Spherical double flag varieties · Flag varieties Schubert varieties 2 Spherical double ﬂags of...

Spherical double flag varieties

Evgeny Smirnov

Higher School of EconomicsDepartment of Mathematics

Independent University of Moscow

Torus Actions in Geometry, Topology, and ApplicationsSkolkovo, February 16, 2015

Evgeny Smirnov (HSE & IUM) Spherical double flag varieties Skolkovo, February 16, 2015 1 / 12

Outline

1 General definitionsFlag varietiesSchubert varieties

2 Spherical double flags of type ADouble GrassmanniansCombinatorics of B-orbits in double Grassmannians

3 Cominuscule flag varietiesDefinitionCombinatorial and geometric results

Flag varieties

NotationG reductive algebraic group;

B fixed Borel subgroup;T maximal torus contained in B;P parabolic subgroup, P ⊃ B;G/P (generalized) partial flag variety.

Main exampleG = GL(n);B upper-triangular matrices;T diagonal matrices;Two important subcases:

P = B: full flag variety G/B = Fl(n).;P = P(k)

max : Grassmannian of k -planes G/Pmax = Gr(k ,n).

Flag varieties

NotationG reductive algebraic group;B fixed Borel subgroup;

T maximal torus contained in B;P parabolic subgroup, P ⊃ B;G/P (generalized) partial flag variety.

Flag varieties

NotationG reductive algebraic group;B fixed Borel subgroup;T maximal torus contained in B;

P parabolic subgroup, P ⊃ B;G/P (generalized) partial flag variety.

Flag varieties

NotationG reductive algebraic group;B fixed Borel subgroup;T maximal torus contained in B;P parabolic subgroup, P ⊃ B;

G/P (generalized) partial flag variety.

Flag varieties

NotationG reductive algebraic group;B fixed Borel subgroup;T maximal torus contained in B;P parabolic subgroup, P ⊃ B;G/P (generalized) partial flag variety.

Flag varieties

Main exampleG = GL(n);

B upper-triangular matrices;T diagonal matrices;Two important subcases:

Flag varieties

Main exampleG = GL(n);B upper-triangular matrices;

T diagonal matrices;Two important subcases:

Flag varieties

Main exampleG = GL(n);B upper-triangular matrices;T diagonal matrices;

Two important subcases:P = B: full flag variety G/B = Fl(n).;P = P(k)

Flag varieties

P = B: full flag variety G/B = Fl(n).;

P = P(k)max : Grassmannian of k -planes G/Pmax = Gr(k ,n).

Flag varieties

Schubert decomposition

Schubert cellsG/P has finitely many B-orbits Ωw(G/P is a spherical variety);

Each Ωw has a unique T -stable point;G/P =

⊔Ωw is a Białynicki-Birula decomposition;

Ωw are indexed by the cosets W/WI ;Xw = Ωw are called Schubert varieties.

Schubert varietiesUseful for enumerative geometry;Have interesting geometry: they are usually singular......however, still nice!

normal;Cohen–Macaulay;have rational singularities...

Schubert cellsG/P has finitely many B-orbits Ωw(G/P is a spherical variety);Each Ωw has a unique T -stable point;

G/P =⊔

Ωw is a Białynicki-Birula decomposition;Ωw are indexed by the cosets W/WI ;Xw = Ωw are called Schubert varieties.

Schubert cellsG/P has finitely many B-orbits Ωw(G/P is a spherical variety);Each Ωw has a unique T -stable point;G/P =

Ωw are indexed by the cosets W/WI ;

Xw = Ωw are called Schubert varieties.

Schubert varietiesUseful for enumerative geometry;

Have interesting geometry: they are usually singular......however, still nice!

Schubert varietiesUseful for enumerative geometry;Have interesting geometry: they are usually singular...

...however, still nice!normal;Cohen–Macaulay;have rational singularities...

Schubert varieties in Grassmannians

Let U ∈ Gr(k ,n), and let (Vi) be the flag fixed by B.

Then Xw ⊂ Gr(k ,n) are given by conditions of type

dim(U ∩ Vi) ≥ di , di − di−1 ∈ 0,1, dn = k .

Such sequences of 0’s and 1’s correspond to Young diagrams inside ak × (n − k)-rectangle.

Example: k = 4, n = 9

di = (0 0 1 0 1 1 0 0 1)

Let U ∈ Gr(k ,n), and let (Vi) be the flag fixed by B.Then Xw ⊂ Gr(k ,n) are given by conditions of type

di = (0 0 1 0 1 1 0 0 1)

Spherical multiple flags

Consider the direct product of several flag varieties:

X = G/P1 × · · · ×G/Pr .

G acts diagonally.X is not G-homogeneous anymore!

Main questionWhen is X a G-spherical variety?I.e., when does B ⊂ G have a finite number of orbits on X?

Answerr = 1: always;r ≥ 3: never;r = 2: sometimes......classification given by P. Littelmann, J. Stembridge.

X = G/P1 × · · · ×G/Pr .

G acts diagonally.

X is not G-homogeneous anymore!

X = G/P1 × · · · ×G/Pr .

Answerr = 1: always;

r ≥ 3: never;r = 2: sometimes......classification given by P. Littelmann, J. Stembridge.

X = G/P1 × · · · ×G/Pr .

Answerr = 1: always;r ≥ 3: never;

r = 2: sometimes......classification given by P. Littelmann, J. Stembridge.

X = G/P1 × · · · ×G/Pr .

Answerr = 1: always;r ≥ 3: never;r = 2: sometimes...

...classification given by P. Littelmann, J. Stembridge.

X = G/P1 × · · · ×G/Pr .

Type A: double Grassmannians

If G = GL(n), all spherical double flag varieties correspond to P1,P2maximal:

X = Gr(k ,n)×Gr(l ,n).

B-orbits correspond to configurations of two subspaces and a full flag,up to GL(n)-action.Combinatorics becomes more involved!

Example: Gr(1,3)×Gr(1,3) = P2 × P2

One B × B-orbit splits into three B-orbits!

B-orbits correspond to configurations of two subspaces and a full flag,up to GL(n)-action.

Combinatorics becomes more involved!

Example: Gr(1,3)×Gr(1,3) = P2 × P2

Combinatorics of B-orbits

Our next goal is to describe B-orbits on Gr(k ,n)×Gr(l ,n).

B × B-orbits are indexed by pairs of Young diagrams.But we need some extra data...

The common diagram of two Young diagrams

0 0 1 0 1 1 0 0 10 1 0 0 1 0 0 0 10 0 1 0 0 1

Consider rook placements in the common diagram.

Our next goal is to describe B-orbits on Gr(k ,n)×Gr(l ,n).B × B-orbits are indexed by pairs of Young diagrams.

But we need some extra data...

0 0 1 0 1 1 0 0 10 1 0 0 1 0 0 0 10 0 1 0 0 1

Our next goal is to describe B-orbits on Gr(k ,n)×Gr(l ,n).B × B-orbits are indexed by pairs of Young diagrams.But we need some extra data...

0 0 1 0 1 1 0 0 10 1 0 0 1 0 0 0 10 0 1 0 0 1

Theorem (S.)B-orbits in Gr(k ,n)×Gr(l ,n) are indexed by triples (Y1,Y2,R), where:

Y1 ⊂ k × (n − k) and Y2 ⊂ l × (n − l) are Young diagrams;R is a rook placement in the common diagram of Y1 and Y2.

Dimension and rank of orbits can be read from this description;This allows to construct resolutions of singularities of orbitclosures à la Bott–Samelson–Demazure–Hansen.

Y1 ⊂ k × (n − k) and Y2 ⊂ l × (n − l) are Young diagrams;

R is a rook placement in the common diagram of Y1 and Y2.

Dimension and rank of orbits can be read from this description;

This allows to construct resolutions of singularities of orbitclosures à la Bott–Samelson–Demazure–Hansen.

Cominuscule double flags

DefinitionA partial flag variety is cominuscule if it belongs to the following list:

Group type VarietyAn−1 Gr(k ,n) GrassmannianBn Q2n−1 quadricCn LGr(n) Lagrangian GrassmannianDn OGr(n) orthogonal Grassmannian

Q2n quadricE6 OP2 Cayley planeE7 Gω(O3,O6) Lagrangian octonion Grassmannian

We will consider double cominuscule flag varieties(they are all spherical).

DefinitionA partial flag variety is cominuscule if it belongs to the following list:Group type VarietyAn−1 Gr(k ,n) GrassmannianBn Q2n−1 quadricCn LGr(n) Lagrangian GrassmannianDn OGr(n) orthogonal Grassmannian

Results on cominuscule double flags

CombinatoricsThere is a combinatorial indexing of B-orbits in doublecominuscule flag varieties of classical groups, similar to the onewe had in the type A. (S.)

Still open for E6 and E7.

GeometryIf G is simply laced, then the B-orbit closures in doublecominuscule flag varieties are normal, Cohen–Macaulay, andhave rational singularities (P.Achinger, N.Perrin, 2013).In type A was proved by methods of quiver theory (G.Bobinski,G.Zwara, 2001)Normality can fail for nonsimply laced G.

CombinatoricsThere is a combinatorial indexing of B-orbits in doublecominuscule flag varieties of classical groups, similar to the onewe had in the type A. (S.)Still open for E6 and E7.

GeometryIf G is simply laced, then the B-orbit closures in doublecominuscule flag varieties are normal, Cohen–Macaulay, andhave rational singularities (P.Achinger, N.Perrin, 2013).

In type A was proved by methods of quiver theory (G.Bobinski,G.Zwara, 2001)Normality can fail for nonsimply laced G.

GeometryIf G is simply laced, then the B-orbit closures in doublecominuscule flag varieties are normal, Cohen–Macaulay, andhave rational singularities (P.Achinger, N.Perrin, 2013).In type A was proved by methods of quiver theory (G.Bobinski,G.Zwara, 2001)

Normality can fail for nonsimply laced G.

That’s all...

Thank you!

Spherical double flag varieties · Flag varieties Schubert varieties 2 Spherical double ﬂags of...

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