Small maximal spaces of non-invertible matrices Jan Draisma - - PowerPoint PPT Presentation

Small maximal spaces of non-invertible matrices

Jan Draisma

jan.draisma@unibas.ch

Mathematisches Institut der Universit¨ at Basel

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Setting

A: vector space of n × n matrices over C

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Setting

A: vector space of n × n matrices over C

Rank-critical: rk(A) = r and rk(A + CB) > r for all B ∈ A

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Setting

A: vector space of n × n matrices over C

Rank-critical: rk(A) = r and rk(A + CB) > r for all B ∈ A Question: how small can dim(A) be?

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Setting

A: vector space of n × n matrices over C

Rank-critical: rk(A) = r and rk(A + CB) > r for all B ∈ A Question: how small can dim(A) be? Special case: r = n − 1, maximal singular space

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Examples

• 1. U, V ⊆ Cn with dim U = k, dim V = k − (n − r) and

A = {M | MU ⊆ V } ‘compression space’

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Examples

• 1. U, V ⊆ Cn with dim U = k, dim V = k − (n − r) and

A = {M | MU ⊆ V } ‘compression space’

• 2. n odd, r = n − 1, A = {skew-symmetric matrices}

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Examples

• 1. U, V ⊆ Cn with dim U = k, dim V = k − (n − r) and

A = {M | MU ⊆ V } ‘compression space’

• 2. n odd, r = n − 1, A = {skew-symmetric matrices}
• 3. r = n − 1, A1, . . . , An generic skew-symmetric matrices

and A = {(A1x| . . . |Anx) | x ∈ Cn} (Bob Paré)

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Examples

• 1. U, V ⊆ Cn with dim U = k, dim V = k − (n − r) and

A = {M | MU ⊆ V } ‘compression space’

• 2. n odd, r = n − 1, A = {skew-symmetric matrices}
• 3. r = n − 1, A1, . . . , An generic skew-symmetric matrices

and A = {(A1x| . . . |Anx) | x ∈ Cn} (Bob Paré)

• 4. ∃ many sufficient conditions for a matrix space A to be

contained in a compression space: dim A large enough,

A spanned by rank one matrices, r = 1, 2, 3 while n

large, etc. (Fillmore, Laurie, and Radjavi; Dieudonné; Eisenbud-Harris; Lovász)

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A sufficient condition for rank-criticality

Setting: A a matrix space, rk(A) = r Goal: decide whether A is rank-critical Theoretically: Groebner basis computations. Not feasible!

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A sufficient condition for rank-criticality

Setting: A a matrix space, rk(A) = r Goal: decide whether A is rank-critical Notation: Xr ⊆ Mn the variety of rank ≤ r matrices Note: if rk(A + CB) = r, then B ∈ TAXr for all A ∈ A

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A sufficient condition for rank-criticality

Setting: A a matrix space, rk(A) = r Goal: decide whether A is rank-critical Notation: Xr ⊆ Mn the variety of rank ≤ r matrices Note: if rk(A + CB) = r, then B ∈ TAXr for all A ∈ A Define: RND(A) :=

A∈A TAXr, rank-neutral directions of A

Conclusion: if RND(A) = A, then A is rank-critical

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Where do Lie-algebras come into play?

Setting: G an algebraic group, ρ : G → GL(V ) a

representation, g the Lie algebra of G

Set: A := ρ(g) ⊆ End(V )

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Where do Lie-algebras come into play?

Setting: G an algebraic group, ρ : G → GL(V ) a

representation, g the Lie algebra of G

Set: A := ρ(g) ⊆ End(V ) Observe: RND(A) is a G-submodule of End(V )

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Where do Lie-algebras come into play?

Setting: G an algebraic group, ρ : G → GL(V ) a

representation, g the Lie algebra of G

Set: A := ρ(g) ⊆ End(V ) Observe: RND(A) is a G-submodule of End(V ) Conclusion: decomposition of End(V ) into irreducible

G-modules can be used for proving that ρ(g) is

rank-critical

Implementation: in GAP

, using Willem de Graaf’s Lie algebra algorithms

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Two results

Small maximal singular spaces: G := SLm, m ≥ 3 acting on

V :=homogeneous polynomials of degree me, e ≥ 1

yields a (m2 − 1)-dimensional maximal space of singular

n × n-matrices, where n = dim V .

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Two results

Small maximal singular spaces: G := SLm, m ≥ 3 acting on

V :=homogeneous polynomials of degree me, e ≥ 1

yields a (m2 − 1)-dimensional maximal space of singular

n × n-matrices, where n = dim V .

Adjoint representation: for any semisimple g, ad(g) ⊆ End(g)

is rank-critical of rank dim g − rk g.

In fact: linear equations ‘cutting out’ ad(g) in End(g):

A ∈ End(g) lies in ad(g) if and only if A maps every

Cartan h into

α∈h∗\{0} gα.

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See you in Basel!?

R W C A 2 0 0 6

Rhine Workshop on Computer Algebra, 16-17 March 2006, Basel Contact: jan.draisma@unibas.ch URL: http://www.math.unibas.ch/ draisma/rwca06/

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