Higher Secants of Sato s Grassmannian. Jan Draisma TU Eindhoven

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1 Higher Secants of Sato s Grassmannian Jan Draisma TU Eindhoven SIAM AG, Fort Collins, August 2, 2013

2 Grassmannians: functoriality and duality V finite-dimensional vector space Gr p (V) := {v 1 v p v i V} p V cone over Grassmannian rank-one alternating tensors

3 Grassmannians: functoriality and duality V finite-dimensional vector space Gr p (V) := {v 1 v p v i V} p V cone over Grassmannian rank-one alternating tensors Two properties: 1. if ϕ : V W linear p ϕ : p V p W maps Gr p (V) Gr p (W)

4 Grassmannians: functoriality and duality V finite-dimensional vector space Gr p (V) := {v 1 v p v i V} p V cone over Grassmannian rank-one alternating tensors Two properties: 1. if ϕ : V W linear p ϕ : p V p W maps Gr p (V) Gr p (W) 2. if dim V =: n + p natural map p V ( n V) n (V ) maps Gr p (V) Gr n p (V )

5 Plücker varieties Definition Rules X 0, X 1, X 2,... with X p : {vector spaces V} {varieties in p V}

6 Plücker varieties Definition Rules X 0, X 1, X 2,... with X p : {vector spaces V} {varieties in p V} form a Plücker variety if 1. ϕ : V W p ϕ maps X p (V) X p (W) 2. p V n (V ) maps X p (V) X n (V ).

7 Plücker varieties Definition Rules X 0, X 1, X 2,... with X p : {vector spaces V} {varieties in p V} form a Plücker variety if 1. ϕ : V W p ϕ maps X p (V) X p (W) 2. p V n (V ) maps X p (V) X n (V ). Constructions X, Y Plücker varieties so are X + Y (join), τx (tangential), X Y, X Y

8 Plücker varieties Definition Rules X 0, X 1, X 2,... with X p : {vector spaces V} {varieties in p V} form a Plücker variety if 1. ϕ : V W p ϕ maps X p (V) X p (W) 2. p V n (V ) maps X p (V) X n (V ). Constructions X, Y Plücker varieties so are X + Y (join), τx (tangential), X Y, X Y skew analogue of Snowden s -varieties

9 Results, with Eggermont Definition Plücker variety {X p } p is bounded if V : X 2 (V) 2 V

10 Results, with Eggermont Definition Plücker variety {X p } p is bounded if V : X 2 (V) 2 V Theorem Any bounded Plücker variety is defined set-theoretically in bounded degree, by finitely many equations up to symmetry.

11 Results, with Eggermont Definition Plücker variety {X p } p is bounded if V : X 2 (V) 2 V Theorem Any bounded Plücker variety is defined set-theoretically in bounded degree, by finitely many equations up to symmetry. Theorem For any fixed bounded Plücker variety there exists a polynomial-time membership test.

12 Results, with Eggermont Definition Plücker variety {X p } p is bounded if V : X 2 (V) 2 V Theorem Any bounded Plücker variety is defined set-theoretically in bounded degree, by finitely many equations up to symmetry. Theorem For any fixed bounded Plücker variety there exists a polynomial-time membership test. Theorems apply, in particular, to kgr = {alternating tensors of alternating rank k}.

13 The infinite wedge V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... V n,p := x n,..., x 1, x 1,..., x p V

14 The infinite wedge V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... V n,p := x n,..., x 1, x 1,..., x p V Diagram V 00 1 V 01 2 V 02 V np +1 V n,p+1 V 10 1 V 11 2 V 12 V n+1,p

15 The infinite wedge V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... V n,p := x n,..., x 1, x 1,..., x p V Diagram V 00 1 V 01 2 V 02 V 10 1 V 11 2 V 12 V np +1 V n,p+1 t t v p+1 V n+1,p

16 The infinite wedge V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... V n,p := x n,..., x 1, x 1,..., x p V Diagram V 00 1 V 01 2 V 02 V 10 1 V 11 2 V 12 V np +1 V n,p+1 t t v p+1 V n+1,p Definition /2 V := lim V n,p the infinite wedge (charge-0 part); basis {x I := x i1 x i2 } I, I = {i 1 < i 2 <...}, i k = k for k 0

17 The infinite wedge V :=..., x 3, x 2, x 1, x 1, x 2, x 3,... V n,p := x n,..., x 1, x 1,..., x p V Diagram V 00 1 V 01 2 V 02 V 10 1 V 11 2 V 12 V np +1 V n,p+1 t t v p+1 V n+1,p Definition /2 V := lim V n,p the infinite wedge (charge-0 part); basis {x I := x i1 x i2 } I, I = {i 1 < i 2 <...}, i k = k for k 0 On /2 V acts GL := n,p GL(V n,p ).

18 The limit of a Plücker variety Dual diagram V 1 V V np +1 V n,p+1 V 10 1 V 11 V n+1,p

19 The limit of a Plücker variety Dual diagram V 1 V V np +1 V n,p+1 V 10 1 V 11 V n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p)

20 The limit of a Plücker variety Dual diagram V 1 V V np +1 V n,p+1 V 10 1 V 11 V n+1,p X n,p X n,p+1 X n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p)

21 The limit of a Plücker variety Dual diagram V 1 V V np +1 V n,p+1 V 10 1 V 11 V n+1,p X n,p X n,p+1 X n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p) X := lim X n,p is GL -stable subvariety of ( /2 V )

22 The limit of a Plücker variety Dual diagram V 1 V V np +1 V n,p+1 V 10 1 V 11 V n+1,p X n,p X n,p+1 X n+1,p {X p } p 0 a Plücker variety varieties X n,p := X p (V n,p) X := lim X n,p is GL -stable subvariety of ( /2 V ) Theorem (implies other theorems) If X bounded X cut out by finitely many GL -orbits of equations.

23 Sato s Grassmannian Definition Gr ( /2 V ) is Sato s Grassmannian defined by polynomials i I ±x I i x J+i = 0 where i k = k 1 for k 0 and j k = k + 1 for k 0

24 Sato s Grassmannian Definition Gr ( /2 V ) is Sato s Grassmannian defined by polynomials i I ±x I i x J+i = 0 where i k = k 1 for k 0 and j k = k + 1 for k 0 not finitely many GL -orbits

25 Sato s Grassmannian Definition Gr ( /2 V ) is Sato s Grassmannian defined by polynomials i I ±x I i x J+i = 0 where i k = k 1 for k 0 and j k = k + 1 for k 0 not finitely many GL -orbits But in fact the GL -orbit of (x 2, 1,3,... x 1,2,3,... ) (x 2,1,3,... x 1,2,3,... ) + (x 2,2,3,... x 1,1,3,... ) defines Gr set-theoretically.

26 Sato s Grassmannian Definition Gr ( /2 V ) is Sato s Grassmannian defined by polynomials i I ±x I i x J+i = 0 where i k = k 1 for k 0 and j k = k + 1 for k 0 not finitely many GL -orbits But in fact the GL -orbit of (x 2, 1,3,... x 1,2,3,... ) (x 2,1,3,... x 1,2,3,... ) + (x 2,2,3,... x 1,1,3,... ) defines Gr set-theoretically. Our theorems imply that also higher secant varieties of Sato s Grassmannian are defined by finitely many GL -orbits of equations... we just don t know which!

27 Poly time Setting X bounded Plücker variety n 0, p 0 such that GL -orbits of equations of X n0,p 0 0 V n 0,p 0 define X ( /2 V ).

28 Poly time Setting X bounded Plücker variety n 0, p 0 such that GL -orbits of equations of X n0,p 0 0 V n 0,p 0 define X ( /2 V ). Shape of randomised algorithm Input: p, V, T p V Output: T X p (V)? n 0 p 0

29 Poly time Setting X bounded Plücker variety n 0, p 0 such that GL -orbits of equations of X n0,p 0 0 V n 0,p 0 define X ( /2 V ). Shape of randomised algorithm Input: p, V, T p V Output: T X p (V)? n 0 p 0 p 1. n := dim V p 2. pick random linear iso ϕ : V V n,p 3. set T := ( p ϕ)t n T

30 Poly time Setting X bounded Plücker variety n 0, p 0 such that GL -orbits of equations of X n0,p 0 0 V n 0,p 0 define X ( /2 V ). Shape of randomised algorithm Input: p, V, T p V Output: T X p (V)? n 0 p 0 T p 1. n := dim V p 2. pick random linear iso ϕ : V V n,p 3. set T := ( p ϕ)t 4. set T := image of T in V n 0,p 0 5. return T X n0,p 0? n T

31 Wrapping up Pfaffians Y k,l := {t ( /2 V ) g GL : image of gt in 2 V 2,2l has rank 2l and image of gt in 2k V 2k,2 has rank 2k}. defined by orbits of two Pfaffians

32 Wrapping up Pfaffians Y k,l := {t ( /2 V ) g GL : image of gt in 2 V 2,2l has rank 2l and image of gt in 2k V 2k,2 has rank 2k}. defined by orbits of two Pfaffians Theorem (implies earlier) Y k,l is GL -Noetherian

33 Wrapping up Pfaffians Y k,l := {t ( /2 V ) g GL : image of gt in 2 V 2,2l has rank 2l and image of gt in 2k V 2k,2 has rank 2k}. defined by orbits of two Pfaffians Theorem (implies earlier) Y k,l is GL -Noetherian Don t get new equations for secant varieties of Grassmannians!

34 Wrapping up Pfaffians Y k,l := {t ( /2 V ) g GL : image of gt in 2 V 2,2l has rank 2l and image of gt in 2k V 2k,2 has rank 2k}. defined by orbits of two Pfaffians Theorem (implies earlier) Y k,l is GL -Noetherian Don t get new equations for secant varieties of Grassmannians! Problem How to make things work ideal-theoretically? Landsberg-Ottaviani s skew flattenings?