Vi
fα : Vsα → Vtα linear map
quiver representation
space:
L
Rep(Q, d) = α∈Q1 Mat(C, dtα × dsα )
Q
quiver group: GL(Q, d) = i∈Q0 GLdi (C)
quiver representation: GL(Q, d) → AutC Rep(Q, d),
−1 .
(g .f )α := gtα ◦ fα ◦ gsα
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Star quiver
Mathias Schulze
Example (Buchweitz-Mond)
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
•1
x1,2
x2,2
•1 x1,1
x2,1
I
I
/
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
•2 o x1,3 •1
x2,3
GL(Q, d) = GL2 (C) × GL1 (C)3 acts
by
 on Rep(Q, d) 
d1−1
0
0
a1,1 a1,2
x1,1 x1,2 x1,3 
0 d2−1
0 .
a2,1 a2,2
x2,1 x2,2 x2,3
0
0 d3−1
Q
D = { i<j (x1,i x2,j − x1,j x2,i ) = 0} is free.
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Quiver category
Mathias Schulze
Basic definitions
Definition
Let M = (V , f ) quiver be a quiver representation. Consider:
L
I V :=
i∈Q0 Vi
I
I
f
α
fα : V Vsα →
Vtα ,→ V ,
fα fsα = fα = ftα fα ,
fi 2
= fi ,
fi : V Vi ,→ V .
other products zero.
Then V becomes a module over the path algebra CQ
defined by above relations.
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Theorem
{quiver representations} and {CQ-modules} are equivalent
categories, and hereditary (i.e. Ext p = 0 for p ≥ 2).
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Ringel, Euler, Tits
Ringel sequence
Mathias Schulze
Let M = (Vi , fα ) and N = (Wi , gα ) be Q-representations.
0 / HomQ (M, N)
/
L
i∈Q0
GF
@A c
L
M,N
/
α∈Q1
where
Hom(Vi , Wi )
ED
BC
Hom(Vsα , Wtα ) / Ext1Q (M, N) / 0
cM,N ((φi )i∈Q0 ) = (φtα fα − gα φsα )α∈Q1 .
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Euler/Tits form
For dim. vectors m = dim(M) and n = dim(N),
hm, niQ = dim HomQ (M, N) − dim Ext1Q (M, N),
qQ (m) = hm, miQ .
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Quiver LFDs
Theorem (Granger, Mond, S.)
The discriminant D in Rep(Q, d) is free iff
I
qQ (d) = 1 (enough for Q Dynkin [Buchweitz, Mond]),
I
d is a Schur root,
I dimC Ext1 (M, M)
Q
= 1 for minimal degenerations M.
Mathias Schulze
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Theorem (Granger, Mond, S.)
If D in Rep(Q, d) is free then Q is a tree.
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Proof.
I
Rep(Q, d) has #Q1 irreducible summands.
I
Z (GL(Q, d)) consists of #Q0 copies of C∗ .
I
Thus, #Q1 = #Q0 − 1 and Q connected.
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Reflection functors
Mathias Schulze
Definition (Bernstein-Gel’fand-Ponomarev)
Basic definitions
Linear free divisors
Groups, Lie algebras
Reflection of (Q, d)
I
I
Structure results
P
at a source i: (i, j) 7→ (j, i), di 7→ (i,j) dj − di .
P
at a sink j: (i, j) 7→ (j, i), dj 7→ (i,j) dj − dj .
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Example (Star quiver revisited)
•1
•1
/ •2 o •1
•O 1
···
•1 / •2 o •1
•O 1
•1 o •2 / •1
•1
•1
/ •1 o •1
Theorem (Granger, Mond, S.)
Quiver free divisors are stable under BGP reflections.
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
LCT and homogeneity
Mathias Schulze
Basic definitions
Cohomology of complement U of D ⊂ X
I
I
Linear free divisors
Groups, Lie algebras
∼
Grothendieck: Ω•X (∗D) → Rj∗ CU , j : U = X \D ,→ X .
def
Log. Comparison “Theorem” ⇔
∼
Ω•X (log D) ,→
Ω•X (∗D)
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Theorem (Castro-Narvaez-Mond)
For free D, locally (weakly) quasihomogeneous ⇒ LCT.
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Conjecture
For free D, LCT implies strong Euler homogeneity.
(True, if dim X ≤ 3 or D Koszul free.)
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Homogeneity results
Lemma
A linear free divisor D = {f = 0} is strongly Euler homog. iff
◦
the GD and AD := GD,f
orbits in D coincide.
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Proposition (Granger, S.)
A LFD D is strongly Euler homogeneous at p ∈ D if the
normal representation of Gp is prehomogeneous.
Theorem (Granger, Mond, S.)
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
Quiver linear free divisors are
I
strongly Euler homogeneous,
I
locally weakly q.h. at non-regular points,
I
locally weakly q.h. if qQ positive non definite (Q tame),
I
locally quasihomogeneous if Q Dynkin.
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Koszul LFDs
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
Corollary (Calderon-Narvaez-Torrelli)
Basic definitions
A Koszul linear free divisor D = {f = 0} satisfies LCT iff
1. D is Euler homogeneous,
2. Der(log f )-stratification of D is finite,
3. −1 is the only integer root of bf (s).
Theorem (Granger, S.)
For any reductive linear free divisor D = {f = 0}, the roots
of bf (s) are symmetric about −1, the only integer root.
Corollary
For reductive Koszul linear free divisors,
LCT ⇔ strong Euler homog.
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Free discriminants
in prehomogeneous
vector spaces
Low dim. b-Functions
Mathias Schulze
n
f
gD
reductive?
b(s)
1
x
C
Yes
−1
2
xy
C2
Yes
−1, −1
3
Yes
No
−1, −1, −1
− 54 , −1, −1, − 43
4
3
3
xyz
(y 2 + xz)z
C
C ⊕ g2
4
4
4
xyzw
(y + xz)zw
(yz + xw )zw
C
C ⊕ g2
2
C ⊕ g2
Yes
No
No
4
x(y 3 − 3xyz + 3x 2 w )
C ⊕ g3
No
gl2 (C)
Yes
2
2 2
4
3
4y 3 w
2 2
y z − 4xz −
18xyzw − 27w x
2
+
−1, −1, −1, −1
− 45 , −1, −1, −1, − 43
− 34 , −1, −1, −1, − 32
− 75 , − 43 , − 65 , −1, −1, −1,
− 54 , − 32 , − 53
− 76 , −1, −1, − 65
Proposition (S.)
Let f be a reduced relative invariant polynomial in a
reductive prehomogeneous vector space. Then the
multiplicity of −1 in bf (s) is at least the number of
irreducible factors of f .
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions
Thank you for your attention!
Free discriminants
in prehomogeneous
vector spaces
Mathias Schulze
Basic definitions
Linear free divisors
Groups, Lie algebras
Structure results
Low dimension
Prehom. vector spaces
Relative invariants
Reductive case
Quiver spaces
Quiver representations
Star quiver
Quiver category
Ringel, Euler, Tits
Quiver LFDs
Reflection functors
Log. cohomology
LCT and homogeneity
Homogeneity results
Koszul LFDs
Low dim. b-Functions