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 Trento 2005 – p.1/7

Setting A : vector space of n × n matrices over C Trento 2005 – p.2/7

Setting A : vector space of n × n matrices over C Rank-critical: rk( A ) = r and rk( A + C B ) > r for all B �∈ A Trento 2005 – p.2/7

Setting A : vector space of n × n matrices over C Rank-critical: rk( A ) = r and rk( A + C B ) > r for all B �∈ A Question: how small can dim( A ) be? Trento 2005 – p.2/7

Setting A : vector space of n × n matrices over C Rank-critical: rk( A ) = r and rk( A + C B ) > r for all B �∈ A Question: how small can dim( A ) be? Special case: r = n − 1 , maximal singular space Trento 2005 – p.2/7

Examples 1. U, V ⊆ C n with dim U = k, dim V = k − ( n − r ) and A = { M | MU ⊆ V } ‘compression space’ Trento 2005 – p.3/7

Examples 1. U, V ⊆ C n 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 } Trento 2005 – p.3/7

Examples 1. U, V ⊆ C n 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 , A 1 , . . . , A n generic skew-symmetric matrices and A = { ( A 1 x | . . . | A n x ) | x ∈ C n } (Bob Paré) Trento 2005 – p.3/7

Examples 1. U, V ⊆ C n 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 , A 1 , . . . , A n generic skew-symmetric matrices and A = { ( A 1 x | . . . | A n x ) | x ∈ C n } (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) Trento 2005 – p.3/7

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! Trento 2005 – p.4/7

A sufficient condition for rank-criticality Setting: A a matrix space, rk( A ) = r Goal: decide whether A is rank-critical Notation: X r ⊆ M n the variety of rank ≤ r matrices Note: if rk( A + C B ) = r , then B ∈ T A X r for all A ∈ A Trento 2005 – p.4/7

A sufficient condition for rank-criticality Setting: A a matrix space, rk( A ) = r Goal: decide whether A is rank-critical Notation: X r ⊆ M n the variety of rank ≤ r matrices Note: if rk( A + C B ) = r , then B ∈ T A X r for all A ∈ A Define: RND( A ) := � A ∈A T A X r , rank-neutral directions of A Conclusion: if RND( A ) = A , then A is rank-critical Trento 2005 – p.4/7

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 ) Trento 2005 – p.5/7

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 ) Trento 2005 – p.5/7

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 Trento 2005 – p.5/7

Two results Small maximal singular spaces: G := SL m , m ≥ 3 acting on V := homogeneous polynomials of degree me, e ≥ 1 yields a ( m 2 − 1) -dimensional maximal space of singular n × n -matrices, where n = dim V . Trento 2005 – p.6/7

Two results Small maximal singular spaces: G := SL m , m ≥ 3 acting on V := homogeneous polynomials of degree me, e ≥ 1 yields a ( m 2 − 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 } g α . Trento 2005 – p.6/7

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/ Trento 2005 – p.7/7