Computing toric degenerations of flag varieties Lara Bossinger SIAM - PowerPoint PPT Presentation

Computing toric degenerations of flag varieties Lara Bossinger SIAM 31. July, 2017 Computing toric degenerations of flag varieties Lara Bossinger 1/ 18

Motivation: Why toric degenerations? For toric varieties we have a dictionary between 8 9 8 9 algebraic and geometric combinatorial < = < = properties ; $ data e.g. smooth, compact e.g. polytope, fan : : ; Computing toric degenerations of flag varieties Lara Bossinger 2/ 18

Motivation: Why toric degenerations? For toric varieties we have a dictionary between 8 9 8 9 algebraic and geometric combinatorial < = < = properties ; $ data e.g. smooth, compact e.g. polytope, fan : : ; Want to use this dictionary for an arbitrary variety X by constructing a flat family ⇡ � 1 (0) ⇠ ⇡ : X ! A 1 , s.t toric variety = T ⇡ � 1 ( t ) ⇠ and = X for t 6 = 0 . Computing toric degenerations of flag varieties Lara Bossinger 2/ 18

Motivation: Why toric degenerations? For toric varieties we have a dictionary between 8 9 8 9 algebraic and geometric combinatorial < = < = properties ; $ data e.g. smooth, compact e.g. polytope, fan : : ; Want to use this dictionary for an arbitrary variety X by constructing a flat family ⇡ � 1 (0) ⇠ ⇡ : X ! A 1 , s.t toric variety = T ⇡ � 1 ( t ) ⇠ and = X for t 6 = 0 . Flatness preserves (some) algebraic and geometric properties, e.g. dimension, degree, Gromov-width.. can use (parts of) the dictionary for X . Computing toric degenerations of flag varieties Lara Bossinger 2/ 18

Motivation: Why flag varieties? • The flag variety F ` n is the set of all flags of C n -vector subspaces V : { 0 } = V 0 ⇢ V 1 ⇢ · · · ⇢ V n � 1 ⇢ V n = C n , dim V i = i . Computing toric degenerations of flag varieties Lara Bossinger 3/ 18

Motivation: Why flag varieties? • The flag variety F ` n is the set of all flags of C n -vector subspaces V : { 0 } = V 0 ⇢ V 1 ⇢ · · · ⇢ V n � 1 ⇢ V n = C n , dim V i = i . • Can also be realized as SL n / B , where B is the subgroup of upper triangular matrices with determinant 1. So we can use representation theory of SL n . Computing toric degenerations of flag varieties Lara Bossinger 3/ 18

Motivation: Why flag varieties? • The flag variety F ` n is the set of all flags of C n -vector subspaces V : { 0 } = V 0 ⇢ V 1 ⇢ · · · ⇢ V n � 1 ⇢ V n = C n , dim V i = i . • Can also be realized as SL n / B , where B is the subgroup of upper triangular matrices with determinant 1. So we can use representation theory of SL n . • Consider U ⇢ B matrices with all diagonal entries being 1. Then SL n / B and SL n / U di ff er only by ( C ⇤ ) n . The homogenous coordinate ring C [ SL n / U ] has the structure of a cluster algebra. lots of additional information to explore di ff erent theories Computing toric degenerations of flag varieties Lara Bossinger 3/ 18

Constructions of toric degenerations Tropical classical Gr¨ obner Geometry degeneration [KM16] trop( F ` n )

Constructions of toric degenerations Tropical classical Gr¨ obner Geometry degeneration [KM16] trop( F ` n ) [Cal02] Representation [AB01] Theory of SL n [FFL17]

Constructions of toric degenerations Tropical classical Gr¨ obner Geometry degeneration [KM16] trop( F ` n ) [BFZ05] Cluster structure [GHKK14] of C [ SL n / U ] [Mag15] [Cal02] Representation [AB01] Theory of SL n [FFL17]

Constructions of toric degenerations Tropical classical Gr¨ obner Geometry degeneration [KM16] trop( F ` n ) [BFZ05] Cluster structure [GHKK14] of C [ SL n / U ] [Mag15] [Cal02] Representation [AB01] Theory of SL n [FFL17]

Constructions of toric degenerations Tropical classical Gr¨ obner Geometry degeneration [KM16] trop( F ` n ) [BFZ05] Cluster structure [GHKK14] jt. S.Lamboglia, of C [ SL n / U ] F.Mohammadi, [Mag15] K.Mincheva [Cal02] Representation [AB01] Theory of SL n [FFL17] jt. G.Fourier Computing toric degenerations of flag varieties Lara Bossinger 4/ 18

A: Tropical Geometry ! P ( n k ) � 1 for Using the Pl¨ ucker embedding Gr( k , n ) , Grassmannians we fix the embedding 1 ) � 1 ⇥ · · · ⇥ P ( n ! P ( n n � 1 ) � 1 . F ` n , ! Gr(1 , n ) ⇥ · · · ⇥ Gr( n � 1 , n ) , Computing toric degenerations of flag varieties Lara Bossinger 5/ 18

A: Tropical Geometry ! P ( n k ) � 1 for Using the Pl¨ ucker embedding Gr( k , n ) , Grassmannians we fix the embedding 1 ) � 1 ⇥ · · · ⇥ P ( n ! P ( n n � 1 ) � 1 . F ` n , ! Gr(1 , n ) ⇥ · · · ⇥ Gr( n � 1 , n ) , As a result we obtain an ideal I n ⇢ C [ p I | I ⇢ { 1 , . . . , n } ] with V ( I n ) = F ` n and I n is generated by Pl¨ ucker relations, e.g. I 3 = h p 1 p 23 � p 2 p 13 + p 3 p 12 i . Computing toric degenerations of flag varieties Lara Bossinger 5/ 18

A: Tropical Geometry ! P ( n k ) � 1 for Using the Pl¨ ucker embedding Gr( k , n ) , Grassmannians we fix the embedding 1 ) � 1 ⇥ · · · ⇥ P ( n ! P ( n n � 1 ) � 1 . F ` n , ! Gr(1 , n ) ⇥ · · · ⇥ Gr( n � 1 , n ) , As a result we obtain an ideal I n ⇢ C [ p I | I ⇢ { 1 , . . . , n } ] with V ( I n ) = F ` n and I n is generated by Pl¨ ucker relations, e.g. I 3 = h p 1 p 23 � p 2 p 13 + p 3 p 12 i . Definition Let I ⇢ C [ x 1 , . . . , x n ] be an ideal and f = P a u x u 2 I . We define with respect to w 2 R n w · u minimal a u x u , and the initial form of f as in w ( f ) = P the initial ideal of I as in w ( I ) = h in w ( f ) | f 2 I i . Computing toric degenerations of flag varieties Lara Bossinger 5/ 18

A: Tropical Geometry Example Take I 3 ⇢ C [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] and w = (0 , 0 , 1 , 0 , 0 , 0) 2 R 6 . Then in w ( p 1 p 23 � p 2 p 13 + p 3 p 12 ) = p 1 p 23 � p 2 p 13 . Computing toric degenerations of flag varieties Lara Bossinger 6/ 18

A: Tropical Geometry Example Take I 3 ⇢ C [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] and w = (0 , 0 , 1 , 0 , 0 , 0) 2 R 6 . Then in w ( p 1 p 23 � p 2 p 13 + p 3 p 12 ) = p 1 p 23 � p 2 p 13 . Let X = V ( I ) for I ⇢ C [ x 1 , . . . , x n ] and w 2 R n arbitrary. Then we have a flat family ⇡ : X ! A 1 with ⇡ � 1 ( t ) ⇠ = V ( I ) for t 6 = 0, and ⇡ � 1 (0) ⇠ = V (in w ( I )) . Computing toric degenerations of flag varieties Lara Bossinger 6/ 18

A: Tropical Geometry Example Take I 3 ⇢ C [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] and w = (0 , 0 , 1 , 0 , 0 , 0) 2 R 6 . Then in w ( p 1 p 23 � p 2 p 13 + p 3 p 12 ) = p 1 p 23 � p 2 p 13 . Let X = V ( I ) for I ⇢ C [ x 1 , . . . , x n ] and w 2 R n arbitrary. Then we have a flat family ⇡ : X ! A 1 with ⇡ � 1 ( t ) ⇠ = V ( I ) for t 6 = 0, and ⇡ � 1 (0) ⇠ = V (in w ( I )) . If in w ( I ) is binomial and prime, then V (in w ( I )) is a toric variety. Hence, the flat family defines a (Gr¨ obner) toric degeneration of X . Computing toric degenerations of flag varieties Lara Bossinger 6/ 18

A: Tropical Geometry Definition The tropicalized flag variety is defined as trop( F ` n ) = { w 2 R ( n 1 ) + ··· + ( n n � 1 ) | in w ( I n ) contains no monomials } . Computing toric degenerations of flag varieties Lara Bossinger 7/ 18

A: Tropical Geometry Definition The tropicalized flag variety is defined as trop( F ` n ) = { w 2 R ( n 1 ) + ··· + ( n n � 1 ) | in w ( I n ) contains no monomials } . It has a fan structure: for w , w 0 in relative interior of a cone C in w ( I n ) = in w 0 ( I n ) =: in C ( I n ). Computing toric degenerations of flag varieties Lara Bossinger 7/ 18

A: Tropical Geometry Definition The tropicalized flag variety is defined as trop( F ` n ) = { w 2 R ( n 1 ) + ··· + ( n n � 1 ) | in w ( I n ) contains no monomials } . It has a fan structure: for w , w 0 in relative interior of a cone C in w ( I n ) = in w 0 ( I n ) =: in C ( I n ). The S n -action on F ` n , for � 2 S n induced by p { i 1 ,..., i k } 7! sgn( � ) p { σ ( i 1 ) ,..., σ ( i k ) } , and the Z 2 -action induced by p I 7! p [ n ] \ I extend to trop( F ` n ). Computing toric degenerations of flag varieties Lara Bossinger 7/ 18

A: Tropical Geometry Definition The tropicalized flag variety is defined as trop( F ` n ) = { w 2 R ( n 1 ) + ··· + ( n n � 1 ) | in w ( I n ) contains no monomials } . It has a fan structure: for w , w 0 in relative interior of a cone C in w ( I n ) = in w 0 ( I n ) =: in C ( I n ). The S n -action on F ` n , for � 2 S n induced by p { i 1 ,..., i k } 7! sgn( � ) p { σ ( i 1 ) ,..., σ ( i k ) } , and the Z 2 -action induced by p I 7! p [ n ] \ I extend to trop( F ` n ). Aim: Find (up to symmetry) all maximal prime cones C ⇢ trop( F ` n ), i.e. in C ( I n ) is binomial and prime. Computing toric degenerations of flag varieties Lara Bossinger 7/ 18

A: Tropical Geometry Kaveh-Manon construction: 8 9 C ⇢ trop( F ` n ) < = maximal prime cone : ; Computing toric degenerations of flag varieties Lara Bossinger 8/ 18

A: Tropical Geometry Kaveh-Manon construction: 8 9 C ⇢ trop( F ` n ) ⇢ full rank � < = maximal ; valuation ⌫ C prime cone : Computing toric degenerations of flag varieties Lara Bossinger 8/ 18

A: Tropical Geometry Kaveh-Manon construction: 8 9 8 9 C ⇢ trop( F ` n ) Newton- ⇢ full rank � < = < = maximal Okounkov ; valuation ⌫ C prime cone polytope NO C : : ; Computing toric degenerations of flag varieties Lara Bossinger 8/ 18