Properties of Wronski map of Grassmannians of
spaces of solutions of linear differential operators
Yanhe Huang
Mentor: Igor Zelenko
School of Mathematics and System Science,
Beihang University and Texas A&M University
May 13, 2015
Wronskian of a space of functions
Λ = span{f1 (t), f2 (t), . . . , fm (t)},



Wr f1 (t), f2 (t), . . . , fm (t) := det 

dim Λ = m
f1 (t)
f10 (t)
..
.
f2 (t)
f20 (t)
..
.
(m−1)
(m−1)
f1
(t) f2
...
...
..
.
fm (t)
0 (t)
fm
..
.
(m−1)
(t) . . . fm





(t)
Change of basis −→ multiplication of Wronskian by a constant
2 / 26
Wronskian of a space of functions
Λ = span{f1 (t), f2 (t), . . . , fm (t)},



Wr f1 (t), f2 (t), . . . , fm (t) := det 

dim Λ = m
f1 (t)
f10 (t)
..
.
f2 (t)
f20 (t)
..
.
(m−1)
(m−1)
f1
(t) f2
...
...
..
.
fm (t)
0 (t)
fm
..
.
(m−1)
(t) . . . fm





(t)
Change of basis −→ multiplication of Wronskian by a constant
3 / 26
Wronskian of a space of functions
Λ = span{f1 (t), f2 (t), . . . , fm (t)},



Wr f1 (t), f2 (t), . . . , fm (t) := det 

dim Λ = m
f1 (t)
f10 (t)
..
.
f2 (t)
f20 (t)
..
.
(m−1)
(m−1)
f1
(t) f2
...
...
..
.
fm (t)
0 (t)
fm
..
.
(m−1)
(t) . . . fm





(t)
Change of basis −→ multiplication of Wronskian by a constant
4 / 26
Classical Wronksi map
If Λ ∈ Grm (Polm+p−1 ) is an (m + p)−dimensional space of
univariant polynomials of degree at most m + p − 1, we will have
Wr : Grm (Polm+p−1 ) −→ P(Polmp )
Schubert in 1886 showed that the Wronski map is surjective and
the general polynomial in P(Polmp ) has
#m,p =
1!2! . . . (p − 1)! · (mp)!
m!(m + 1)! . . . (m + p − 1)!
preimages.
Polm+p−1 is a space of solution of x(m+p) = 0.
Natural question: What about the preimage of Wronski map for a
space of solution of a general homogeneous linear differential
operator?
5 / 26
Classical Wronksi map
If Λ ∈ Grm (Polm+p−1 ) is an (m + p)−dimensional space of
univariant polynomials of degree at most m + p − 1, we will have
Wr : Grm (Polm+p−1 ) −→ P(Polmp )
Schubert in 1886 showed that the Wronski map is surjective and
the general polynomial in P(Polmp ) has
#m,p =
1!2! . . . (p − 1)! · (mp)!
m!(m + 1)! . . . (m + p − 1)!
preimages.
Polm+p−1 is a space of solution of x(m+p) = 0.
Natural question: What about the preimage of Wronski map for a
space of solution of a general homogeneous linear differential
operator?
6 / 26
Classical Wronksi map
If Λ ∈ Grm (Polm+p−1 ) is an (m + p)−dimensional space of
univariant polynomials of degree at most m + p − 1, we will have
Wr : Grm (Polm+p−1 ) −→ P(Polmp )
Schubert in 1886 showed that the Wronski map is surjective and
the general polynomial in P(Polmp ) has
#m,p =
1!2! . . . (p − 1)! · (mp)!
m!(m + 1)! . . . (m + p − 1)!
preimages.
Polm+p−1 is a space of solution of x(m+p) = 0.
Natural question: What about the preimage of Wronski map for a
space of solution of a general homogeneous linear differential
operator?
7 / 26
Classical Wronksi map
If Λ ∈ Grm (Polm+p−1 ) is an (m + p)−dimensional space of
univariant polynomials of degree at most m + p − 1, we will have
Wr : Grm (Polm+p−1 ) −→ P(Polmp )
Schubert in 1886 showed that the Wronski map is surjective and
the general polynomial in P(Polmp ) has
#m,p =
1!2! . . . (p − 1)! · (mp)!
m!(m + 1)! . . . (m + p − 1)!
preimages.
Polm+p−1 is a space of solution of x(m+p) = 0.
Natural question: What about the preimage of Wronski map for a
space of solution of a general homogeneous linear differential
operator?
8 / 26
Natural generalization
Lx = x(2m) (t) + a2m−1 (t)x(2m−1) (t) + . . . + a0 (t)x(t)
Let VL be the space of solution of Lx = 0.
Wr : Grm (VL ) −→ P(C ∞ ) .
Wronski map on Grm (V ) is strongly non-injective if the preimage
of a general point in the image contains more than one point.
Wronski map on Grm (V ) is essentially injective if the preimage of
a general point in the image contains only one point.
9 / 26
Natural generalization
Lx = x(2m) (t) + a2m−1 (t)x(2m−1) (t) + . . . + a0 (t)x(t)
Let VL be the space of solution of Lx = 0.
Wr : Grm (VL ) −→ P(C ∞ ) .
Wronski map on Grm (V ) is strongly non-injective if the preimage
of a general point in the image contains more than one point.
Wronski map on Grm (V ) is essentially injective if the preimage of
a general point in the image contains only one point.
10 / 26
Natural generalization
Lx = x(2m) (t) + a2m−1 (t)x(2m−1) (t) + . . . + a0 (t)x(t)
Let VL be the space of solution of Lx = 0.
Wr : Grm (VL ) −→ P(C ∞ ) .
Wronski map on Grm (V ) is strongly non-injective if the preimage
of a general point in the image contains more than one point.
Wronski map on Grm (V ) is essentially injective if the preimage of
a general point in the image contains only one point.
11 / 26
Case of Self-adjoint operator
L is called self-adjoint if
Z
b
Z
Lu v dt =
a
b
u Lv dt + Aa,b (u, v)
a
where Aa,b (u, v) = σb (u, v) − σa (u, v).
The restriction of σ to the space of solution VL is a skew symmetric
form on VL .
Proposition
(Hein-Sottile-Zelenko ,2012, for Lx = x(2m) ; Zelenko for general
self-adjoint L) If L is self-adjoint, then Wr(Λ) = Wr(Λ∠ ) where Λ∠ is the
skew-symmetric complement w.r.t the form σ ⇒strong noninjectivity
Question: Is there a non-self adjoint operator for which the
corresponding Wronski map is strongly non-injective?
12 / 26
Case of Self-adjoint operator
L is called self-adjoint if
Z
b
Z
Lu v dt =
a
b
u Lv dt + Aa,b (u, v)
a
where Aa,b (u, v) = σb (u, v) − σa (u, v).
The restriction of σ to the space of solution VL is a skew symmetric
form on VL .
Proposition
(Hein-Sottile-Zelenko ,2012, for Lx = x(2m) ; Zelenko for general
self-adjoint L) If L is self-adjoint, then Wr(Λ) = Wr(Λ∠ ) where Λ∠ is the
skew-symmetric complement w.r.t the form σ ⇒strong noninjectivity
Question: Is there a non-self adjoint operator for which the
corresponding Wronski map is strongly non-injective?
13 / 26
Case of Self-adjoint operator
L is called self-adjoint if
Z
b
Z
Lu v dt =
a
b
u Lv dt + Aa,b (u, v)
a
where Aa,b (u, v) = σb (u, v) − σa (u, v).
The restriction of σ to the space of solution VL is a skew symmetric
form on VL .
Proposition
(Hein-Sottile-Zelenko ,2012, for Lx = x(2m) ; Zelenko for general
self-adjoint L) If L is self-adjoint, then Wr(Λ) = Wr(Λ∠ ) where Λ∠ is the
skew-symmetric complement w.r.t the form σ ⇒strong noninjectivity
Question: Is there a non-self adjoint operator for which the
corresponding Wronski map is strongly non-injective?
14 / 26
Geometric setting
Let C(t) be a natural curve in PV ∗ ,
C(t) := {p ∈ V ∗ : hp, x(·)i = 0 ∀ x(·) ∈ V , such that x(t) = 0} ,
Let c(t) ⊂ V ∗ such that C(t) = span{c(t)},
C (i) (t) := span c(t), c0 (t), . . . , c(i) (t)
m
Plücker embedding PlX
m : Grm (X) → P(∧ X) :
span(x1 , . . . , xm ) → x1 ∧ x2 . . . ∧ xm
15 / 26
Geometric setting
Let C(t) be a natural curve in PV ∗ ,
C(t) := {p ∈ V ∗ : hp, x(·)i = 0 ∀ x(·) ∈ V , such that x(t) = 0} ,
Let c(t) ⊂ V ∗ such that C(t) = span{c(t)},
C (i) (t) := span c(t), c0 (t), . . . , c(i) (t)
m
Plücker embedding PlX
m : Grm (X) → P(∧ X) :
span(x1 , . . . , xm ) → x1 ∧ x2 . . . ∧ xm
16 / 26
Geometric setting
Let C(t) be a natural curve in PV ∗ ,
C(t) := {p ∈ V ∗ : hp, x(·)i = 0 ∀ x(·) ∈ V , such that x(t) = 0} ,
Let c(t) ⊂ V ∗ such that C(t) = span{c(t)},
C (i) (t) := span c(t), c0 (t), . . . , c(i) (t)
m
Plücker embedding PlX
m : Grm (X) → P(∧ X) :
span(x1 , . . . , xm ) → x1 ∧ x2 . . . ∧ xm
17 / 26
Wronski map and Plücker embedding
SL := spant PlVm∗ (C (m−1) (t)) ⊂ P(∧m V ∗ )
AL := {ω ∈ (∧m V ∗ )∗ = ∧m V : ω|SL = 0} = SL⊥ .
Let π : ∧m V → ∧m V /AL be the canonical projection and
V
f := π ◦ P lm
Wr
: Grm (V ) → ∧m V /AL .
Proposition
f
Injectivity properties of Wr are the same as of Wr.
Proposition
If SL = P(∧m V ∗ ) or, equivalently AL = 0 ⇒ Wr is injective.
18 / 26
Reduction to the study of orbits
If m = 2, dimAL ≥ 1 ⇔ there ∃ a 2-form (symplectic form) ω s.t.
ω|C (m−1) (t) = 0 ⇔L is self-adjoint.
What about m = 3?
If L is self-adjoint, then dim AL ≥ 6.
If L is trivial Lx = x(6) , then dim AL = 10 (iff).
The method is to study classical injectivity properties of πK ◦ PlVm
where πK : ∧m V → ∧m V /K with dim K = 1, K ⊂ AL .
Gl2m (V ) acts on ∧m V , or to be specific, P(∧m V ).
It is enough to consider this problem for a representative of each orbit
under this action.
19 / 26
Reduction to the study of orbits
If m = 2, dimAL ≥ 1 ⇔ there ∃ a 2-form (symplectic form) ω s.t.
ω|C (m−1) (t) = 0 ⇔L is self-adjoint.
What about m = 3?
If L is self-adjoint, then dim AL ≥ 6.
If L is trivial Lx = x(6) , then dim AL = 10 (iff).
The method is to study classical injectivity properties of πK ◦ PlVm
where πK : ∧m V → ∧m V /K with dim K = 1, K ⊂ AL .
Gl2m (V ) acts on ∧m V , or to be specific, P(∧m V ).
It is enough to consider this problem for a representative of each orbit
under this action.
20 / 26
Reduction to the study of orbits
If m = 2, dimAL ≥ 1 ⇔ there ∃ a 2-form (symplectic form) ω s.t.
ω|C (m−1) (t) = 0 ⇔L is self-adjoint.
What about m = 3?
If L is self-adjoint, then dim AL ≥ 6.
If L is trivial Lx = x(6) , then dim AL = 10 (iff).
The method is to study classical injectivity properties of πK ◦ PlVm
where πK : ∧m V → ∧m V /K with dim K = 1, K ⊂ AL .
Gl2m (V ) acts on ∧m V , or to be specific, P(∧m V ).
It is enough to consider this problem for a representative of each orbit
under this action.
21 / 26
Reduction to the study of orbits
If m = 2, dimAL ≥ 1 ⇔ there ∃ a 2-form (symplectic form) ω s.t.
ω|C (m−1) (t) = 0 ⇔L is self-adjoint.
What about m = 3?
If L is self-adjoint, then dim AL ≥ 6.
If L is trivial Lx = x(6) , then dim AL = 10 (iff).
The method is to study classical injectivity properties of πK ◦ PlVm
where πK : ∧m V → ∧m V /K with dim K = 1, K ⊂ AL .
Gl2m (V ) acts on ∧m V , or to be specific, P(∧m V ).
It is enough to consider this problem for a representative of each orbit
under this action.
22 / 26
Reduction to the study of orbits
If m = 2, dimAL ≥ 1 ⇔ there ∃ a 2-form (symplectic form) ω s.t.
ω|C (m−1) (t) = 0 ⇔L is self-adjoint.
What about m = 3?
If L is self-adjoint, then dim AL ≥ 6.
If L is trivial Lx = x(6) , then dim AL = 10 (iff).
The method is to study classical injectivity properties of πK ◦ PlVm
where πK : ∧m V → ∧m V /K with dim K = 1, K ⊂ AL .
Gl2m (V ) acts on ∧m V , or to be specific, P(∧m V ).
It is enough to consider this problem for a representative of each orbit
under this action.
23 / 26
Classification of orbits
Image of Plücker embedding = Plücker manifold.
For m = 3, there are 5 different orbits under this action
ω0 = e123 + e456 : open orbit O0 ;
ω1 = e126 + e135 + e234 : codim 1 orbit O1 , the closure is tangential
variety of Plücker manifold;
ω5 = e1 ∧ (e23 + e45 ): codim 5 orbit O5 , the union of lines
connecting two points in Plücker manifold such that the 3-planes
corresponding to these two points intersect in a line;
ω10 = e123 : codim 10 orbit O10 , Plücker manifold;
ω20 = 0.
24 / 26
Results
If K ⊂ O0 → one pair of 3-planes with the same Wronskian.
If K ⊂ O1 → classically injective.
If K ⊂ O5 → 4-parametric family of pairs of 3-planes with same
Wronskian.
Theorem
If dim AL < 10 and dim AL ∩ O5 ≤ 5,then Wr is essentially injective.
Corollary
If dim AL ≤ 5,then Wr is essentially injective.
Theorem
If dim AL = 6 and Wr is strongly non-injective, then L is self-adjoint.
25 / 26
Thanks for your attention.
26 / 26