The motivation Various examples When M has a small codimension Techniques Open issues
Range-compatible homomorphisms on matrix spaces Clément de Seguins Pazzis
Preservers everywhere conference, Szeged, 2017
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Starting point: Range preservers Let f : Mn,p (F) → Mn,p (F) be a linear range preserver i.e. ∀M, Im f (M) = Im M. Then, for some Q ∈ GLp (F). f : M 7→ MQ. Easy application of results on rank preserving maps.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Starting point: Range preservers Let f : Mn,p (F) → Mn,p (F) be a linear range preserver i.e. ∀M, Im f (M) = Im M. Then, for some Q ∈ GLp (F). f : M 7→ MQ. Easy application of results on rank preserving maps.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Starting point: Range preservers Let f : Mn,p (F) → Mn,p (F) be a linear range preserver i.e. ∀M, Im f (M) = Im M. Then, for some Q ∈ GLp (F). f : M 7→ MQ. Easy application of results on rank preserving maps.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
A generalization
More generally, let M linear subspace of Mn,p (F) (say, with codim M ”small"). Which (semi-)linear maps f : M → Mn,p (F) are range preservers? Note: any such map is injective.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
A generalization
More generally, let M linear subspace of Mn,p (F) (say, with codim M ”small"). Which (semi-)linear maps f : M → Mn,p (F) are range preservers? Note: any such map is injective.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
An approach: range restricters Determine the (semi)-linear maps f : M → Mn,p (F) that are range restricters i.e. ∀M ∈ M, Im f (M) ⊆ Im M. Key point: if f linear and injective, f range preserver iff f and f −1 range restricters.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
An approach: range restricters Determine the (semi)-linear maps f : M → Mn,p (F) that are range restricters i.e. ∀M ∈ M, Im f (M) ⊆ Im M. Key point: if f linear and injective, f range preserver iff f and f −1 range restricters.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
From range restricters to range-compatibility
Let f : M → Mn,p (F). Then, ∀M,
f (M) = f1 (M) · · ·
where fk : M → Mn,1 (F) = Fn .
fp (M)
Then, f range restricter iff each fk is range-compatible, i.e. ∀M ∈ M, fk (M) ∈ Im M.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
From range restricters to range-compatibility
Let f : M → Mn,p (F). Then, ∀M,
f (M) = f1 (M) · · ·
where fk : M → Mn,1 (F) = Fn .
fp (M)
Then, f range restricter iff each fk is range-compatible, i.e. ∀M ∈ M, fk (M) ∈ Im M.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The problem What are the range-compatible additive/semi-linear/linear maps f : M → Fn
?
Obvious solutions: the local maps M 7→ MX for fixed X ∈ Fp . Remark: the RC homomorphisms constitute a linear subspace of Hom(M, Fn ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The problem What are the range-compatible additive/semi-linear/linear maps f : M → Fn
?
Obvious solutions: the local maps M 7→ MX for fixed X ∈ Fp . Remark: the RC homomorphisms constitute a linear subspace of Hom(M, Fn ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The problem What are the range-compatible additive/semi-linear/linear maps f : M → Fn
?
Obvious solutions: the local maps M 7→ MX for fixed X ∈ Fp . Remark: the RC homomorphisms constitute a linear subspace of Hom(M, Fn ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
In terms of operator spaces Let S linear subspace of L(U, V ). F :S→V is range-compatible iff ∀s ∈ S, F (s) ∈ Im s. It is local iff F : s 7→ s(x) (evaluation at x).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
In terms of operator spaces Let S linear subspace of L(U, V ). F :S→V is range-compatible iff ∀s ∈ S, F (s) ∈ Im s. It is local iff F : s 7→ s(x) (evaluation at x).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Basic question
Is every range-compatible additive/semi-linear/linear map F : S → V local? S is called RC-complete (respectively, RCL-complete) if every RC homomorphism (resp. RC linear map) on S is local.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Basic question
Is every range-compatible additive/semi-linear/linear map F : S → V local? S is called RC-complete (respectively, RCL-complete) if every RC homomorphism (resp. RC linear map) on S is local.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Connexions
The problem of detecting RC-complete spaces is connected to: range-preservers; the structure of large spaces of matrices with bounded rank; algebraic reflexivity.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Connexions
The problem of detecting RC-complete spaces is connected to: range-preservers; the structure of large spaces of matrices with bounded rank; algebraic reflexivity.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Connexions
The problem of detecting RC-complete spaces is connected to: range-preservers; the structure of large spaces of matrices with bounded rank; algebraic reflexivity.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Connexions
The problem of detecting RC-complete spaces is connected to: range-preservers; the structure of large spaces of matrices with bounded rank; algebraic reflexivity.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The Basic Theorem
Theorem (Various independent authors?) The space Mn,p (F) is RCL-complete. If n ≥ 2 it is also RC-complete. Special case n = 1: every additive map on Mn,p (F) is range-compatible; a map on M1,p (F) is local iff it is linear.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The Basic Theorem
Theorem (Various independent authors?) The space Mn,p (F) is RCL-complete. If n ≥ 2 it is also RC-complete. Special case n = 1: every additive map on Mn,p (F) is range-compatible; a map on M1,p (F) is local iff it is linear.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
RC maps on spaces of nonsingular matrices
Set M=
a −b 2 | (a, b) ∈ R ≃ C. b a
Then,
a −b a F : 7 → b a 0 linear, range-compatible, non-local.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
RC maps on spaces of nonsingular matrices
Set M=
a −b 2 | (a, b) ∈ R ≃ C. b a
Then,
a −b a F : 7 → b a 0 linear, range-compatible, non-local.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
More generally, assume that n = p, dim M ≥ 2 and M ⊂ {0} ∪ GLn (F). Then: Every linear map F : M → Fn is local.
The space of all local maps on M has dimension ≤ n < dim L(M, Fn ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
More generally, assume that n = p, dim M ≥ 2 and M ⊂ {0} ∪ GLn (F). Then: Every linear map F : M → Fn is local.
The space of all local maps on M has dimension ≤ n < dim L(M, Fn ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
More generally, assume that n = p, dim M ≥ 2 and M ⊂ {0} ∪ GLn (F). Then: Every linear map F : M → Fn is local.
The space of all local maps on M has dimension ≤ n < dim L(M, Fn ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
A space of 2 × 2 triangular matrices Set M :=
a b 2 | (a, b) ∈ F . 0 a
Then,
a b 0 F : 7 → 0 a a linear, range-compatible, non-local.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Symmetric matrices over F2 Set M = Sn (F). For S = (si,j )1≤i,j≤n ∈ M, set
s1,1 s2,2 ∆(S) = . ..
(diagonal vector).
sn,n
If F = F2 , then ∆ linear, non-local (if n > 1), range-compatible.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Symmetric matrices over F2 Set M = Sn (F). For S = (si,j )1≤i,j≤n ∈ M, set
s1,1 s2,2 ∆(S) = . ..
(diagonal vector).
sn,n
If F = F2 , then ∆ linear, non-local (if n > 1), range-compatible.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is ∆ RC? Let X ∈ Ker S. Prove that ∆(S)⊥X . Compute 0 = X T SX =
n X
si,i xi2
i=1
=
n X
2 2 si,i xi
i=1
=
X n i=1
si,i xi
2
= (∆(S) | X )2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is ∆ RC? Let X ∈ Ker S. Prove that ∆(S)⊥X . Compute 0 = X T SX =
n X
si,i xi2
i=1
=
n X
2 2 si,i xi
i=1
=
X n i=1
si,i xi
2
= (∆(S) | X )2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is ∆ RC? Let X ∈ Ker S. Prove that ∆(S)⊥X . Compute 0 = X T SX =
n X
si,i xi2
i=1
=
n X
2 2 si,i xi
i=1
=
X n i=1
si,i xi
2
= (∆(S) | X )2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is ∆ RC? Let X ∈ Ker S. Prove that ∆(S)⊥X . Compute 0 = X T SX =
n X
si,i xi2
i=1
=
n X
2 2 si,i xi
i=1
=
X n i=1
si,i xi
2
= (∆(S) | X )2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is ∆ RC? Let X ∈ Ker S. Prove that ∆(S)⊥X . Compute 0 = X T SX =
n X
si,i xi2
i=1
=
n X
2 2 si,i xi
i=1
=
X n i=1
si,i xi
2
= (∆(S) | X )2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is ∆ RC? Let X ∈ Ker S. Prove that ∆(S)⊥X . Compute 0 = X T SX =
n X
si,i xi2
i=1
=
n X
2 2 si,i xi
i=1
=
X n i=1
si,i xi
2
= (∆(S) | X )2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Generalization to perfect fields with characteristic 2
If F perfect field with char(F) = 2, √ s1,1 p S ∈ Sn (F) 7→ ... = ∆(S) √ sn,n
is semi-linear and RC.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Generalization to fields with characteristic 2 Further generalization: with char(F) = 2, let f : F → F be root-linear i.e. ∀(α, t) ∈ F2 , f (α2 t) = α f (t). Then, f (s1,1 ) f (s2,2 ) f S ∈ Sn (F) 7→ ∆(S) := . ..
f (sn,n )
is additive and RC (not obvious!). Note: if char(F) 6= 2 and n ≥ 2 then Sn (F) is RC-complete. If n ≥ 3 then An (F) is RC-complete. Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Generalization to fields with characteristic 2 Further generalization: with char(F) = 2, let f : F → F be root-linear i.e. ∀(α, t) ∈ F2 , f (α2 t) = α f (t). Then, f (s1,1 ) f (s2,2 ) f S ∈ Sn (F) 7→ ∆(S) := . ..
f (sn,n )
is additive and RC (not obvious!). Note: if char(F) 6= 2 and n ≥ 2 then Sn (F) is RC-complete. If n ≥ 3 then An (F) is RC-complete. Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem Theorem (First classification theorem) Let M ⊂ Mn,p (F) with codim M ≤ n − 2. Then, M is RC-complete. for linear maps: C. de Seguins Pazzis, The classification of large spaces of matrices with bounded rank, Israel J. Math. 208 (2015) 219–259.
for homomorphisms: C. de Seguins Pazzis, Range-compatible homomorphisms on matrix spaces, Linear Algebra Appl. 484 (2015) 237–289.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem Theorem (First classification theorem) Let M ⊂ Mn,p (F) with codim M ≤ n − 2. Then, M is RC-complete. for linear maps: C. de Seguins Pazzis, The classification of large spaces of matrices with bounded rank, Israel J. Math. 208 (2015) 219–259.
for homomorphisms: C. de Seguins Pazzis, Range-compatible homomorphisms on matrix spaces, Linear Algebra Appl. 484 (2015) 237–289.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem Theorem (First classification theorem) Let M ⊂ Mn,p (F) with codim M ≤ n − 2. Then, M is RC-complete. for linear maps: C. de Seguins Pazzis, The classification of large spaces of matrices with bounded rank, Israel J. Math. 208 (2015) 219–259.
for homomorphisms: C. de Seguins Pazzis, Range-compatible homomorphisms on matrix spaces, Linear Algebra Appl. 484 (2015) 237–289.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is the bound n − 2 optimal? → Case n = 1 there are non-linear homomorphisms M1,p (F) → F iff F nonprime. Generalization: assume ϕ : F → F nonlinear homomorphism. Then ϕ(x) x [?]1×p−1 7→ F : . [0](n−1)×1 [?](n−1)×(p−1) [0](n−1)×1 is a non-linear RC homomorphism.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Why is the bound n − 2 optimal? → Case n = 1 there are non-linear homomorphisms M1,p (F) → F iff F nonprime. Generalization: assume ϕ : F → F nonlinear homomorphism. Then ϕ(x) x [?]1×p−1 7→ F : . [0](n−1)×1 [?](n−1)×(p−1) [0](n−1)×1 is a non-linear RC homomorphism.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The Second Classification Theorem
Bound n − 2 non-optimal for linear maps! Theorem (Second classification theorem) Assume |F| > 2. Let M ⊂ Mn,p (F) with codim M ≤ 2n − 3. Then M is RCL-complete. For |F| = 2, the proper upper bound is 2n − 4 (counter-example connected to the RC maps on S2 (F2 )).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The Second Classification Theorem
Bound n − 2 non-optimal for linear maps! Theorem (Second classification theorem) Assume |F| > 2. Let M ⊂ Mn,p (F) with codim M ≤ 2n − 3. Then M is RCL-complete. For |F| = 2, the proper upper bound is 2n − 4 (counter-example connected to the RC maps on S2 (F2 )).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
The Second Classification Theorem
Bound n − 2 non-optimal for linear maps! Theorem (Second classification theorem) Assume |F| > 2. Let M ⊂ Mn,p (F) with codim M ≤ 2n − 3. Then M is RCL-complete. For |F| = 2, the proper upper bound is 2n − 4 (counter-example connected to the RC maps on S2 (F2 )).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Optimality of the 2n − 3 bound
Assume that p ≥ 2. Take a b [?]1×(p−2) 0 . 0 a [?]1×(p−2) 7→ a F : [0](n−2)×1 [0](n−2)×1 [0](n−2)×1 [?](n−2)×(p−2)
Then F is linear, RC, non-local.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Split spaces Let M1 ⊂ Mn,p (F) and M2 ⊂ Mn,q (F). Set a M2 = M1 M2 | (M1 , M2 ) ∈ M1 × M2 . M1 Let F : M1
`
M2 → Fn additive. Then, F : M1 M2 7→ F1 (M1 ) + F2 (M2 )
with F1 : M1 → Fn and F2 : M2 → Fn additive. We write a F = F1 F2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Split spaces Let M1 ⊂ Mn,p (F) and M2 ⊂ Mn,q (F). Set a M2 = M1 M2 | (M1 , M2 ) ∈ M1 × M2 . M1
Let F : M1
`
M2 → Fn additive. Then, F : M1 M2 7→ F1 (M1 ) + F2 (M2 )
with F1 : M1 → Fn and F2 : M2 → Fn additive. We write a F = F1 F2 . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Moreover: ` F1 F2 linear ⇔ F1 and F2 linear; ` F1 F2 RC ⇔ F1 and F2 RC; ` F1 F2 local ⇔ F1 and F2 local. Bottom line:
M RC-complete iff M1 and M2 RC-complete.
M RCL-complete iff M1 and M2 RCL-complete.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Moreover: ` F1 F2 linear ⇔ F1 and F2 linear; ` F1 F2 RC ⇔ F1 and F2 RC; ` F1 F2 local ⇔ F1 and F2 local.
Bottom line:
M RC-complete iff M1 and M2 RC-complete.
M RCL-complete iff M1 and M2 RCL-complete.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Proof of the basic theorem ` ` ` We have Mn,p (F) = Fn Fn · · · Fn (with p summands). It suffices to consider the case p = 1!! Lemma Any linear subspace M ⊂ Fn with dim M ≥ 2 is RC-complete! Proof. Let F : M → Fn RC homomorphism. Then, ∀X ∈ M, ∃λX ∈ F : F (X ) = λX X . Classically, F : X 7→ λX for some fixed λ ∈ F, i.e. F : X 7→ X × λ . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Proof of the basic theorem ` ` ` We have Mn,p (F) = Fn Fn · · · Fn (with p summands). It suffices to consider the case p = 1!! Lemma Any linear subspace M ⊂ Fn with dim M ≥ 2 is RC-complete! Proof. Let F : M → Fn RC homomorphism. Then, ∀X ∈ M, ∃λX ∈ F : F (X ) = λX X . Classically, F : X 7→ λX for some fixed λ ∈ F, i.e. F : X 7→ X × λ . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Proof of the basic theorem ` ` ` We have Mn,p (F) = Fn Fn · · · Fn (with p summands). It suffices to consider the case p = 1!! Lemma Any linear subspace M ⊂ Fn with dim M ≥ 2 is RC-complete! Proof. Let F : M → Fn RC homomorphism. Then, ∀X ∈ M, ∃λX ∈ F : F (X ) = λX X . Classically, F : X 7→ λX for some fixed λ ∈ F, i.e. F : X 7→ X × λ . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Quotient space technique Basic idea: induction on the number p of rows. Delete a row (projection). Let F : M → Fn RC homomorphism. K (M) C(M) M= 7→ F (M) = . [?]1×p ? Then C(M) ∈ Im K (M). Moreover, C(M) additive function of K (M)! Indeed, K (M) = 0 ⇒ C(M) = 0. Thus, G(K (M)) F (M) = . ? G : K (M) → Fn−1 is RC and additive. If we know G then we almost know F . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Quotient space technique Basic idea: induction on the number p of rows. Delete a row (projection). Let F : M → Fn RC homomorphism. K (M) C(M) M= 7→ F (M) = . [?]1×p ? Then C(M) ∈ Im K (M). Moreover, C(M) additive function of K (M)! Indeed, K (M) = 0 ⇒ C(M) = 0. Thus, G(K (M)) F (M) = . ? G : K (M) → Fn−1 is RC and additive. If we know G then we almost know F . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Quotient space technique Basic idea: induction on the number p of rows. Delete a row (projection). Let F : M → Fn RC homomorphism. K (M) C(M) M= 7→ F (M) = . [?]1×p ? Then C(M) ∈ Im K (M). Moreover, C(M) additive function of K (M)! Indeed, K (M) = 0 ⇒ C(M) = 0. Thus, G(K (M)) F (M) = . ? G : K (M) → Fn−1 is RC and additive. If we know G then we almost know F . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Quotient space technique Basic idea: induction on the number p of rows. Delete a row (projection). Let F : M → Fn RC homomorphism. K (M) C(M) M= 7→ F (M) = . [?]1×p ? Then C(M) ∈ Im K (M). Moreover, C(M) additive function of K (M)! Indeed, K (M) = 0 ⇒ C(M) = 0. Thus, G(K (M)) F (M) = . ? G : K (M) → Fn−1 is RC and additive. If we know G then we almost know F . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Quotient space technique Basic idea: induction on the number p of rows. Delete a row (projection). Let F : M → Fn RC homomorphism. K (M) C(M) M= 7→ F (M) = . [?]1×p ? Then C(M) ∈ Im K (M). Moreover, C(M) additive function of K (M)! Indeed, K (M) = 0 ⇒ C(M) = 0. Thus, G(K (M)) F (M) = . ? G : K (M) → Fn−1 is RC and additive. If we know G then we almost know F . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Quotient space technique Basic idea: induction on the number p of rows. Delete a row (projection). Let F : M → Fn RC homomorphism. K (M) C(M) M= 7→ F (M) = . [?]1×p ? Then C(M) ∈ Im K (M). Moreover, C(M) additive function of K (M)! Indeed, K (M) = 0 ⇒ C(M) = 0. Thus, G(K (M)) F (M) = . ? G : K (M) → Fn−1 is RC and additive. If we know G then we almost know F . Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
In terms of operators Choose x ∈ V \ {0}, set D := Fx (line spanned by x). For s ∈ L(U, V ), set s mod D : x 7→ s(x) ∈ V /D.
Then, S mod D is a linear subspace of L(U, V /D), and any RC additive F : S → V induces F mod D : S mod D → V /D
so that the diagram commutes S
F
/V
s7→s mod D
S mod D Clément de Seguins Pazzis
F mod D
/ V /D.
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
In terms of operators Choose x ∈ V \ {0}, set D := Fx (line spanned by x). For s ∈ L(U, V ), set s mod D : x 7→ s(x) ∈ V /D.
Then, S mod D is a linear subspace of L(U, V /D), and any RC additive F : S → V induces F mod D : S mod D → V /D
so that the diagram commutes S
F
/V
s7→s mod D
S mod D Clément de Seguins Pazzis
F mod D
/ V /D.
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
In terms of operators Choose x ∈ V \ {0}, set D := Fx (line spanned by x). For s ∈ L(U, V ), set s mod D : x 7→ s(x) ∈ V /D.
Then, S mod D is a linear subspace of L(U, V /D), and any RC additive F : S → V induces F mod D : S mod D → V /D
so that the diagram commutes S
F
/V
s7→s mod D
S mod D Clément de Seguins Pazzis
F mod D
/ V /D.
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Idea 1: F is known if F mod D1 and F mod D2 are known for distinct lines D1 and D2 . Idea 2: codim(S mod D) ≤ codim(S) (rank theorem). Equality holds iff ∀s ∈ L(U, V ), Im s ⊂ D ⇒ s ∈ S. When, codim(S mod D) < codim(S), say that D is adapted to S.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Idea 1: F is known if F mod D1 and F mod D2 are known for distinct lines D1 and D2 . Idea 2: codim(S mod D) ≤ codim(S) (rank theorem). Equality holds iff ∀s ∈ L(U, V ), Im s ⊂ D ⇒ s ∈ S. When, codim(S mod D) < codim(S), say that D is adapted to S.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Application to the first classification theorem Inductive proof. Theorem is trivial if n < 2. Assume now that n ≥ 2. Let S ⊂ L(U, V ) with codim S ≤ dim V − 2. Assume that there are distinct lines D1 = Fy1 and D2 = Fy2 adapted to S. Let F : S → V a RC homomorphism. Then, F mod D1 and F mod D2 are local (by induction). Yields x1 , x2 ∈ U s.t. for all s ∈ S, F (s) = s(x1 ) mod D1
and F (s) = s(x2 ) mod D2.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Application to the first classification theorem Inductive proof. Theorem is trivial if n < 2. Assume now that n ≥ 2. Let S ⊂ L(U, V ) with codim S ≤ dim V − 2. Assume that there are distinct lines D1 = Fy1 and D2 = Fy2 adapted to S. Let F : S → V a RC homomorphism. Then, F mod D1 and F mod D2 are local (by induction). Yields x1 , x2 ∈ U s.t. for all s ∈ S, F (s) = s(x1 ) mod D1
and F (s) = s(x2 ) mod D2.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Application to the first classification theorem Inductive proof. Theorem is trivial if n < 2. Assume now that n ≥ 2. Let S ⊂ L(U, V ) with codim S ≤ dim V − 2. Assume that there are distinct lines D1 = Fy1 and D2 = Fy2 adapted to S. Let F : S → V a RC homomorphism. Then, F mod D1 and F mod D2 are local (by induction). Yields x1 , x2 ∈ U s.t. for all s ∈ S, F (s) = s(x1 ) mod D1
and F (s) = s(x2 ) mod D2.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Application to the first classification theorem Inductive proof. Theorem is trivial if n < 2. Assume now that n ≥ 2. Let S ⊂ L(U, V ) with codim S ≤ dim V − 2. Assume that there are distinct lines D1 = Fy1 and D2 = Fy2 adapted to S. Let F : S → V a RC homomorphism. Then, F mod D1 and F mod D2 are local (by induction). Yields x1 , x2 ∈ U s.t. for all s ∈ S, F (s) = s(x1 ) mod D1
and F (s) = s(x2 ) mod D2.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Application to the first classification theorem Inductive proof. Theorem is trivial if n < 2. Assume now that n ≥ 2. Let S ⊂ L(U, V ) with codim S ≤ dim V − 2. Assume that there are distinct lines D1 = Fy1 and D2 = Fy2 adapted to S. Let F : S → V a RC homomorphism. Then, F mod D1 and F mod D2 are local (by induction). Yields x1 , x2 ∈ U s.t. for all s ∈ S, F (s) = s(x1 ) mod D1
and F (s) = s(x2 ) mod D2.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem, continued
If x1 = x2 then F : s 7→ s(x1 ) QED. Assume x1 6= x2 . Then, for all s ∈ S, F (s) = s(x1 ) mod D1 + D2
and F (s) = s(x2 )
mod D1 + D2
whence ∀s ∈ S, s(x1 − s2 ) ∈ D1 + D2 = Span(y1 , y2 ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem, continued
If x1 = x2 then F : s 7→ s(x1 ) QED. Assume x1 6= x2 . Then, for all s ∈ S, F (s) = s(x1 ) mod D1 + D2
and F (s) = s(x2 )
mod D1 + D2
whence ∀s ∈ S, s(x1 − s2 ) ∈ D1 + D2 = Span(y1 , y2 ).
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem, continued Extend x1 − x2 into a basis B = (x1 − x2 , e2 , . . . , ep ) of U, (y1 , y2 ) into a basis C = (y1 , y2 , . . . , yn ) of V . In those bases ) ( ? [?]1×(p−1) S ←→ ? [?]2×(p−1) =: M. [0](n−2)×1 [?](n−2)×(p−1) Now,
M = (F2 × {0})
a
Fn
a
···
a
Fn
and each summand is RC-complete. Hence, S is RC-complete. Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem, continued Extend x1 − x2 into a basis B = (x1 − x2 , e2 , . . . , ep ) of U, (y1 , y2 ) into a basis C = (y1 , y2 , . . . , yn ) of V . In those bases ) ( ? [?]1×(p−1) S ←→ ? [?]2×(p−1) =: M. [0](n−2)×1 [?](n−2)×(p−1)
Now,
M = (F2 × {0})
a
Fn
a
···
a
Fn
and each summand is RC-complete. Hence, S is RC-complete. Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem: The existence of two adapted lines
Suppose that there is no more than 1 adapted line. Then, for some basis (y1 , . . . , yn ) ∈ V n s.t. ∀i, {s ∈ L(U, V ) : Im s ⊂ Fyi } ⊂ S. By summing, S = L(U, V ) and S is RC-complete. QED.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
First classification theorem: The existence of two adapted lines
Suppose that there is no more than 1 adapted line. Then, for some basis (y1 , . . . , yn ) ∈ V n s.t. ∀i, {s ∈ L(U, V ) : Im s ⊂ Fyi } ⊂ S. By summing, S = L(U, V ) and S is RC-complete. QED.
Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Some open problems What happens beyond the critical bound 2n − 3? → probably too difficult! Structured case: classify RC homomorphisms on subspaces of Sn (F) and An (F). Theorem Let M ⊂ Sn (F) with char(F) 6= 2. If codim M ≤ n − 2 then M is RC-complete. Let M ⊂ An (F). If codim M ≤ n − 3 then M is RC-complete. Optimal bounds! Open problem: what is the least k such that every subspace M ⊂ Sn (F) with codim M ≤ k is RCL-complete? Ditto for An (F). Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Some open problems What happens beyond the critical bound 2n − 3? → probably too difficult! Structured case: classify RC homomorphisms on subspaces of Sn (F) and An (F). Theorem Let M ⊂ Sn (F) with char(F) 6= 2. If codim M ≤ n − 2 then M is RC-complete. Let M ⊂ An (F). If codim M ≤ n − 3 then M is RC-complete. Optimal bounds! Open problem: what is the least k such that every subspace M ⊂ Sn (F) with codim M ≤ k is RCL-complete? Ditto for An (F). Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Some open problems What happens beyond the critical bound 2n − 3? → probably too difficult! Structured case: classify RC homomorphisms on subspaces of Sn (F) and An (F). Theorem Let M ⊂ Sn (F) with char(F) 6= 2. If codim M ≤ n − 2 then M is RC-complete. Let M ⊂ An (F). If codim M ≤ n − 3 then M is RC-complete. Optimal bounds! Open problem: what is the least k such that every subspace M ⊂ Sn (F) with codim M ≤ k is RCL-complete? Ditto for An (F). Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
The motivation Various examples When M has a small codimension Techniques Open issues
Some open problems What happens beyond the critical bound 2n − 3? → probably too difficult! Structured case: classify RC homomorphisms on subspaces of Sn (F) and An (F). Theorem Let M ⊂ Sn (F) with char(F) 6= 2. If codim M ≤ n − 2 then M is RC-complete. Let M ⊂ An (F). If codim M ≤ n − 3 then M is RC-complete. Optimal bounds! Open problem: what is the least k such that every subspace M ⊂ Sn (F) with codim M ≤ k is RCL-complete? Ditto for An (F). Clément de Seguins Pazzis
Range-compatible homomorphisms on matrix spaces
