Algorithms for the Separation of Orbit Closures of Matrices (arXiv:1801.02043) Harm Derksen (University of Michigan) joint work with Visu Makam (IAS) SIAM conference on Applied Algebraic Geometry July 12, 2019 Harm Derksen Algorithms for Orbit Closure Separation
Invariant Theory K algebraically closed base field G reductive algebraic group over K e.g., GL n , SL n , O n , finite, or products of these Harm Derksen Algorithms for Orbit Closure Separation
Invariant Theory K algebraically closed base field G reductive algebraic group over K e.g., GL n , SL n , O n , finite, or products of these V n -dimensional representation of G K [ V ] ring of polynomial functions on V G acts by polynomial automorphisms on K [ V ] Harm Derksen Algorithms for Orbit Closure Separation
Invariant Theory K algebraically closed base field G reductive algebraic group over K e.g., GL n , SL n , O n , finite, or products of these V n -dimensional representation of G K [ V ] ring of polynomial functions on V G acts by polynomial automorphisms on K [ V ] Definition invariant ring K [ V ] G = { f ∈ K [ V ] | ∀ g ∈ G g · f = f } = { f ∈ K [ V ] | f constant on G -orbits } . Harm Derksen Algorithms for Orbit Closure Separation
Invariant Theory K algebraically closed base field G reductive algebraic group over K e.g., GL n , SL n , O n , finite, or products of these V n -dimensional representation of G K [ V ] ring of polynomial functions on V G acts by polynomial automorphisms on K [ V ] Definition invariant ring K [ V ] G = { f ∈ K [ V ] | ∀ g ∈ G g · f = f } = { f ∈ K [ V ] | f constant on G -orbits } . Theorem (Hilbert, Nagata, Haboush) K [ V ] G is a finitely generated K-algebra Harm Derksen Algorithms for Orbit Closure Separation
Geometry of Orbits Definition an invariant f ∈ K [ V ] G separates v , w ∈ V if f ( v ) � = f ( w ) Harm Derksen Algorithms for Orbit Closure Separation
Geometry of Orbits Definition an invariant f ∈ K [ V ] G separates v , w ∈ V if f ( v ) � = f ( w ) G · v is Zariski closure of orbit G · v . Proposition G · v ∩ G · w = ∅ ⇔ f ( v ) � = f ( w ) for some f ∈ K [ V ] G ⇐ is easy to see Harm Derksen Algorithms for Orbit Closure Separation
Geometry of Orbits Definition an invariant f ∈ K [ V ] G separates v , w ∈ V if f ( v ) � = f ( w ) G · v is Zariski closure of orbit G · v . Proposition G · v ∩ G · w = ∅ ⇔ f ( v ) � = f ( w ) for some f ∈ K [ V ] G ⇐ is easy to see Orbit Closure Problem given v , w ∈ W determine whether G · v ∩ G · w = ∅ if so, find explicit f ∈ K [ V ] G with f ( v ) � = f ( w ) Harm Derksen Algorithms for Orbit Closure Separation
Geometry of Orbits Definition an invariant f ∈ K [ V ] G separates v , w ∈ V if f ( v ) � = f ( w ) G · v is Zariski closure of orbit G · v . Proposition G · v ∩ G · w = ∅ ⇔ f ( v ) � = f ( w ) for some f ∈ K [ V ] G ⇐ is easy to see Orbit Closure Problem given v , w ∈ W determine whether G · v ∩ G · w = ∅ if so, find explicit f ∈ K [ V ] G with f ( v ) � = f ( w ) N := { v ∈ V | 0 ∈ G · v } Null cone v ∈ N ⇔ G · v ∩ G · 0 � = ∅ ⇔ ∀ f ∈ K [ V ] G , f ( v ) = f (0) Harm Derksen Algorithms for Orbit Closure Separation
Matrix Conjugation Example: V = Mat n , n n × n matrices G = GL n acts on V by conjugation: g · A = gAg − 1 Harm Derksen Algorithms for Orbit Closure Separation
Matrix Conjugation Example: V = Mat n , n n × n matrices G = GL n acts on V by conjugation: g · A = gAg − 1 characteristic polynomial of A ∈ Mat n , n : χ A ( t ) := det( tI − A ) = t n + f 1 ( A ) t n − 1 + · · · + f n ( A ) K [ V ] G = K [ f 1 , f 2 , . . . , f n ] Harm Derksen Algorithms for Orbit Closure Separation
Matrix Conjugation Example: V = Mat n , n n × n matrices G = GL n acts on V by conjugation: g · A = gAg − 1 characteristic polynomial of A ∈ Mat n , n : χ A ( t ) := det( tI − A ) = t n + f 1 ( A ) t n − 1 + · · · + f n ( A ) K [ V ] G = K [ f 1 , f 2 , . . . , f n ] G · A ∩ G · B � = ∅ ⇔ χ A ( t ) = χ B ( t ) Harm Derksen Algorithms for Orbit Closure Separation
Matrix Conjugation Example: V = Mat n , n n × n matrices G = GL n acts on V by conjugation: g · A = gAg − 1 characteristic polynomial of A ∈ Mat n , n : χ A ( t ) := det( tI − A ) = t n + f 1 ( A ) t n − 1 + · · · + f n ( A ) K [ V ] G = K [ f 1 , f 2 , . . . , f n ] G · A ∩ G · B � = ∅ ⇔ χ A ( t ) = χ B ( t ) A ∈ N ⇔ f 1 ( A ) = · · · = f n ( A ) = 0 ⇔ χ A ( t ) = t n ⇔ A is nilpotent Harm Derksen Algorithms for Orbit Closure Separation
Simultaneous Matrix Conjugation Example: V = Mat m n , n m -tuples n × n matrices G = GL n acts on V by simultaneous conjugation: g · ( A 1 , . . . , A m ) = ( gA 1 g − 1 , . . . , gA m g − 1 ) Harm Derksen Algorithms for Orbit Closure Separation
Simultaneous Matrix Conjugation Example: V = Mat m n , n m -tuples n × n matrices G = GL n acts on V by simultaneous conjugation: g · ( A 1 , . . . , A m ) = ( gA 1 g − 1 , . . . , gA m g − 1 ) for a word w = w 1 w 2 · · · w r with w 1 , . . . , w r ∈ { 1 , 2 , . . . , m } define A w = A w 1 A w 2 · · · A w r the length ℓ ( w ) of w is r Harm Derksen Algorithms for Orbit Closure Separation
Simultaneous Matrix Conjugation Example: V = Mat m n , n m -tuples n × n matrices G = GL n acts on V by simultaneous conjugation: g · ( A 1 , . . . , A m ) = ( gA 1 g − 1 , . . . , gA m g − 1 ) for a word w = w 1 w 2 · · · w r with w 1 , . . . , w r ∈ { 1 , 2 , . . . , m } define A w = A w 1 A w 2 · · · A w r the length ℓ ( w ) of w is r Theorem (Procesi, Razmyslov, char( K ) = 0) K [ V ] G generated by all A = ( A 1 , . . . , A m ) �→ Trace( A w ) for all w of length ≤ n 2 Harm Derksen Algorithms for Orbit Closure Separation
Simultaneous Matrix Conjugation Example: V = Mat m n , n m -tuples n × n matrices G = GL n acts on V by simultaneous conjugation: g · ( A 1 , . . . , A m ) = ( gA 1 g − 1 , . . . , gA m g − 1 ) for a word w = w 1 w 2 · · · w r with w 1 , . . . , w r ∈ { 1 , 2 , . . . , m } define A w = A w 1 A w 2 · · · A w r the length ℓ ( w ) of w is r Theorem (Procesi, Razmyslov, char( K ) = 0) K [ V ] G generated by all A = ( A 1 , . . . , A m ) �→ Trace( A w ) for all w of length ≤ n 2 Theorem (Donkin, char( K ) arbitrary) K [ V ] G generated by all coefficients of χ A w ( t ) for all w D.-Makam: only need w with ℓ ( w ) ≤ ( m + 1) n 4 Harm Derksen Algorithms for Orbit Closure Separation
Simultaneous Matrix Conjugation Algorithm Forbes and Shpilka (2013) gave a (parallel) polynomial time algorithm for the orbit closure problem if char( K ) = 0 but algorithm does not explicitly construct a separating invariant if orbit closures are disjoint Harm Derksen Algorithms for Orbit Closure Separation
Simultaneous Matrix Conjugation Algorithm Forbes and Shpilka (2013) gave a (parallel) polynomial time algorithm for the orbit closure problem if char( K ) = 0 but algorithm does not explicitly construct a separating invariant if orbit closures are disjoint Algorithm D. and Makam (2018) gave a polynomial time algorithm for orbit closure problem in arbitary characteristic that also explicitly constructs a separating invariant when orbit closures are disjoint Harm Derksen Algorithms for Orbit Closure Separation
Orbit Closures for Simultaneous Conjugation given A = ( A 1 , . . . , A m ) , B = ( B 1 , . . . , B m ) ∈ V = Mat m n , n � A i 0 � define C i = , i = 1 , 2 , . . . , m 0 B i Harm Derksen Algorithms for Orbit Closure Separation
Orbit Closures for Simultaneous Conjugation given A = ( A 1 , . . . , A m ) , B = ( B 1 , . . . , B m ) ∈ V = Mat m n , n � A i 0 � define C i = , i = 1 , 2 , . . . , m 0 B i C = K � C 1 , . . . , C m � = Span { C w | w word } Harm Derksen Algorithms for Orbit Closure Separation
Orbit Closures for Simultaneous Conjugation given A = ( A 1 , . . . , A m ) , B = ( B 1 , . . . , B m ) ∈ V = Mat m n , n � A i 0 � define C i = , i = 1 , 2 , . . . , m 0 B i C = K � C 1 , . . . , C m � = Span { C w | w word } order all words lexicographically ∅ , 1 , 2 , . . . , m , 11 , 12 , . . . , 1 m , 21 , . . . , 2 m , . . . , 111 , 112 , . . . Definition w is called a pivot if C w �∈ Span { C u | u < w } Lemma { C w | w is a pivot } is basis of C Harm Derksen Algorithms for Orbit Closure Separation
Orbit Closures for Simultaneous Conjugation Lemma every subword of a pivot is also a pivot so # of pivots is at most dim C ≤ 2 n 2 largest pivot has length < 2 n 2 (actually O ( n log( n )) by Shitov) Harm Derksen Algorithms for Orbit Closure Separation
Orbit Closures for Simultaneous Conjugation Lemma every subword of a pivot is also a pivot so # of pivots is at most dim C ≤ 2 n 2 largest pivot has length < 2 n 2 (actually O ( n log( n )) by Shitov) suppose we found all pivots of length d to find pivots of length d + 1 we only have to check all words wi where w is a pivot of length d and 1 ≤ i ≤ m we can find all pivots in polynomial time Harm Derksen Algorithms for Orbit Closure Separation
Orbit Closures for Simultaneous Conjugation Theorem (char( K ) = 0) G · A ∩ G · B � = ∅ ⇔ Trace( A w ) = Trace( B w ) for all pivots w Harm Derksen Algorithms for Orbit Closure Separation
Recommend
More recommend
