Extremal Problems for Spaces of Matrices

Transcription

1 Extremal Problems for Spaces of Matrices Roy Meshulam Technion Israel Institute of Technology Optimization, Complexity and Invariant Theory Princeton, June 2018

2 Plan Maximal Rank and Matching Numbers Flanders theorem and its extensions Maximal rank in the exterior algebra Edmonds and Lovász min-max theorems Minimal Rank: between Algebra and Topology Algebraically closed vs. finite field cases Nonsingular spaces of real matrices Linear Spaces in Nilpotent varieties The matrix case: Gerstenhaber theorems Some Lie algebra generalizations

3 Notations M m n (F) - the space of m n matrices over a field F. M n (F) = M n n (F). Sym n (F) = {A M n (F) : A = A T }, Alt n (F) = {A M n (F) : A = A T & A(i,i) = 0 for all i}. For u F m,v F n let For S M m n (F) let: u v = u v T M m n (F). ρ(s) = max{rk(a) : A S}, ρ(s) = min{rk(a) : 0 A S}.

4 Notations M m n (F) - the space of m n matrices over a field F. M n (F) = M n n (F). Sym n (F) = {A M n (F) : A = A T }, Alt n (F) = {A M n (F) : A = A T & A(i,i) = 0 for all i}. For u F m,v F n let For S M m n (F) let: u v = u v T M m n (F). ρ(s) = max{rk(a) : A S}, ρ(s) = min{rk(a) : 0 A S}.

5 Subspaces of M n (F) with bounded ρ Theorem [Flanders ( F n), M (any F)] Let S M n (F) be a linear subspace such that ρ(s) k. Then: dims kn. dims = kn iff either S = V F n or S = F n V for some k-dimensional linear subspace V F n. n k n k k n k n

6 Subspaces of Alt n (F) with bounded ρ Theorem [M ( F n), de Seguins Pazzis (any F)] Let S Alt n (F) be a linear subspace such that ρ(s) k = 2t. Then: {( ) ( )} 2t +1 t +1 dims max,tn t +1 t t 2t +1 dims = ( 2t+1) 2 dims = tn ( ) t+1 2

7 An Extremal Problem for Graph Matchings A Matching in a graph is a family of pairwise disjoint edges. The Matching Number of G = (V,E): ν(g) = max{ M : M E is a matching }. Theorem [Erdős-Gallai]: If G = (V,E) satisfies ν(g) t then {( ) ( )} 2t +1 t +1 E max,t V. 2 2 K 2t+1 K t (n t)k 1

8 An Extremal Problem for Graph Matchings A Matching in a graph is a family of pairwise disjoint edges. The Matching Number of G = (V,E): ν(g) = max{ M : M E is a matching }. Theorem [Erdős-Gallai]: If G = (V,E) satisfies ν(g) t then {( ) ( )} 2t +1 t +1 E max,t V. 2 2 K 2t+1 K t (n t)k 1

9 Proof Idea Let S Alt n (F) such that ρ(s) k = 2t. Associate with S a graph G S = ([n],e S ), with E S = dims. Main point: The matching number of G S satisfies ν(g S ) ρ(s) 2. Use extremal graph theory to bound from above dims = E S.

10 Proof Idea Let S Alt n (F) such that ρ(s) k = 2t. Associate with S a graph G S = ([n],e S ), with E S = dims. Main point: The matching number of G S satisfies ν(g S ) ρ(s) 2. Use extremal graph theory to bound from above dims = E S.

11 Proof Idea Let S Alt n (F) such that ρ(s) k = 2t. Associate with S a graph G S = ([n],e S ), with E S = dims. Main point: The matching number of G S satisfies ν(g S ) ρ(s) 2. Use extremal graph theory to bound from above dims = E S.

12 From Matrix Spaces to Graphs The Colexicographic Order [n] = {1,...,n}, [n] 2 < = {(i,j) [n] 2 : i < j}. (i,j) (i,j ) j < j or (j = j & i < i ). The Leading Entry of a Matrix For 0 A = ( A(i,j) ) n i,j=1 Alt n(f) let q(a) = max{(i,j) [n] 2 < : A(i,j) 0}. The Graph of an Alternating Space For a subspace S Alt n (F) let G S = ([n],e S ), where E S = { {i,j} : (i,j) = q(a) for some 0 A S }.

13 Examples The Colexicographic Order: The Leading Entry of a Matrix A = q(a) = (2,4). The Graph of an Alternating Space { S = 0 x 0 x 0 y 0 y 0 : x,y F } G S = 1 2 3

14 Examples The Colexicographic Order: The Leading Entry of a Matrix A = q(a) = (2,4). The Graph of an Alternating Space { S = 0 x 0 x 0 y 0 y 0 : x,y F } G S = 1 2 3

15 Examples The Colexicographic Order: The Leading Entry of a Matrix A = q(a) = (2,4). The Graph of an Alternating Space { S = 0 x 0 x 0 y 0 y 0 : x,y F } G S = 1 2 3

16 The Pfaffian of an Alternating Matrix C = ( C(i,j) ) n i,j=1 Alt n(f) of even order n = 2t. M n - all perfect matchings in K n. For let The Pfaffian of C is: M = { {k 1 < l 1 },...,{k t < l t } } M n ( 1 2 n 1 n θ(m) = sgn k 1 l 1 k t l t Pf(C) = Fact: det(c) = Pf(C) 2. M M n θ(m) t C(k i,l i ). i=1 ).

17 The Pfaffian of an Alternating Matrix C = ( C(i,j) ) n i,j=1 Alt n(f) of even order n = 2t. M n - all perfect matchings in K n. For let The Pfaffian of C is: M = { {k 1 < l 1 },...,{k t < l t } } M n ( 1 2 n 1 n θ(m) = sgn k 1 l 1 k t l t Pf(C) = Fact: det(c) = Pf(C) 2. M M n θ(m) t C(k i,l i ). i=1 ).

18 The Pfaffian of an Alternating Matrix C = ( C(i,j) ) n i,j=1 Alt n(f) of even order n = 2t. M n - all perfect matchings in K n. For let The Pfaffian of C is: M = { {k 1 < l 1 },...,{k t < l t } } M n ( 1 2 n 1 n θ(m) = sgn k 1 l 1 k t l t Pf(C) = Fact: det(c) = Pf(C) 2. M M n θ(m) t C(k i,l i ). i=1 ).

19 Combinatorial Nullstellensatz g(x 1,...,x t ) = (α 1,...,α t) c(α 1,...,α t )x α 1 1 xαt t F[x 1,...,x t ]. deg(g) = max{ t i=1 α i : c(α 1,...,α t ) 0}. Theorem [Alon] Assume: deg(g) = t d i & c(d 1,...,d t ) 0. i=1 Let Λ 1,...,Λ t F such that Λ i > d i. Then there exist λ 1 Λ 1,...,λ t Λ t such that g(λ 1,...,λ t ) 0.

20 Combinatorial Nullstellensatz g(x 1,...,x t ) = (α 1,...,α t) c(α 1,...,α t )x α 1 1 xαt t F[x 1,...,x t ]. deg(g) = max{ t i=1 α i : c(α 1,...,α t ) 0}. Theorem [Alon] Assume: deg(g) = t d i & c(d 1,...,d t ) 0. i=1 Let Λ 1,...,Λ t F such that Λ i > d i. Then there exist λ 1 Λ 1,...,λ t Λ t such that g(λ 1,...,λ t ) 0.

21 Main Step Proposition Let n = 2m and B 1,...,B m Alt n (F). If {q(b 1 ),...,q(b m )} is a perfect matching in K n, then: ρ ( B 1,...,B m ) = n. Sketch of Proof: It can be shown that the monomial x 1 x m appears in f(x 1,...,x m ) = Pf ( m ) x i B i with a nonzero coefficient. By the Combinatorial Nullstellensatz there exists a λ = (λ 1,...,λ m ) F m such that i=1 det ( m ) λ i B i = f(λ1,...,λ m ) 2 0. i=1

22 Main Step Proposition Let n = 2m and B 1,...,B m Alt n (F). If {q(b 1 ),...,q(b m )} is a perfect matching in K n, then: ρ ( B 1,...,B m ) = n. Sketch of Proof: It can be shown that the monomial x 1 x m appears in f(x 1,...,x m ) = Pf ( m ) x i B i with a nonzero coefficient. By the Combinatorial Nullstellensatz there exists a λ = (λ 1,...,λ m ) F m such that i=1 det ( m ) λ i B i = f(λ1,...,λ m ) 2 0. i=1

23 Subspaces Sym n (F) with bounded ρ Theorem [M ( F n), de Seguins Pazzis (any F)] Let S Sym n (F) be a linear subspace such that ρ(s) k. If k = 2t then dims max {( ) 2t +1 2 If k = 2t +1 then dims max {( ) 2t +2 2,tn,tn ( ) t }. 2 ( ) t +1 }. 2

24 Exterior Powers and the Plücker Embedding The p-th Exterior Power I p (V)= subspace of V p generated by v 1 v p sgn(π)v π(1) v π(p) where v 1,...,v p V, π S p. p V = V p /I p (V). The Plücker Embedding G p (V) - the Grassmannian of p-dimensional linear subspaces of V, embeds into P( p V) by the map u 1,...,u p [u 1 u p ].

25 Exterior Powers and the Plücker Embedding The p-th Exterior Power I p (V)= subspace of V p generated by v 1 v p sgn(π)v π(1) v π(p) where v 1,...,v p V, π S p. p V = V p /I p (V). The Plücker Embedding G p (V) - the Grassmannian of p-dimensional linear subspaces of V, embeds into P( p V) by the map u 1,...,u p [u 1 u p ].

26 Maximal Linear Subspaces of G p (V) Classical Fact Let M be a maximal linear space in G p (V) P( p V). Then either: or M = {Λ G p (V) : Λ U} for some U G p+1 (V), M = {Λ G p (V) : Λ U } for some U G p 1 (V). Equivalent Formulation: If W p V is a linear space of decomposable p-vectors then either (i) W p U for some U G p+1 (V), or (ii) W {α v : v V} for some α = v 1 v p 1 p V. In particular: dimw max{p +1,n p +1}.

27 Maximal Linear Subspaces of G p (V) Classical Fact Let M be a maximal linear space in G p (V) P( p V). Then either: or M = {Λ G p (V) : Λ U} for some U G p+1 (V), M = {Λ G p (V) : Λ U } for some U G p 1 (V). Equivalent Formulation: If W p V is a linear space of decomposable p-vectors then either (i) W p U for some U G p+1 (V), or (ii) W {α v : v V} for some α = v 1 v p 1 p V. In particular: dimw max{p +1,n p +1}.

28 The Rank of a p-vector Enveloping Space of w p V: E(w) = {U V : w p U}. Rank of w: rk(w) = dime(w). Example: w = e 1 e 2 e 3 +e 1 e 2 e 4 +e 3 e 5 e 6 +e 4 e 5 e 6 = e 1 e 2 (e 3 +e 4 )+(e 3 +e 4 ) e 5 e 6. E(w) = e 1,e 2,e 3 +e 4,e 5,e 6, rk(w) = 5.

29 The Rank of a p-vector Enveloping Space of w p V: E(w) = {U V : w p U}. Rank of w: rk(w) = dime(w). Example: w = e 1 e 2 e 3 +e 1 e 2 e 4 +e 3 e 5 e 6 +e 4 e 5 e 6 = e 1 e 2 (e 3 +e 4 )+(e 3 +e 4 ) e 5 e 6. E(w) = e 1,e 2,e 3 +e 4,e 5,e 6, rk(w) = 5.

30 Subspaces of p V of Bounded Rank For p k n = dimv let { 1 p = k or p = 2 k, ǫ p (k) = 0 otherwise. m p (k) = the unique m such that ( ) ( ) m 1 m +m k +m. p 1 p 1 Theorem [Gelbord,M]: If W p V satisfies ρ(w) k, then: {( ) ( ) } k +ǫp (k) mp (k) dimw max, +(k m p (k))(n m p (k)). p p

31 Subspaces of p V of Bounded Rank For p k n = dimv let { 1 p = k or p = 2 k, ǫ p (k) = 0 otherwise. m p (k) = the unique m such that ( ) ( ) m 1 m +m k +m. p 1 p 1 Theorem [Gelbord,M]: If W p V satisfies ρ(w) k, then: {( ) ( ) } k +ǫp (k) mp (k) dimw max, +(k m p (k))(n m p (k)). p p

32 Large Spaces W p V with ρ(w) k Fix U G k+ǫp(k)(v) and let W 1 (n,p,k) = p U. Then: ρ(w 1 ) = k and dimw 1 = ( k+ǫ p(k) ) p. Fix U G m (V) and k m decomposable elements z 1,...,z k m p 1 (U), and let W 2 (n,p,k) = p U + z 1,...,z k m V. Then: ρ(w 2 ) = k and dimw 2 = ( m p(k) ) p +(k mp (k))(n m p (k)).

33 Large Spaces W p V with ρ(w) k Fix U G k+ǫp(k)(v) and let W 1 (n,p,k) = p U. Then: ρ(w 1 ) = k and dimw 1 = ( k+ǫ p(k) ) p. Fix U G m (V) and k m decomposable elements z 1,...,z k m p 1 (U), and let W 2 (n,p,k) = p U + z 1,...,z k m V. Then: ρ(w 2 ) = k and dimw 2 = ( m p(k) ) p +(k mp (k))(n m p (k)).

34 The PIT Problem Given A 1,...,A m M n (F), can ρ( A 1,...,A m ) be computed in deterministic polynomial time? Remarks The problem admits a simple probabilistic solution: Choose random x 1,...,x m F. Then with high probability m rk( x i A i ) = ρ( A 1,...,A m ). i=1 There are (highly nontrivial) deterministic algorithms for PIT, when all A i s are of rank 1, or skew-symmetric of rank 2.

35 Spaces Generated by Rank 1 Matrices Let u 1,,u t F m, v 1,...,v t F n and let S = u 1 v 1,...,u t v t M m n (F). Matroid Intersection Theorem [Edmonds] ( ρ(s) = min dim ui : i I +dim v j : j I ). I [t] Computational Result [Edmonds] There is a polynomial time algorithm to determine ρ(s).

36 Spaces Generated by Rank 1 Matrices Let u 1,,u t F m, v 1,...,v t F n and let S = u 1 v 1,...,u t v t M m n (F). Matroid Intersection Theorem [Edmonds] ( ρ(s) = min dim ui : i I +dim v j : j I ). I [t] Computational Result [Edmonds] There is a polynomial time algorithm to determine ρ(s).

37 Spaces Generated by Rank 1 Matrices Let u 1,,u t F m, v 1,...,v t F n and let S = u 1 v 1,...,u t v t M m n (F). Matroid Intersection Theorem [Edmonds] ( ρ(s) = min dim ui : i I +dim v j : j I ). I [t] Computational Result [Edmonds] There is a polynomial time algorithm to determine ρ(s).

38 Spaces Generated by Alternating Decomposables For u,v F n let u v = u v v u Alt n (F). Let u 1,v 1,...,u t,v t F n and let S = u 1 v 1,...,u t v t Alt n (F). Linear Matroid Parity Theorem [Lovász] ρ(s) = min { 2dimA+2 } k dimb i 2 where the minimum ranges over all subspaces A,B 1,...,B k F n such that k S A V + 2 B i. i=1 i=1

39 Spaces Generated by Alternating Decomposables For u,v F n let u v = u v v u Alt n (F). Let u 1,v 1,...,u t,v t F n and let S = u 1 v 1,...,u t v t Alt n (F). Linear Matroid Parity Theorem [Lovász] ρ(s) = min { 2dimA+2 } k dimb i 2 where the minimum ranges over all subspaces A,B 1,...,B k F n such that k S A V + 2 B i. i=1 i=1

40 A Dual Problem and Graph Rigidity Let u 1,v 1,...,u t,v t F n and let S = {[ (u 1 v 1 )x... (u t v t )x ] : x F n} Theorem [ Raz,Wigderson] There is a polynomial time algorithm to determine ρ(s). Remark As a very special case, this provides a new polynomial algorithm for deciding graph rigidity in the plane.

41 A Dual Problem and Graph Rigidity Let u 1,v 1,...,u t,v t F n and let S = {[ (u 1 v 1 )x... (u t v t )x ] : x F n} Theorem [ Raz,Wigderson] There is a polynomial time algorithm to determine ρ(s). Remark As a very special case, this provides a new polynomial algorithm for deciding graph rigidity in the plane.

42 Weak Duality Switching Roles Let U,V be linear spaces over F and let S Hom(U,V). View U as a subspace of Hom(S,V), by u(s) = s(u). Theorem [M,Šemrl] ρ(s) ρ(u). Let F be infinite. If ρ(u) dims 1 then: either or ρ(s) dims ρ(s) dims 2.

43 Weak Duality Switching Roles Let U,V be linear spaces over F and let S Hom(U,V). View U as a subspace of Hom(S,V), by u(s) = s(u). Theorem [M,Šemrl] ρ(s) ρ(u). Let F be infinite. If ρ(u) dims 1 then: either or ρ(s) dims ρ(s) dims 2.

44 Maximal Singular Spaces Rank 1 Generation The minimal dimension of a maximal singular subspace W M n (C) generated by rank 1 matrices is 3n2 2n 4. Example: W = U 1 F n +F n U 2 where dimu 1 = n 2,dimU 2 = n 2 1. Alternating Rank 2 Generation The minimal dimension of a maximal singular subspace W 2 C n generated by decomposable elements is 3n 2 3. Example: W = n 2 1 i=1 2 U i where the U i s are 3-dimensional spaces in general position. Theorem [Draisma] For infinitely many n s there exist 8-dimensional maximal singular spaces in M n (C).

45 Maximal Singular Spaces Rank 1 Generation The minimal dimension of a maximal singular subspace W M n (C) generated by rank 1 matrices is 3n2 2n 4. Example: W = U 1 F n +F n U 2 where dimu 1 = n 2,dimU 2 = n 2 1. Alternating Rank 2 Generation The minimal dimension of a maximal singular subspace W 2 C n generated by decomposable elements is 3n 2 3. Example: W = n 2 1 i=1 2 U i where the U i s are 3-dimensional spaces in general position. Theorem [Draisma] For infinitely many n s there exist 8-dimensional maximal singular spaces in M n (C).

46 Maximal Singular Spaces Rank 1 Generation The minimal dimension of a maximal singular subspace W M n (C) generated by rank 1 matrices is 3n2 2n 4. Example: W = U 1 F n +F n U 2 where dimu 1 = n 2,dimU 2 = n 2 1. Alternating Rank 2 Generation The minimal dimension of a maximal singular subspace W 2 C n generated by decomposable elements is 3n 2 3. Example: W = n 2 1 i=1 2 U i where the U i s are 3-dimensional spaces in general position. Theorem [Draisma] For infinitely many n s there exist 8-dimensional maximal singular spaces in M n (C).

47 Subspaces of M n (F) with Bounded ρ Rank Varieties R n,k (F) = {A M n (F) : rk(a) k}. f F (n,k) = max{dims : S M n (F),ρ(S) k} = max{dims : S R n,k 1 (F) = {0}. If F is algebraically closed then R n,k (F) is an irreducible (2nk k 2 )-dimensional affine variety. Hence: f F (n,k) codimr n,k 1 (F) = (n k +1) 2. k 1 Example: There exists an (n k +1) 2 -dim subspace S W that satisfies ρ(s) = k. k 1

48 Subspaces of M n (F) with Bounded ρ Rank Varieties R n,k (F) = {A M n (F) : rk(a) k}. f F (n,k) = max{dims : S M n (F),ρ(S) k} = max{dims : S R n,k 1 (F) = {0}. If F is algebraically closed then R n,k (F) is an irreducible (2nk k 2 )-dimensional affine variety. Hence: f F (n,k) codimr n,k 1 (F) = (n k +1) 2. k 1 Example: There exists an (n k +1) 2 -dim subspace S W that satisfies ρ(s) = k. k 1

49 Subspaces of M n (F) with Bounded ρ Rank Varieties R n,k (F) = {A M n (F) : rk(a) k}. f F (n,k) = max{dims : S M n (F),ρ(S) k} = max{dims : S R n,k 1 (F) = {0}. If F is algebraically closed then R n,k (F) is an irreducible (2nk k 2 )-dimensional affine variety. Hence: f F (n,k) codimr n,k 1 (F) = (n k +1) 2. k 1 Example: There exists an (n k +1) 2 -dim subspace S W that satisfies ρ(s) = k. k 1

50 Bounded ρ: the Finite Field Case Theorem [Roth] f Fq (n,k) = n(n k +1). A Simple Construction [M]: n k W q (n,k) = g(x) = a j x qj : a j F q n End F q (F q n) = M n (F q ). j=0 W q (n,k) = q n(n k+1) dim Fq W q (n,k) = n(n k +1). 0 g(x) W q (n,k) ker(g) q n k dimker(g) n k rk(g) k.

51 Spaces of Non-Singular Real Matrices Hurwitz-Radon Number Write n = (2α 1)2 γ+4δ where 0 γ 3, and let HR(n) = 2 γ +8δ. Theorem [Hurwitz-Radon, Adams] f R (n,n) = HR(n). Comments The lower bound f R (n,n) HR(n) follows from a construction of Hurwitz and Radon, using Clifford algebras. The upper bound f R (n,n) HR(n) follows from Adams theorem on the maximal number of vector fields on the sphere.

52 Spaces of Non-Singular Real Matrices Hurwitz-Radon Number Write n = (2α 1)2 γ+4δ where 0 γ 3, and let HR(n) = 2 γ +8δ. Theorem [Hurwitz-Radon, Adams] f R (n,n) = HR(n). Comments The lower bound f R (n,n) HR(n) follows from a construction of Hurwitz and Radon, using Clifford algebras. The upper bound f R (n,n) HR(n) follows from Adams theorem on the maximal number of vector fields on the sphere.

53 The Clifford Algebra C t C t = R-algebra generated by e 1,...,e t modulo the relations ei 2 = 1 & e i e j = e j e i for i j. C t is 2 t -dimensional algebra with basis: t {e i1 e ik : 1 i 1 < < i k t}. k=0 Periodicity: C t+8 = M16 (C t ). C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 R C H H H M 2 (H) M 4 (C) M 8 (R) M 8 (R) M 8 (R)

54 The Clifford Algebra C t C t = R-algebra generated by e 1,...,e t modulo the relations ei 2 = 1 & e i e j = e j e i for i j. C t is 2 t -dimensional algebra with basis: t {e i1 e ik : 1 i 1 < < i k t}. k=0 Periodicity: C t+8 = M16 (C t ). C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 R C H H H M 2 (H) M 4 (C) M 8 (R) M 8 (R) M 8 (R)

55 Nonsingular Spaces via Clifford Algebras A Representation of C t on an n-dimensional real vector space W is an algebra homomorphism ρ : C t End(W). Claim: S = { t x i ρ(e i ) : x i R } i=1 is a t-dimensional nonsingular space in End(W). Proof: Let 0 w W. If t i=1 x iρ(e i )(w) = 0 then ( t ) ( t ) 2 xi 2 w = ρ x i e i w i=1 i=1 ( t 2 = x i ρ(e i )) w = 0. i=1 Therefore x 1 = = x t = 0.

56 Nonsingular Spaces via Clifford Algebras A Representation of C t on an n-dimensional real vector space W is an algebra homomorphism ρ : C t End(W). Claim: S = { t x i ρ(e i ) : x i R } i=1 is a t-dimensional nonsingular space in End(W). Proof: Let 0 w W. If t i=1 x iρ(e i )(w) = 0 then ( t ) ( t ) 2 xi 2 w = ρ x i e i w i=1 i=1 ( t 2 = x i ρ(e i )) w = 0. i=1 Therefore x 1 = = x t = 0.

57 Hurwitz-Radon Construction Fix n 1. Using the classification of Clifford algebras, it can be checked that for t = HR(n), there is a representation of C t on R n. Therefore: f R (n,n) HR(n). Examples n = 2 [ x1 x S 2 = 2 x 2 x 1 ] S 4 = n = 4 x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 4 x 3 x 2 x 1

58 Hurwitz-Radon Construction Fix n 1. Using the classification of Clifford algebras, it can be checked that for t = HR(n), there is a representation of C t on R n. Therefore: f R (n,n) HR(n). Examples n = 2 [ x1 x S 2 = 2 x 2 x 1 ] S 4 = n = 4 x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 4 x 3 x 2 x 1

59 Adams Theorem Vector fields on the (n 1)-Sphere S n 1 = {x R n : x = 1}. A vector field on S n 1 is a continuous map such that for all x S n 1 : φ : S n 1 R n φ(x) x = 0. Theorem [Adams]: Let φ 1,...,φ k be vector fields on S n 1 such that φ 1 (x),...,φ k (x) are linearly independent for all x S n 1. Then: k HR(n) 1.

60 Adams Theorem Vector fields on the (n 1)-Sphere S n 1 = {x R n : x = 1}. A vector field on S n 1 is a continuous map such that for all x S n 1 : φ : S n 1 R n φ(x) x = 0. Theorem [Adams]: Let φ 1,...,φ k be vector fields on S n 1 such that φ 1 (x),...,φ k (x) are linearly independent for all x S n 1. Then: k HR(n) 1.

61 Nonsingular Spaces and Vector Fields Theorem [Adams] f R (n,n) HR(n). Proof: Let t = f R (n,n) and let S = A 1,...,A t be a nonsingular space in M n (R). We may assume A t = I. For 1 i t 1 let φ i (x) = A i x ((A i x) x)x. Then φ 1,...,φ t 1 are linearly independent vector fields on S n 1. The result then follows from Adams vector fields theorem. Remark: For the proof of this special case, one only needs the additive structure of K R (RP n 1 ).

62 Nonsingular Spaces and Vector Fields Theorem [Adams] f R (n,n) HR(n). Proof: Let t = f R (n,n) and let S = A 1,...,A t be a nonsingular space in M n (R). We may assume A t = I. For 1 i t 1 let φ i (x) = A i x ((A i x) x)x. Then φ 1,...,φ t 1 are linearly independent vector fields on S n 1. The result then follows from Adams vector fields theorem. Remark: For the proof of this special case, one only needs the additive structure of K R (RP n 1 ).

63 K-Theory of the Real Projective Space Notation ǫ - the trivial line bundle. η r 1 - the tautological line bundle over RP r 1 : E(η r 1 ) = { ([x],v) : [x] RP r 1,v x }. Theorem [Adams] K R (RP r 1 ) = Z/(2 λ(r 1) ) where λ(r 1) = {1 k r 1 : k 0,1,2,4(mod 8)}. K R (RP r 1 ) is generated by η r 1 ǫ.

64 K-Theory of the Real Projective Space Notation ǫ - the trivial line bundle. η r 1 - the tautological line bundle over RP r 1 : E(η r 1 ) = { ([x],v) : [x] RP r 1,v x }. Theorem [Adams] K R (RP r 1 ) = Z/(2 λ(r 1) ) where λ(r 1) = {1 k r 1 : k 0,1,2,4(mod 8)}. K R (RP r 1 ) is generated by η r 1 ǫ.

65 Proof of f R (n,n) HR(n) Let S = A 1,...,A r be a nonsingular subspace of M n (R). Suppose for contradiction that r = HR(n)+1. Define a bundle map g : η n r 1 ǫ n by g ([x],θ 1 x,...θ n x) = [x], ( r i=1 ) x i A i θ 1.. θ n. The nonsingularity of S implies that g is an isomorphism. Therefore n(η r 1 ǫ) = 0 in K R (RP r 1 ). Hence: 2 λ(2γ +8δ) = 2 λ(r 1) n = (2α 1)2 γ+4δ Thus γ +1+4δ = λ(2 γ +8δ) γ +4δ, a contradiction.

66 Proof of f R (n,n) HR(n) Let S = A 1,...,A r be a nonsingular subspace of M n (R). Suppose for contradiction that r = HR(n)+1. Define a bundle map g : η n r 1 ǫ n by g ([x],θ 1 x,...θ n x) = [x], ( r i=1 ) x i A i θ 1.. θ n. The nonsingularity of S implies that g is an isomorphism. Therefore n(η r 1 ǫ) = 0 in K R (RP r 1 ). Hence: 2 λ(2γ +8δ) = 2 λ(r 1) n = (2α 1)2 γ+4δ Thus γ +1+4δ = λ(2 γ +8δ) γ +4δ, a contradiction.

67 Proof of f R (n,n) HR(n) Let S = A 1,...,A r be a nonsingular subspace of M n (R). Suppose for contradiction that r = HR(n)+1. Define a bundle map g : η n r 1 ǫ n by g ([x],θ 1 x,...θ n x) = [x], ( r i=1 ) x i A i θ 1.. θ n. The nonsingularity of S implies that g is an isomorphism. Therefore n(η r 1 ǫ) = 0 in K R (RP r 1 ). Hence: 2 λ(2γ +8δ) = 2 λ(r 1) n = (2α 1)2 γ+4δ Thus γ +1+4δ = λ(2 γ +8δ) γ +4δ, a contradiction.

68 Spaces of Nilpotent Matrices Theorem [Gerstenhaber ( F n), Serežkin (any F)] Let S M n (F) be a linear space of nilpotent matrices. Then: dims ( n 2). dims = ( n 2) iff S is similar to the space of strictly upper diagonal matrices.

69 Spaces of Symmetric Nilpotent Matrices Theorem [M,Radwan] The maximal dimension of a linear subspace of Sym n (C) consisting of nilpotent matrices is n2 4. Example For n = 2m let [ X +X S = t +Y i(x t ] X +Y) i(x X t +Y) X +X t Sym Y n (C) where X M m (C) is strictly upper triangular and Y Sym m (C). Then dims = m 2 = n2 4 and S is nilpotent.

70 Nilpotent Subspaces of Lie Algebras g - a complex semi-simple Lie algebra. h - a fixed Cartan subalgebra of g. g = n h n + - a fixed Cartan decomposition. Theorem [Radwan, M] If W g is a linear subspace of ad nilpotent elements then dimw 1 2 (dimg rkg) = dimn +. Theorem [Draisma, Kraft, Kuttler] W nilpotent & dimw = dimn + W is conjugate to n +.

71 Nilpotent Subspaces of Lie Algebras g - a complex semi-simple Lie algebra. h - a fixed Cartan subalgebra of g. g = n h n + - a fixed Cartan decomposition. Theorem [Radwan, M] If W g is a linear subspace of ad nilpotent elements then dimw 1 2 (dimg rkg) = dimn +. Theorem [Draisma, Kraft, Kuttler] W nilpotent & dimw = dimn + W is conjugate to n +.

72 Partitions A partition of n: p = (p 1,...,p t ) such that p 1 p t 0 & t p i = n. i=1 The conjugate partition: p = q = (q 1,...,q s ) where q i = {j : p j i}. p =(4,3,1) p =(3,2,2,1)

73 Nilpotent Orbits Jordan Form J p J p J p = J pt where J k is the k k Jordan block. The Orbit of J p O p = {gj p g 1 : g GL n (C)}. The closure O p is an algebraic variety of dimension s dimo p = n 2 qj 2. j=1

74 Nilpotent Orbits Jordan Form J p J p J p = J pt where J k is the k k Jordan block. The Orbit of J p O p = {gj p g 1 : g GL n (C)}. The closure O p is an algebraic variety of dimension s dimo p = n 2 qj 2. j=1

75 Linear Spaces in Nilpotent Varieties Theorem [Gerstenhaber] If S O p is a linear space then dims 1 2 dimo p. Problem What is the analogues statement for Lie Algebras?

76 Linear Spaces in Nilpotent Varieties Theorem [Gerstenhaber] If S O p is a linear space then dims 1 2 dimo p. Problem What is the analogues statement for Lie Algebras?

77 THANK YOU!