Isomorphism type of Schubert varieties Ed Richmond 1 William Slofstra - PowerPoint PPT Presentation

Isomorphism type of Schubert varieties Ed Richmond 1 William Slofstra 2 1 Oklahoma State University 2 University of Waterloo June 4th, 2018 Isomorphism type of Schubert varieties W Slofstra

Generalized Cartan matrices A GCM is an n × n matrix A such that • A ii = 2 for all i = 1 , . . . , n , • A ij ≤ 0 if i � = j , and • if A ij = 0 then A ji = 0. Examples:     2 − 3 − 7 2 − 1 0 0         − 1 2 0 − 1 2 − 1 0     type A :     − 2 0 2 0 − 1 2 − 1   0 0 − 1 2 Isomorphism type of Schubert varieties W Slofstra

Weyl groups Starting with n × n GCM A , the Weyl group W ( A ) is the group i = 1 , ( s i s j ) m ij = 1 for 1 ≤ i � = j ≤ n � � s 1 , . . . , s n : s 2 with    2 A ij A ji = 0       3 A ij A ji = 1    m ij = 4 A ij A ji = 2      6 A ij A ji = 3       ∞ A ij A ji ≥ 4 Example: W ( A n ) = S n +1 , the permutation group Isomorphism type of Schubert varieties W Slofstra

Flag varieties From a GCM A , can also construct: • A Kac-Moody group G = G ( A ), including Cartan and Borel subgroups T ⊆ B . Example: G ( A n ) = GL n C , T = diagonal invertible matrices, B = upper triangular invertible matrices. • The full flag variety X ( A ) = G / B . For A n , get the space Fl ( n ) = { 0 = E 0 � E 1 � · · · � E n � E n +1 = C n +1 } . In general, X ( A ) can be infinite-dimensional. Isomorphism type of Schubert varieties W Slofstra

Schubert varieties From a GCM A , can also construct: • A Kac-Moody group G = G ( A ), including Cartan and Borel subgroups T ⊆ B . • The full flag variety X ( A ) of A , defined by G / B . In general, X ( A ) can be infinite-dimensional. • Schubert varieties X ( w ; A ) indexed by w ∈ W ( A ). These are finite-dimensional normal projective T -varieties stratifying X ( A ). X ( w ; A n ) is the closure of B F w , where F w = ( E 0 , . . . , E n ) is defined by E i = span { e w (1) , . . . , e w ( i ) } . Isomorphism type of Schubert varieties W Slofstra

Natural question: When are the Schubert varieties X ( w ; A ) and X ( w ′ ; A ′ ) isomorphic as algebraic varieties? Motivation: are there smooth varieties in affine type � A n that do not appear in finite type? Isomorphism type of Schubert varieties W Slofstra

Natural question: When are the Schubert varieties X ( w ; A ) and X ( w ′ ; A ′ ) isomorphic as algebraic varieties? Motivation: are there smooth varieties in affine type � A n that do not appear in finite type? Possible answer: diagram isomorphism (Example: X ( s i ) ∼ = P 1 ) Isomorphism type of Schubert varieties W Slofstra

When are X ( w ; A ) and X ( w ′ ; A ′ ) isomorphic as algebraic varieties? s i 1 · · · s i k is a reduced word if there no way to write w as a product of fewer simple reflections s i S ( w ) = { 1 ≤ i ≤ n : s i appears in reduced word for w } Suppose w ∈ W ( A ), w ′ ∈ W ( A ′ ), and there is a bijection σ : S ( w ) → S ( w ′ ) such that • A st = A ′ σ ( s ) σ ( t ) for all s , t ∈ S ( w ) • the iso W ( A ) S ( w ) → W ( A ′ ) S ( w ′ ) sends w �→ w ′ . Then X ( w ; A ) ∼ = X ( w ′ ; A ′ ). Example: In A 3 , X ( s 1 s 2 s 1 ) ∼ = X ( s 2 s 3 s 2 ). Isomorphism type of Schubert varieties W Slofstra

Are diagram isomorphisms the only isomorphisms? � � 2 − a Look at X ( s 1 s 2 ; A ) with A = − b 2 P 1 ∼ = X ( s 2 ) X ( s 1 s 2 ) X ( s 1 s 2 ) is a Hirzebruch surface Σ n X ( s 2 ) ∼ = P 1 (Σ n ∼ = Σ m if and only if m = n ) Multiplication table on H 2 : ξ s 1 ξ s 2 X ( s 1 s 2 ) is Σ a . ξ s 1 ξ s 1 s 2 0 b is irrelevant ξ s 2 ξ s 1 s 2 a ξ s 1 s 2 Conclusion: no! Isomorphism type of Schubert varieties W Slofstra

When are X ( w ; A ) and X ( w ′ ; A ′ ) isomorphic as algebraic varieties? Theorem (Richmond-S) The following are equivalent: (1) X ( w ; A ) ∼ = X ( w ′ ; A ′ ) (2) there is an isomorphism H ∗ ( X ( w ; A )) → H ∗ ( X ( w ′ ; A ′ )) which preserves the Schubert basis (3) there is a bijection σ : S ( w ) → S ( w ′ ) and a reduced expression w = s i 1 · · · s i k such that • s ′ σ ( i 1 ) · · · s ′ σ ( i k ) is a reduced expression for w ′ , and • A i j i j ′ = A σ ( i j ) σ ( i j ′ ) for all j < j ′ Isomorphism type of Schubert varieties W Slofstra

Hard direction: (3) implies (1) Theorem (Richmond-S) The following are equivalent: (1) X ( w ; A ) ∼ = X ( w ′ ; A ′ ) (2) ... (3) there is a bijection σ : S ( w ) → S ( w ′ ) and a reduced expression w = s i 1 · · · s i k such that • s ′ σ ( i 1 ) · · · s ′ σ ( i k ) is a reduced expression for w ′ , and • A i j i j ′ = A σ ( i j ) σ ( i j ′ ) for all j < j ′ Why? No T -variety structure Isomorphism type of Schubert varieties W Slofstra

Proof: (1) implies (2)? Theorem (Richmond-S) The following are equivalent: (1) X ( w ; A ) ∼ = X ( w ′ ; A ′ ) (2) there is an isomorphism H ∗ ( X ( w ; A )) → H ∗ ( X ( w ′ ; A ′ )) which preserves the Schubert basis H ∗ ( X ( w ; A ) spanned by Schubert classes ξ v , v ≤ w in Bruhat order These classes are the extremal rays of the effective cone of X ( w ; A ) Isomorphism type of Schubert varieties W Slofstra

Proof: (2) implies (3)? Theorem (Richmond-S) The following are equivalent: (1) X ( w ; A ) ∼ = X ( w ′ ; A ′ ) (2) there is an isomorphism H ∗ ( X ( w ; A )) → H ∗ ( X ( w ′ ; A ′ )) which preserves the Schubert basis (3) there is a bijection σ : S ( w ) → S ( w ′ ) and a reduced expression w = s i 1 · · · s i k such that • s ′ σ ( i 1 ) · · · s ′ σ ( i k ) is a reduced expression for w ′ , and • A i j i j ′ = A σ ( i j ) σ ( i j ′ ) for all j < j ′ Isomorphism type of Schubert varieties W Slofstra

Proof: (2) implies (3)? Given an algebraic variety X which is promised to be a Schubert variety, can we construct A and w ∈ W ( A ) such that X ∼ = X ( w ; A )? Answer: yes, identify extremal rays of effective cone in H ∗ ( X ) with Schubert classes and recover w from rules for Schubert calculus H 2 ( X ) spanned by ξ s i , i ∈ S ( w ) = ⇒ can identity S ( w ) Can recover Bruhat order and right descents from Chevalley-Monk formula for ξ s i ξ v = ⇒ can get reduced expression A ij shows up in structure constants if and only if s i s j ≤ w in Bruhat order Isomorphism type of Schubert varieties W Slofstra

Further questions Does the same answer apply to parabolic Schubert varieties X J ( w ; A )? Difficulty: H 2 ( X J ( w ; A )) no longer spanned by ξ s i , i ∈ S ( w ) Isomorphism type of Schubert varieties W Slofstra

Further questions Does the same answer apply to parabolic Schubert varieties X J ( w ; A )? Difficulty: H 2 ( X J ( w ; A )) no longer spanned by ξ s i , i ∈ S ( w ) The end! Isomorphism type of Schubert varieties W Slofstra