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

Isomorphism type of Schubert varieties

Ed Richmond1 William Slofstra2

1Oklahoma State University 2University 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

• Aii = 2 for all i = 1, . . . , n,
• Aij ≤ 0 if i = j, and
• if Aij = 0 then Aji = 0.

Examples: type A :        2 −1 −1 2 −1 −1 2 −1 −1 2            2 −3 −7 −1 2 −2 2    

Isomorphism type of Schubert varieties W Slofstra

Weyl groups

Starting with n × n GCM A, the Weyl group W (A) is the group s1, . . . , sn : s2

i = 1, (sisj)mij = 1 for 1 ≤ i = j ≤ n

with mij =                        2 AijAji = 0 3 AijAji = 1 4 AijAji = 2 6 AijAji = 3 ∞ AijAji ≥ 4 Example: W (An) = Sn+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(An) = GLn C, T = diagonal invertible matrices, B = upper triangular invertible matrices.

• The full flag variety X(A) = G/B.

For An, get the space Fl(n) = {0 = E0 E1 · · · En En+1 = Cn+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; An) is the closure of BFw, where Fw = (E0, . . . , En) is defined by Ei = span{ew(1), . . . , ew(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 An 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 An that do not appear in finite type? Possible answer: diagram isomorphism (Example: X(si) ∼ = P1)

Isomorphism type of Schubert varieties W Slofstra

When are X(w; A) and X(w ′; A′) isomorphic as algebraic varieties?

si1 · · · sik is a reduced word if there no way to write w as a product

• f fewer simple reflections si

S(w) = {1 ≤ i ≤ n : si 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

• Ast = 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 A3, X(s1s2s1) ∼ = X(s2s3s2).

Isomorphism type of Schubert varieties W Slofstra

Are diagram isomorphisms the only isomorphisms?

Look at X(s1s2; A) with A =

• 2

−a −b 2

• P1 ∼

= X(s2) X(s1s2) X(s2) ∼ = P1 Multiplication table on H2: ξs1 ξs2 ξs1 ξs1s2 ξs2 ξs1s2 aξs1s2 X(s1s2) is a Hirzebruch surface Σn (Σn ∼ = Σm if and only if m = n) X(s1s2) is Σa. b is irrelevant 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 = si1 · · · sik such that

• s′

σ(i1) · · · s′ σ(ik) is a reduced expression for w ′, and

• Aijij′ = Aσ(ij)σ(ij′) 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 = si1 · · · sik such that

• s′

σ(i1) · · · s′ σ(ik) is a reduced expression for w ′, and

• Aijij′ = Aσ(ij)σ(ij′) 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 ξv, v ≤ w in Bruhat

• rder

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 = si1 · · · sik such that

• s′

σ(i1) · · · s′ σ(ik) is a reduced expression for w ′, and

• Aijij′ = Aσ(ij)σ(ij′) 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 H2(X) spanned by ξsi, i ∈ S(w) = ⇒ can identity S(w) Can recover Bruhat order and right descents from Chevalley-Monk formula for ξsiξv = ⇒ can get reduced expression Aij shows up in structure constants if and only if sisj ≤ 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: H2(X J(w; A)) no longer spanned by ξsi, 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: H2(X J(w; A)) no longer spanned by ξsi, i ∈ S(w)

The end!

Isomorphism type of Schubert varieties W Slofstra