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Criteria for rational smoothness of some symmetric orbit closures - - PowerPoint PPT Presentation

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Criteria for rational smoothness of some symmetric orbit closures Axel Hultman KTH, Stockholm Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 1 Schubert varieties Axel Hultman Criteria for rational smoothness

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Criteria for rational smoothness of some symmetric

  • rbit closures

Axel Hultman

KTH, Stockholm

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 1

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Schubert varieties

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Schubert varieties

  • G – reductive algebraic group over C

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Schubert varieties

  • G – reductive algebraic group over C

Example:

  • GLn(C)

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Schubert varieties

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Schubert varieties

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Schubert varieties

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn
  • B-orbits in B ↔ Weyl group W
  • W = Sn (symmetric group)

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Schubert varieties

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn
  • B-orbits in B ↔ Weyl group W
  • W = Sn (symmetric group)

The closure of the orbit indexed by w ∈ W is the Schubert variety X(w).

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Schubert varieties

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn
  • B-orbits in B ↔ Weyl group W
  • W = Sn (symmetric group)

The closure of the orbit indexed by w ∈ W is the Schubert variety X(w). Well-studied problem: Describe geometric properties, such as the (rationally) singular locus, of X(w) in terms of the combinatorics of W .

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

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Rational smoothness

Definition: A complex variety X is rationally smooth at x if Hm(X, X \ {y}; Q) =

  • Q

if m = 2·topdim,

  • therwise,

for all y in some neighbourhood of x.

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Rational smoothness

Definition: A complex variety X is rationally smooth at x if Hm(X, X \ {y}; Q) =

  • Q

if m = 2·topdim,

  • therwise,

for all y in some neighbourhood of x. Theorem (Carrell-Kuttler): For Schubert varieties in simply laced types, rational smoothness is equivalent to smoothness.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 3

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Symmetric orbits

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

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Symmetric orbits

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn
  • θ – an involution on G
  • g → (g−1)T

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

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Symmetric orbits

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn
  • θ – an involution on G
  • g → (g−1)T
  • K – the fixed point subgroup
  • On(C)

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

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Symmetric orbits

  • G – reductive algebraic group over C

Example:

  • GLn(C)
  • B – Borel subgroup
  • upper triangular matrices
  • B = G/B – the flag variety
  • complete flags in Cn
  • θ – an involution on G
  • g → (g−1)T
  • K – the fixed point subgroup
  • On(C)

Fact: K acts on B with finitely many orbits.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

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Twisted involutions

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Twisted involutions

The map θ restricts to W as a Coxeter diagram involution.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

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Twisted involutions

The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W :

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

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Twisted involutions

The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I(θ) = {w ∈ W | θ(w) = w−1} (twisted involutions)

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

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Twisted involutions

The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I(θ) = {w ∈ W | θ(w) = w−1} (twisted involutions) ι(θ) = {θ(x−1)x | x ∈ W } ⊆ I(θ) (twisted identities)

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

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Twisted involutions

The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I(θ) = {w ∈ W | θ(w) = w−1} (twisted involutions) ι(θ) = {θ(x−1)x | x ∈ W } ⊆ I(θ) (twisted identities) Richardson-Springer constructed an order preserving map from K-orbit closures to twisted involutions ordered by Bruhat order.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

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Twisted involutions

The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I(θ) = {w ∈ W | θ(w) = w−1} (twisted involutions) ι(θ) = {θ(x−1)x | x ∈ W } ⊆ I(θ) (twisted identities) Richardson-Springer constructed an order preserving map from K-orbit closures to twisted involutions ordered by Bruhat order. Problem: Describe geometric properties, such as the (rationally) singular locus, of K-orbit closures in terms of the combinatorics of I(θ).

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

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Restricting the setup

Being outsmarted by the general problem, we put restrictions on G and θ:

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

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Restricting the setup

Being outsmarted by the general problem, we put restrictions on G and θ: Hypotheses:

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

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Restricting the setup

Being outsmarted by the general problem, we put restrictions on G and θ: Hypotheses: The image of the dense K-orbit is in ι(θ). (essential)

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

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Restricting the setup

Being outsmarted by the general problem, we put restrictions on G and θ: Hypotheses: The image of the dense K-orbit is in ι(θ). (essential) K is connected. (convenient)

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

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Restricting the setup

Being outsmarted by the general problem, we put restrictions on G and θ: Hypotheses: The image of the dense K-orbit is in ι(θ). (essential) K is connected. (convenient) Consequence: The RS map is injective with image ι(θ).

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Examples

Four classes of examples satisfy the hypotheses:

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

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Examples

Four classes of examples satisfy the hypotheses: Class 1. (∼Schubert varieties) With G = G ′ × G ′ and θ((x, y)) = (y, x), ι(θ) is in bijection with the Weyl group W ′.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

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Examples

Four classes of examples satisfy the hypotheses: Class 1. (∼Schubert varieties) With G = G ′ × G ′ and θ((x, y)) = (y, x), ι(θ) is in bijection with the Weyl group W ′. Class 2. With G = SL2n(C), K = Sp2n(C), ι(θ) is in bijection with fixed point free involutions in W = S2n (with dual Bruhat order).

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

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Examples

Four classes of examples satisfy the hypotheses: Class 1. (∼Schubert varieties) With G = G ′ × G ′ and θ((x, y)) = (y, x), ι(θ) is in bijection with the Weyl group W ′. Class 2. With G = SL2n(C), K = Sp2n(C), ι(θ) is in bijection with fixed point free involutions in W = S2n (with dual Bruhat order). Class 3. G of type Dn+1, K of type Bn.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

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Examples

Four classes of examples satisfy the hypotheses: Class 1. (∼Schubert varieties) With G = G ′ × G ′ and θ((x, y)) = (y, x), ι(θ) is in bijection with the Weyl group W ′. Class 2. With G = SL2n(C), K = Sp2n(C), ι(θ) is in bijection with fixed point free involutions in W = S2n (with dual Bruhat order). Class 3. G of type Dn+1, K of type Bn. Class 4. G of type E6, K of type F4.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

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Bruhat graphs

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Bruhat graphs

Definition: Given w ∈ ι(θ), the Bruhat graph Γ(w) has vertex set {u ∈ ι(θ) : u ≤ w} (Bruhat order) and an edge {x, θ(t)xt} if t ∈ W is a reflection such that θ(t)xt = x.

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Bruhat graphs

Definition: Given w ∈ ι(θ), the Bruhat graph Γ(w) has vertex set {u ∈ ι(θ) : u ≤ w} (Bruhat order) and an edge {x, θ(t)xt} if t ∈ W is a reflection such that θ(t)xt = x. Example: If W = S2n and θ is nontrivial, we may model ι(θ) using fixed point free involutions. Then, “≤” is the dual Bruhat order and edges in Γ(w) correspond to conjugation with transpositions.

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A Bruhat graph

Example: The Bruhat graph of the twisted identity represented by the fixed point free involution 532614 ∈ S6:

  • Axel Hultman

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Criteria for rational smoothness

Theorem: Let u ≤ w be twisted identities. The following are equivalent:

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Criteria for rational smoothness

Theorem: Let u ≤ w be twisted identities. The following are equivalent: The K-orbit closure indexed by w is rationally smooth at the orbit indexed by u.

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Criteria for rational smoothness

Theorem: Let u ≤ w be twisted identities. The following are equivalent: The K-orbit closure indexed by w is rationally smooth at the orbit indexed by u. The degree of x in Γ(w) is rank(w) for all x ∈ [u, w].

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 10

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Criteria for rational smoothness

Theorem: Let u ≤ w be twisted identities. The following are equivalent: The K-orbit closure indexed by w is rationally smooth at the orbit indexed by u. The degree of x in Γ(w) is rank(w) for all x ∈ [u, w]. Moreover, if u = id, these are equivalent to the interval [id, w] being rank symmetric.

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Criteria for rational smoothness

Theorem: Let u ≤ w be twisted identities. The following are equivalent: The K-orbit closure indexed by w is rationally smooth at the orbit indexed by u. The degree of x in Γ(w) is rank(w) for all x ∈ [u, w]. Moreover, if u = id, these are equivalent to the interval [id, w] being rank symmetric. Remark: In the Schubert variety setting (Class 1, remember?) these are “Carrell-Peterson’s criteria”.

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A (rationally) smooth example

Example: In the flag variety of SL6(C), the closure of the Sp6(C)-orbit which is represented by the fixed point free involution 532614 is rationally smooth...

  • Axel Hultman

Criteria for rational smoothness of some symmetric orbit closures 11

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A singular example

...whereas the orbit closure represented by 351624 is singular.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 12

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A word about the proof

The proof uses Kazhdan-Lusztig-Vogan (KLV) polynomials Pu,w ∈ N[q]. Our hypotheses on G, θ have two nice implications:

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 13

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A word about the proof

The proof uses Kazhdan-Lusztig-Vogan (KLV) polynomials Pu,w ∈ N[q]. Our hypotheses on G, θ have two nice implications: The indices u, w can be thought of as twisted identities in W .

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 13

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A word about the proof

The proof uses Kazhdan-Lusztig-Vogan (KLV) polynomials Pu,w ∈ N[q]. Our hypotheses on G, θ have two nice implications: The indices u, w can be thought of as twisted identities in W . Vogan’s overwhelming recursive formula for KLV R-polynomials becomes slightly less overwhelming.

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 13

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A word about the proof

The proof uses Kazhdan-Lusztig-Vogan (KLV) polynomials Pu,w ∈ N[q]. Our hypotheses on G, θ have two nice implications: The indices u, w can be thought of as twisted identities in W . Vogan’s overwhelming recursive formula for KLV R-polynomials becomes slightly less overwhelming. Carrell-Peterson’s proof uses Kazhdan-Lusztig polynomials. It extends nicely once several technical facts have been established for KLV

  • polynomials. Key steps include:

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 13

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A word about the proof

The proof uses Kazhdan-Lusztig-Vogan (KLV) polynomials Pu,w ∈ N[q]. Our hypotheses on G, θ have two nice implications: The indices u, w can be thought of as twisted identities in W . Vogan’s overwhelming recursive formula for KLV R-polynomials becomes slightly less overwhelming. Carrell-Peterson’s proof uses Kazhdan-Lusztig polynomials. It extends nicely once several technical facts have been established for KLV

  • polynomials. Key steps include:

Pu,w(0) = 1 (relies on M¨

  • bius function of ι(θ)).

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 13

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A word about the proof

The proof uses Kazhdan-Lusztig-Vogan (KLV) polynomials Pu,w ∈ N[q]. Our hypotheses on G, θ have two nice implications: The indices u, w can be thought of as twisted identities in W . Vogan’s overwhelming recursive formula for KLV R-polynomials becomes slightly less overwhelming. Carrell-Peterson’s proof uses Kazhdan-Lusztig polynomials. It extends nicely once several technical facts have been established for KLV

  • polynomials. Key steps include:

Pu,w(0) = 1 (relies on M¨

  • bius function of ι(θ)).

A lower degree bound for Bruhat graphs (derived from work of Brion).

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 13

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A word about the proof

The proof uses Kazhdan-Lusztig-Vogan (KLV) polynomials Pu,w ∈ N[q]. Our hypotheses on G, θ have two nice implications: The indices u, w can be thought of as twisted identities in W . Vogan’s overwhelming recursive formula for KLV R-polynomials becomes slightly less overwhelming. Carrell-Peterson’s proof uses Kazhdan-Lusztig polynomials. It extends nicely once several technical facts have been established for KLV

  • polynomials. Key steps include:

Pu,w(0) = 1 (relies on M¨

  • bius function of ι(θ)).

A lower degree bound for Bruhat graphs (derived from work of Brion). KLV R-polynomials detect edges in “Bruhat graphs” (established using Vogan’s recursion).

Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 13

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Concluding remarks

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Concluding remarks

In the special case G = SL2n(C), K = Sp2n(C), more can be said:

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Concluding remarks

In the special case G = SL2n(C), K = Sp2n(C), more can be said: Theorem: The degree in Γ(w) cannot decrease as we move down in the

  • graph. Thus, to test for rational smoothness at a vertex, it suffices to

inspect the degree of that vertex only. (Extends a result of Deodhar on type A Schubert varieties.)

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Concluding remarks

In the special case G = SL2n(C), K = Sp2n(C), more can be said: Theorem: The degree in Γ(w) cannot decrease as we move down in the

  • graph. Thus, to test for rational smoothness at a vertex, it suffices to

inspect the degree of that vertex only. (Extends a result of Deodhar on type A Schubert varieties.) Theorem (McGovern ’10): The rationally singular loci coincide with the singular loci. (McGovern also gives a pattern avoidance type criterion for detecting smoothness.)

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