Spherical Schubert Varieties

Mahir Bilen Can

September 29, 2020 PART 1

MUKASHIBANASHI Then R is an associative commutative graded ring w.r.t. the product χV · χW = indSn χV × χW Sk ×Sn−k

V where χ is the character of the Sk -representation V . Likewise, W χ is the character of the Sn−k -representation W . ! Sk : the stabilizer of {1,...,k} in Sn Sn−k : the stabilizer of {k + 1,...,n} in Sn

n R : the free Z-module on the irreducible characters of Sn n 0 R : the direct sum of all R , n ∈ N (where R = Z) n R : the free Z-module on the irreducible characters of Sn n 0 R : the direct sum of all R , n ∈ N (where R = Z)

Then R is an associative commutative graded ring w.r.t. the product χV · χW = indSn χV × χW Sk ×Sn−k

λ λ χ : the character of the Specht module S of Sk µ µ χ : the character of the Specht module S of Sn−k

In R, we have λ µ M τ τ χ · χ = cλ,µ χ τ`n τ for some nonnegative integers cλ,µ ∈ N, called the Littlewood- Richardson numbers.

It is well-known that R is isomorphic to the algebra of symmetric τ functions in inﬁnitely many commuting variables; cλ,µ’s can be computed via Schur’s symmetric functions. τ • For which k ∈ N, do we have c(k),(n−k) ∈ {0,1} for every τ ` n? Equivalently, when is the induced trivial character indSn 1 multiplicity free? Sk ×Sn−k Sk ×Sn−k

Deﬁnition A subgroup H 6 G is called a Gelfand subgroup if the induced G trivial character indH 1H is multiplicity free. Remark G The induced character indH 1H can be viewed as the character of the left G-module of C-valued functions on G/H,

C[G/H] = {f : G/H → C}.

To really understand an algebra, one needs to understand its structure constants. In particular, we want to know when the Littlewood-Richardson numbers are as simple as 0 or 1. Of course, as a function of λ,µ, and τ, this question is already too general, so, we ask a more speciﬁc question: Deﬁnition A subgroup H 6 G is called a Gelfand subgroup if the induced G trivial character indH 1H is multiplicity free. Remark G The induced character indH 1H can be viewed as the character of the left G-module of C-valued functions on G/H,

C[G/H] = {f : G/H → C}.

τ • For which k ∈ N, do we have c(k),(n−k) ∈ {0,1} for every τ ` n? Equivalently, when is the induced trivial character indSn 1 multiplicity free? Sk ×Sn−k Sk ×Sn−k To really understand an algebra, one needs to understand its structure constants. In particular, we want to know when the Littlewood-Richardson numbers are as simple as 0 or 1. Of course, as a function of λ,µ, and τ, this question is already too general, so, we ask a more speciﬁc question:

τ • For which k ∈ N, do we have c(k),(n−k) ∈ {0,1} for every τ ` n? Equivalently, when is the induced trivial character indSn 1 multiplicity free? Sk ×Sn−k Sk ×Sn−k

C[G/H] = {f : G/H → C}. How many lines are there that intersect four lines in 3-space?

L1 L2

Littlewood-Richardson numbers appear in the solutions of some perceivable geometric problems. L1 L2

Littlewood-Richardson numbers appear in the solutions of some perceivable geometric problems.

How many lines are there that intersect four lines in 3-space? Littlewood-Richardson numbers appear in the solutions of some perceivable geometric problems.

How many lines are there that intersect four lines in 3-space?

L1 L2

L4 Sometimes there are no solutions! Sometimes there are no solutions! • •

There can be one solution! • •

There can be one solution! •

There can be one solution!

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There can be one solution!

• There can be one solution!

• • There can be one solution!

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There can be two solutions! • •

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There can be two solutions! •

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There can be two solutions!

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There can be two solutions!

• • There can be two solutions!

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• • There can be two solutions!

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• • There can be two solutions!

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There can be inﬁnitely many solutions! There can be inﬁnitely many solutions!

• There can be inﬁnitely many solutions!

• Anarchism?

3 3 No! If we consider ourselves living inside P instead of C , 3 all of that disorder disappears.. Here, P is the set of all 4 1-dimensional vector subspaces of C .

At ﬁrst, it appears as if there is no uniform answer. 3 3 No! If we consider ourselves living inside P instead of C , 3 all of that disorder disappears.. Here, P is the set of all 4 1-dimensional vector subspaces of C .

At ﬁrst, it appears as if there is no uniform answer.

Anarchism? 3 3 If we consider ourselves living inside P instead of C , 3 all of that disorder disappears.. Here, P is the set of all 4 1-dimensional vector subspaces of C .

At ﬁrst, it appears as if there is no uniform answer.

Anarchism?

No! 3 Here, P is the set of all 4 1-dimensional vector subspaces of C .

At ﬁrst, it appears as if there is no uniform answer.

Anarchism?

3 3 No! If we consider ourselves living inside P instead of C , all of that disorder disappears.. At ﬁrst, it appears as if there is no uniform answer.

Anarchism?

3 3 No! If we consider ourselves living inside P instead of C , 3 all of that disorder disappears.. Here, P is the set of all 4 1-dimensional vector subspaces of C . 3 3 P C = {(1,u1,u2,u3): ui ∈ C} (1,0,0,0)

x1 (0,1,v2,v3) 2 3 3 P = P \ C x2

4 C \{0}

x3 x1

x2 3 P

(0,1,v2,v3) 2 3 3 P = P \ C

4 C \{0}

3 C = {(1,u1,u2,u3): ui ∈ C} (1,0,0,0)

x3 x1

x2 3 P

(0,1,v2,v3) 2 3 3 P = P \ C

4 C \{0}

3 C = {(1,u1,u2,u3): ui ∈ C} (1,0,0,0)

x3 x1

x2 (0,1,v2,v3) 2 3 3 P = P \ C

4 C \{0}

3 3 P C = {(1,u1,u2,u3): ui ∈ C} (1,0,0,0)

x2 (0,1,v2,v3)

4 C \{0}

3 3 P C = {(1,u1,u2,u3): ui ∈ C} (1,0,0,0)

x1 2 3 3 P = P \ C x2 4 C \{0}

3 3 P C = {(1,u1,u2,u3): ui ∈ C} (1,0,0,0)

x1 (0,1,v2,v3) 2 3 3 P = P \ C x2 4 C \{0}

3 3 P C = {(1,u1,u2,u3): ui ∈ C} (1,0,0,0)

x1 (0,1,v2,v3) 2 3 3 P = P \ C x2 2 2 P C = {(0,1,v2,v3): vi ∈ C}

(0,0,d,e)

1 = 2 \ 2 x3 P P C

3 C \{0}

x3 (0,0,d,e)

1 2 2 P = P \ C

3 C \{0} 2 2 P C = {(0,1,v2,v3): vi ∈ C}

x3 1 2 2 P = P \ C

3 C \{0} 2 2 P C = {(0,1,v2,v3): vi ∈ C}

(0,0,d,e)

x3 3 C \{0} 2 2 P C = {(0,1,v2,v3): vi ∈ C}

(0,0,d,e)

1 = 2 \ 2 x3 P P C n n+1 I Identify C with {(1,u1,...,un): ui ∈ C }. The map n n ι : C ,→ P deﬁned by (1,u1,...,un) 7→ `(1,u1,...,un) is an open n n embedding. Then, the lines in C are lines in P . I There is a natural chain of closed imbeddings,

1 2 n {pt} ,→ P ,→ P ,→ ··· ,→ P .

Furthermore, this chain of subvarieties give a natural cellular decomposition,

n n n−1 n n−1 1 P = C t P = C t C t ··· t C t {pt}.

Summary: n n+1 I P is the set of 1-dimensional subspaces of C .

n+1 n π : C \{0} −→ P

(u0,...,un) 7−→ `(u0,...,un) I There is a natural chain of closed imbeddings,

1 2 n {pt} ,→ P ,→ P ,→ ··· ,→ P .

Furthermore, this chain of subvarieties give a natural cellular decomposition,

n n n−1 n n−1 1 P = C t P = C t C t ··· t C t {pt}.

Summary: n n+1 I P is the set of 1-dimensional subspaces of C .

n+1 n π : C \{0} −→ P

(u0,...,un) 7−→ `(u0,...,un)

n n+1 I Identify C with {(1,u1,...,un): ui ∈ C }. The map n n ι : C ,→ P deﬁned by (1,u1,...,un) 7→ `(1,u1,...,un) is an open n n embedding. Then, the lines in C are lines in P . Furthermore, this chain of subvarieties give a natural cellular decomposition,

n n n−1 n n−1 1 P = C t P = C t C t ··· t C t {pt}.

Summary: n n+1 I P is the set of 1-dimensional subspaces of C .

n+1 n π : C \{0} −→ P

(u0,...,un) 7−→ `(u0,...,un)

n n+1 I Identify C with {(1,u1,...,un): ui ∈ C }. The map n n ι : C ,→ P deﬁned by (1,u1,...,un) 7→ `(1,u1,...,un) is an open n n embedding. Then, the lines in C are lines in P . I There is a natural chain of closed imbeddings,

1 2 n {pt} ,→ P ,→ P ,→ ··· ,→ P . Summary: n n+1 I P is the set of 1-dimensional subspaces of C .

n+1 n π : C \{0} −→ P

(u0,...,un) 7−→ `(u0,...,un)

1 2 n {pt} ,→ P ,→ P ,→ ··· ,→ P .

Furthermore, this chain of subvarieties give a natural cellular decomposition,

n n n−1 n n−1 1 P = C t P = C t C t ··· t C t {pt}. An important generalization: There is a natural generalization n of P ; it is called the Grassmann variety of d-dimensional m m subspaces of C , denoted by Gr(d,C ). n n+1 I P = Gr(1,C ). m m+1 I Gr(d,P ) := Gr(d + 1,C ). m I Gr(d,C ) has a natural closed imbedding into a large dimensional projective space. For example, we know that

4 5 Gr(2,C ) ,→ P .

In other words, all Grassmann varieties are projective varieties. 1 2 n m I Similarly to {pt} ,→ P ,→ P ,→ ··· ,→ P , Gr(d,P ) has a natural poset of imbeddings of some subspecial varieties. The Schubert variety corresponding to A• is deﬁned by

n X(A•) := {W ∈ Gr(k,P ) : dim(W ∩ Ai ) ≥ i ∀i ∈ {0,...,k}}.

n Let A• = (A0,...,Ak ) be a reference ﬂag in P ,

n A0 ( A1 ( ··· ( Ak ⊆ P .

Assume that dim(∅) = −∞. n Note: If B• : B0 ( B1 ( ··· ( Bk ⊆ P is another reference ﬂag such that dimBi = dimAi for i ∈ {0,...,k}, then there exists an n automorphism of Gr(k,P ) that maps X(A•) isomorphically onto X(B•).

n Let A• = (A0,...,Ak ) be a reference ﬂag in P ,

n A0 ( A1 ( ··· ( Ak ⊆ P .

Assume that dim(∅) = −∞. The Schubert variety corresponding to A• is deﬁned by

n X(A•) := {W ∈ Gr(k,P ) : dim(W ∩ Ai ) ≥ i ∀i ∈ {0,...,k}}. n Let A• = (A0,...,Ak ) be a reference ﬂag in P ,

n A0 ( A1 ( ··· ( Ak ⊆ P .

Assume that dim(∅) = −∞. The Schubert variety corresponding to A• is deﬁned by

n X(A•) := {W ∈ Gr(k,P ) : dim(W ∩ Ai ) ≥ i ∀i ∈ {0,...,k}}.

n Note: If B• : B0 ( B1 ( ··· ( Bk ⊆ P is another reference ﬂag such that dimBi = dimAi for i ∈ {0,...,k}, then there exists an n automorphism of Gr(k,P ) that maps X(A•) isomorphically onto X(B•). Let 3 A• = (A0,A1) = (`,P ). Then 3 X(A•) = {L ∈ Gr(1,P ): ` ∩ L 6= ∅}. Recall that we have a closed imbedding (the Plücker embedding), 3 5 p : Gr(1,P ) ,→ P . 5 1. The image of p is a quadratic hypersurface in P . 5 2. There is a hyperplane H in P , uniquely determined by `, such that 3 p(X(A•)) = H ∩ Gr(1,P ). We will apply this observation to our situation.

3 It is time to go back to our problem on the given four lines in P . Recall that we have a closed imbedding (the Plücker embedding), 3 5 p : Gr(1,P ) ,→ P . 5 1. The image of p is a quadratic hypersurface in P . 5 2. There is a hyperplane H in P , uniquely determined by `, such that 3 p(X(A•)) = H ∩ Gr(1,P ). We will apply this observation to our situation.

3 It is time to go back to our problem on the given four lines in P . Let 3 A• = (A0,A1) = (`,P ). Then 3 X(A•) = {L ∈ Gr(1,P ): ` ∩ L 6= ∅}. 5 1. The image of p is a quadratic hypersurface in P . 5 2. There is a hyperplane H in P , uniquely determined by `, such that 3 p(X(A•)) = H ∩ Gr(1,P ). We will apply this observation to our situation.

3 It is time to go back to our problem on the given four lines in P . Let 3 A• = (A0,A1) = (`,P ). Then 3 X(A•) = {L ∈ Gr(1,P ): ` ∩ L 6= ∅}. Recall that we have a closed imbedding (the Plücker embedding), 3 5 p : Gr(1,P ) ,→ P . 5 1. The image of p is a quadratic hypersurface in P . 5 2. There is a hyperplane H in P , uniquely determined by `, such that 3 p(X(A•)) = H ∩ Gr(1,P ). 3 It is time to go back to our problem on the given four lines in P . Let 3 A• = (A0,A1) = (`,P ). Then 3 X(A•) = {L ∈ Gr(1,P ): ` ∩ L 6= ∅}. Recall that we have a closed imbedding (the Plücker embedding), 3 5 p : Gr(1,P ) ,→ P . 5 1. The image of p is a quadratic hypersurface in P . 5 2. There is a hyperplane H in P , uniquely determined by `, such that 3 p(X(A•)) = H ∩ Gr(1,P ). We will apply this observation to our situation. Then the number of lines that intersect all four Li is given by the equality

4 3 3 | ∩i=1 X(Li ⊂ P )| = |H1 ∩ H2 ∩ H3 ∩ H4 ∩ Gr(1,P )|.

5 Let Hi (i ∈ {1,2,3,4}) denote the hyperplanes in P that 3 correspond to our original four lines Li (i ∈ {1,2,3,4}) in P . 4 I If Hi ’s are linearly independent, then ∩i=1Hi is a line. In this 4 3 case, we have | ∩i=1 Hi ∩ Gr(1,P )| = 2 since in a projective space a line intersects a quadratic hypersurface at exactly two points, or at one point with multiplicity 2. 4 I If Hi ’s are linearly dependent, then dim∩i=1Hi ≥ 2, hence, 4 3 there are inﬁnite number of points in | ∩i=1 Hi ∩ Gr(1,P )|.

5 Let Hi (i ∈ {1,2,3,4}) denote the hyperplanes in P that 3 correspond to our original four lines Li (i ∈ {1,2,3,4}) in P . Then the number of lines that intersect all four Li is given by the equality

4 3 3 | ∩i=1 X(Li ⊂ P )| = |H1 ∩ H2 ∩ H3 ∩ H4 ∩ Gr(1,P )|. 4 I If Hi ’s are linearly dependent, then dim∩i=1Hi ≥ 2, hence, 4 3 there are inﬁnite number of points in | ∩i=1 Hi ∩ Gr(1,P )|.

4 3 3 | ∩i=1 X(Li ⊂ P )| = |H1 ∩ H2 ∩ H3 ∩ H4 ∩ Gr(1,P )|.

4 I If Hi ’s are linearly independent, then ∩i=1Hi is a line. In this 4 3 case, we have | ∩i=1 Hi ∩ Gr(1,P )| = 2 since in a projective space a line intersects a quadratic hypersurface at exactly two points, or at one point with multiplicity 2. 5 Let Hi (i ∈ {1,2,3,4}) denote the hyperplanes in P that 3 correspond to our original four lines Li (i ∈ {1,2,3,4}) in P . Then the number of lines that intersect all four Li is given by the equality

4 3 3 | ∩i=1 X(Li ⊂ P )| = |H1 ∩ H2 ∩ H3 ∩ H4 ∩ Gr(1,P )|.

4 I If Hi ’s are linearly independent, then ∩i=1Hi is a line. In this 4 3 case, we have | ∩i=1 Hi ∩ Gr(1,P )| = 2 since in a projective space a line intersects a quadratic hypersurface at exactly two points, or at one point with multiplicity 2. 4 I If Hi ’s are linearly dependent, then dim∩i=1Hi ≥ 2, hence, 4 3 there are inﬁnite number of points in | ∩i=1 Hi ∩ Gr(1,P )|. Let A• = (A0,...,Ad ) be a reference ﬂag, and denote the corresponding Schubert variety by X(a0,...,ad ), where ai = dimAi for i ∈ {0,...,d}. Theorem (The Schubert basis theorem.) n+1 2p If Y denotes Gr(d + 1,C ), then H (Y ,Z) is a free abelian group. Furthermore, we have 1. The duals of the homology classes corresponding to the Schubert varieties X(a0,...,ad ), where Pd 2p [(d +1)(n −d)− i=0(ai −i)] = p, form a basis of H (Y ,Z). 2. The odd dimensional cohomology groups of Y are trivial. Moreover, in the Schubert basis, the structure constants of the Ln 2p algebra p=0 H (Y ,Z) are given by the Littlewood-Richardson numbers.

I want to brieﬂy explain the Schubert Basis Theorem. Theorem (The Schubert basis theorem.) n+1 2p If Y denotes Gr(d + 1,C ), then H (Y ,Z) is a free abelian group. Furthermore, we have 1. The duals of the homology classes corresponding to the Schubert varieties X(a0,...,ad ), where Pd 2p [(d +1)(n −d)− i=0(ai −i)] = p, form a basis of H (Y ,Z). 2. The odd dimensional cohomology groups of Y are trivial. Moreover, in the Schubert basis, the structure constants of the Ln 2p algebra p=0 H (Y ,Z) are given by the Littlewood-Richardson numbers.

I want to briefly explain the Schubert Basis Theorem. Let A• = (A0,...,Ad ) be a reference flag, and denote the corresponding Schubert variety by X(a0,...,ad ), where ai = dimAi for i ∈ {0,...,d}. I want to briefly explain the Schubert Basis Theorem. Let A• = (A0,...,Ad ) be a reference flag, and denote the corresponding Schubert variety by X(a0,...,ad ), where ai = dimAi for i ∈ {0,...,d}. Theorem (The Schubert basis theorem.) n+1 2p If Y denotes Gr(d + 1,C ), then H (Y ,Z) is a free abelian group. Furthermore, we have 1. The duals of the homology classes corresponding to the Schubert varieties X(a0,...,ad ), where Pd 2p [(d +1)(n −d)− i=0(ai −i)] = p, form a basis of H (Y ,Z). 2. The odd dimensional cohomology groups of Y are trivial. Moreover, in the Schubert basis, the structure constants of the Ln 2p algebra p=0 H (Y ,Z) are given by the Littlewood-Richardson numbers. Lemma If d ∈ {1,...,n}, then there is a natural isomorphism

n+1 φ : Gr(d,C ) → GLn+1/Pd .

n+1 Furthermore, the Schubert varieties in Gr(d,C ) correspond to the closures of the orbits of Bn+1 in GLn+1/Pd .

Let d ∈ {1,...,n}. The d-th standard maximal parabolic subgroup of GLn+1 is (" # ) A11 A12 Pd := : A11 ∈ GLd , A22 ∈ GLn+1−d ⊂ GLn+1. 0 A22

The standard Borel subgroup of GLn+1, denoted by Bn+1, is the subgroup that consists of upper triangular matrices in GLn+1. Let d ∈ {1,...,n}. The d-th standard maximal parabolic subgroup of GLn+1 is (" # ) A11 A12 Pd := : A11 ∈ GLd , A22 ∈ GLn+1−d ⊂ GLn+1. 0 A22

The standard Borel subgroup of GLn+1, denoted by Bn+1, is the subgroup that consists of upper triangular matrices in GLn+1. Lemma If d ∈ {1,...,n}, then there is a natural isomorphism

n+1 φ : Gr(d,C ) → GLn+1/Pd .

n+1 Furthermore, the Schubert varieties in Gr(d,C ) correspond to the closures of the orbits of Bn+1 in GLn+1/Pd . Associated with each composition of n + 1 is a subgroup of the form A A ··· A    11 12 A1r   0 A ··· A  r r   22 A2r  Y Y  Pv :=  . . .  : Aii ∈ GLv ⊂ GLn+1.  . . .. .  i  . . . .  i=1 i=1     0 0 ... Arr 

Any subgroup in GLn+1 that is conjugate to Pv will be called a parabolic subgorup.

A (weak) composition of n + 1 is a sequence of (nonnegative) Pr positive integers v := (v1,...,vr ) such that i=1 vi = n + 1. For example, v = (1,3,1) is a composition of 5. Any subgroup in GLn+1 that is conjugate to Pv will be called a parabolic subgorup.

A (weak) composition of n + 1 is a sequence of (nonnegative) Pr positive integers v := (v1,...,vr ) such that i=1 vi = n + 1. For example, v = (1,3,1) is a composition of 5.

Associated with each composition of n + 1 is a subgroup of the form A A ··· A    11 12 A1r   0 A ··· A  r r   22 A2r  Y Y  Pv :=  . . .  : Aii ∈ GLv ⊂ GLn+1.  . . .. .  i  . . . .  i=1 i=1     0 0 ... Arr  A (weak) composition of n + 1 is a sequence of (nonnegative) Pr positive integers v := (v1,...,vr ) such that i=1 vi = n + 1. For example, v = (1,3,1) is a composition of 5.

Any subgroup in GLn+1 that is conjugate to Pv will be called a parabolic subgorup. A coset space of the form GLn+1/H is a projective variety if and only if H is a parabolic subgroup.

A projective homogeneous space is called a partial ﬂag variety. In particular, GLn+1/Bn+1 is called the (full) ﬂag variety of GLn+1.

Just as in the case of Grassmann varieties, the cohomology ring of a (partial) ﬂag variety is completely determined by the Zariski closures of the orbits of Bn+1. These subvarieties are also called the Schubert varieties.

Clearly, if v = (1,...,1), then Pv = B. The parabolic subgroups containing Bn+1 are called standard with respect to Bn+1. A projective homogeneous space is called a partial ﬂag variety. In particular, GLn+1/Bn+1 is called the (full) ﬂag variety of GLn+1.

Clearly, if v = (1,...,1), then Pv = B. The parabolic subgroups containing Bn+1 are called standard with respect to Bn+1.

A coset space of the form GLn+1/H is a projective variety if and only if H is a parabolic subgroup. Just as in the case of Grassmann varieties, the cohomology ring of a (partial) ﬂag variety is completely determined by the Zariski closures of the orbits of Bn+1. These subvarieties are also called the Schubert varieties.

Clearly, if v = (1,...,1), then Pv = B. The parabolic subgroups containing Bn+1 are called standard with respect to Bn+1.

A coset space of the form GLn+1/H is a projective variety if and only if H is a parabolic subgroup.

A projective homogeneous space is called a partial ﬂag variety. In particular, GLn+1/Bn+1 is called the (full) ﬂag variety of GLn+1. Clearly, if v = (1,...,1), then Pv = B. The parabolic subgroups containing Bn+1 are called standard with respect to Bn+1.

A coset space of the form GLn+1/H is a projective variety if and only if H is a parabolic subgroup.

A projective homogeneous space is called a partial ﬂag variety. In particular, GLn+1/Bn+1 is called the (full) ﬂag variety of GLn+1.

Bn+1 GLn+1 × C −→ GLn+1/Bn+1. (1)

Theorem (Borel-Weil-Bott Theorem) The space of global sections of (1) is isomorphic to the dual of the irreducible rational representation of GLn+1 corresponding to λ.

Let λ be a character of Bn+1. There is a natural action of Bn+1 on the space GLn+1 × C:

−1 b · (g,z) = (gb ,λ(b)z)(b ∈ Bn+1,g ∈ GLn+1, z ∈ C). Theorem (Borel-Weil-Bott Theorem) The space of global sections of (1) is isomorphic to the dual of the irreducible rational representation of GLn+1 corresponding to λ.

Let λ be a character of Bn+1. There is a natural action of Bn+1 on the space GLn+1 × C:

−1 b · (g,z) = (gb ,λ(b)z)(b ∈ Bn+1,g ∈ GLn+1, z ∈ C).

The quotient by this action deﬁnes a line bundle on the ﬂag variety,

Bn+1 GLn+1 × C −→ GLn+1/Bn+1. (1) Let λ be a character of Bn+1. There is a natural action of Bn+1 on the space GLn+1 × C:

−1 b · (g,z) = (gb ,λ(b)z)(b ∈ Bn+1,g ∈ GLn+1, z ∈ C).

The quotient by this action deﬁnes a line bundle on the ﬂag variety,

Bn+1 GLn+1 × C −→ GLn+1/Bn+1. (1)

Theorem (Borel-Weil-Bott Theorem) The space of global sections of (1) is isomorphic to the dual of the irreducible rational representation of GLn+1 corresponding to λ. ∗ Every irreducible rational representation of GLn+1 × C appears exactly once in the total coordinate ring of GLn+1/Bn+1.

Deﬁnition Let G be a connected reductive group, and let V be rational G-module. V is said to be a multiplicity free G-module if every irreducible G-module appears at most once in V .

A related representation theoretic fact is the following: There is a natural projective embedding

n Y i n+1 GLn+1/Bn+1 ,→ P(∧ C ). i=1 Passing to the cone over the image of this closed embedding, we get the total coordinate ring of GLn+1/Bn+1. A related representation theoretic fact is the following: There is a natural projective embedding

n Y i n+1 GLn+1/Bn+1 ,→ P(∧ C ). i=1 Passing to the cone over the image of this closed embedding, we get the total coordinate ring of GLn+1/Bn+1.

∗ Every irreducible rational representation of GLn+1 × C appears exactly once in the total coordinate ring of GLn+1/Bn+1.

Deﬁnition Let G be a connected reductive group, and let V be rational G-module. V is said to be a multiplicity free G-module if every irreducible G-module appears at most once in V . PART 2

SPHERICAL VARIETIES Theorem (Kimelfeld-Vinberg) Let G be a connected reductive group, and let X be a quasi-aﬃne homogeneous space of G. Then a Borel subgroup of G has an open orbit in X if and only if the coordinate ring C[X] is a multiplicity free G-module.

Deﬁnition An irreducible normal G-variety X is called a spherical G-variety if a Borel subgroup of G has an open orbit in X.

Multiplicity-freeness and Borel subgroups.

It turns out that the multiplicity free property is closely related to the orbits of a Borel subgroup. Deﬁnition An irreducible normal G-variety X is called a spherical G-variety if a Borel subgroup of G has an open orbit in X.

Multiplicity-freeness and Borel subgroups.

It turns out that the multiplicity free property is closely related to the orbits of a Borel subgroup.

Theorem (Kimelfeld-Vinberg) Let G be a connected reductive group, and let X be a quasi-aﬃne homogeneous space of G. Then a Borel subgroup of G has an open orbit in X if and only if the coordinate ring C[X] is a multiplicity free G-module. Multiplicity-freeness and Borel subgroups.

It turns out that the multiplicity free property is closely related to the orbits of a Borel subgroup.

Deﬁnition An irreducible normal G-variety X is called a spherical G-variety if a Borel subgroup of G has an open orbit in X. Examples of spherical varieties include I all partial ﬂag varieties, I normal toric varieties, I normal reductive monoids, I symmetric spaces as well as their equivariant embeddings.

Spherical varieties.

The following theorem is proved by Brion in arbitrary characteristic, and around the same time by Vinberg in characteristic 0. Theorem A normal G-variety X is spherical iff the number of orbits of a Borel subgroup is finite. Examples of spherical varieties include I all partial flag varieties, I normal toric varieties, I normal reductive monoids, I symmetric spaces as well as their equivariant embeddings. Recall that a Schubert variety is a Bn+1-orbit closure in GLn+1/Bn+1. So, it is not stable under GLn+1. But we can consider the maximal reductive subgroups of the stabilizing subgroup.

Let X be a G-variety, and let Y ⊆ X be a B-stable irreducible subvariety. Then StabG (Y ) is a parabolic subgroup of X.

What About Spherical Schubert Varieties?

The ﬁrst question to answer here is “spherical with respect to which group action?” Let X be a G-variety, and let Y ⊆ X be a B-stable irreducible subvariety. Then StabG (Y ) is a parabolic subgroup of X.

What About Spherical Schubert Varieties?

The ﬁrst question to answer here is “spherical with respect to which group action?”

Recall that a Schubert variety is a Bn+1-orbit closure in GLn+1/Bn+1. So, it is not stable under GLn+1. But we can consider the maximal reductive subgroups of the stabilizing subgroup. What About Spherical Schubert Varieties?

The ﬁrst question to answer here is “spherical with respect to which group action?”

Recall that a Schubert variety is a Bn+1-orbit closure in GLn+1/Bn+1. So, it is not stable under GLn+1. But we can consider the maximal reductive subgroups of the stabilizing subgroup.

Let X be a G-variety, and let Y ⊆ X be a B-stable irreducible subvariety. Then StabG (Y ) is a parabolic subgroup of X. In this notation, for every Schubert variety Y in GLn+1/Bn+1, there exists a subset I = I(Y ) ⊂ S such that the stabilizing parabolic subgroup of Y is of the form

PI := Bn+1WI Bn+1.

Stabilizers of Schubert Varieties

Let S denote the set of simple transpositions

S = {s1 = (1,2),s2 = (2,3),...,sn = (n,n + 1)}.

The elements of S generate Sn+1. For I ⊆ S, let WI denote the subgroup of W = Sn+1 that is generated by the elements of I. Stabilizers of Schubert Varieties

Let S denote the set of simple transpositions

S = {s1 = (1,2),s2 = (2,3),...,sn = (n,n + 1)}.

The elements of S generate Sn+1. For I ⊆ S, let WI denote the subgroup of W = Sn+1 that is generated by the elements of I.

In this notation, for every Schubert variety Y in GLn+1/Bn+1, there exists a subset I = I(Y ) ⊂ S such that the stabilizing parabolic subgroup of Y is of the form

PI := Bn+1WI Bn+1. The parabolic subgroup corresponding I ⊆ S.

Example

Let I be the subset {s1,s2,s4,s5,s6} in S8. Then the corresponding parabolic subgroup is given by

∗∗∗∗∗∗∗∗    ∗∗∗∗∗∗∗∗      ∗∗∗∗∗∗∗∗    0 0 0 ∗∗∗∗∗  P(3,4,1) :=   0 0 0 ∗∗∗∗∗    0 0 0 ∗∗∗∗∗    0 0 0 ∗∗∗∗∗     0 0 0 0 0 0 0 ∗  First, note that the Schubert varieties in GLn+1/Bn+1 are parametrized by the elements of the symmetric group. Theorem (Bruhat-Chevalley decomposition)

Let W denote the symmetric group Sn+1, Then F GLn+1 = w∈W BwB. In particular, every Schubert variety Y in GLn+1/Bn+1 is of the form Y = Bn+1wBn+1/Bn+1 for some permutation w ∈ Sn+1.

Stabilizers of Schubert Varieties

Given a Schubert variety Y in GLn+1/Bn+1, there is a precise rule for determining the corresponding set of simple roots I = I(Y ). Theorem (Bruhat-Chevalley decomposition)

Let W denote the symmetric group Sn+1, Then F GLn+1 = w∈W BwB. In particular, every Schubert variety Y in GLn+1/Bn+1 is of the form Y = Bn+1wBn+1/Bn+1 for some permutation w ∈ Sn+1.

Stabilizers of Schubert Varieties

Given a Schubert variety Y in GLn+1/Bn+1, there is a precise rule for determining the corresponding set of simple roots I = I(Y ).

First, note that the Schubert varieties in GLn+1/Bn+1 are parametrized by the elements of the symmetric group. Stabilizers of Schubert Varieties

Given a Schubert variety Y in GLn+1/Bn+1, there is a precise rule for determining the corresponding set of simple roots I = I(Y ).

First, note that the Schubert varieties in GLn+1/Bn+1 are parametrized by the elements of the symmetric group. Theorem (Bruhat-Chevalley decomposition)

Let W denote the symmetric group Sn+1, Then F GLn+1 = w∈W BwB. In particular, every Schubert variety Y in GLn+1/Bn+1 is of the form Y = Bn+1wBn+1/Bn+1 for some permutation w ∈ Sn+1. Stabilizers of Schubert Varieties

If Y is the Schubert variety corresponding to the permutation

w ∈ Sn+1 as in the theorem, then let us write XwBn+1 instead of Y .

More generally, if Y is Schubert variety corresponding to the Weyl group element w ∈ W in G/B, then we will write Y = XwB.

−1 The stabilizer of XwB in G is found by the descent set of w . The rank function (W ,≤) is given by

` : W −→ N

w 7−→ dimXwBn+1 .

The interval between two elements v,w ∈ W will be denoted by [v,w]BC .

The strong order.

The Bruhat-Chevalley order on W (Sn+1) is deﬁned by

v ≤ w ⇐⇒ XvBn+1 ⊆ XwBn+1 .

Note that (W ,≤) is a graded poset. The interval between two elements v,w ∈ W will be denoted by [v,w]BC .

The strong order.

The Bruhat-Chevalley order on W (Sn+1) is deﬁned by

v ≤ w ⇐⇒ XvBn+1 ⊆ XwBn+1 .

Note that (W ,≤) is a graded poset. The rank function (W ,≤) is given by

` : W −→ N

w 7−→ dimXwBn+1 . The strong order.

The Bruhat-Chevalley order on W (Sn+1) is deﬁned by

v ≤ w ⇐⇒ XvBn+1 ⊆ XwBn+1 .

Note that (W ,≤) is a graded poset. The rank function (W ,≤) is given by

` : W −→ N

w 7−→ dimXwBn+1 .

The interval between two elements v,w ∈ W will be denoted by [v,w]BC . The interval between two elements v,w ∈ W will be denoted by [v,w]R .

The (right) weak order.

Let v and w be two elements from W . The right weak order on W , denoted by ≤R , is deﬁned by the transitive closure of the following relation:

v lR w ⇐⇒ ∃ s ∈ S such that vs = w and `(v) + 1 = `(w).

Remark A permutation t which is conjugate to a simple transposition is called a reﬂection (or a transposition). The covering relations of the strong order are given by

v lw ⇐⇒ ∃ a reﬂection t such that vt = w and `(v) + 1 = `(w). The interval between two elements v,w ∈ W will be denoted by [v,w]R .

The (right) weak order.

Let v and w be two elements from W . The right weak order on W , denoted by ≤R , is deﬁned by the transitive closure of the following relation:

v lR w ⇐⇒ ∃ s ∈ S such that vs = w and `(v) + 1 = `(w).

Remark A permutation t which is conjugate to a simple transposition is called a reﬂection (or a transposition). The covering relations of the strong order are given by

v lw ⇐⇒ ∃ a reﬂection t such that vt = w and `(v) + 1 = `(w). The (right) weak order.

Let v and w be two elements from W . The right weak order on W , denoted by ≤R , is deﬁned by the transitive closure of the following relation:

v lR w ⇐⇒ ∃ s ∈ S such that vs = w and `(v) + 1 = `(w).

Remark A permutation t which is conjugate to a simple transposition is called a reﬂection (or a transposition). The covering relations of the strong order are given by

v lw ⇐⇒ ∃ a reﬂection t such that vt = w and `(v) + 1 = `(w).

The interval between two elements v,w ∈ W will be denoted by [v,w]R . 4321 {s1,s2,s3}

4312 4231 3421 {s2,s3} {s1,s3} {s1,s2}

4132 4213 3412 2431 3241 {s2,s3} {s1,s3} {s2} {s1,s3} {s1,s2}

1432 4123 2413 3142 3214 2341 {s2,s3} {s3} {s1,s3} {s2} {s1,s2} {s1}

1423 1342 2143 3124 2314 {s3} {s2} {s1,s3} {s2} {s1}

1243 1324 2134 {s3} {s2} {s1}

1234 ∅ ∗∗∗ 0 0 0 0 0   ∗∗∗ 0 0 0 0 0     ∗∗∗ 0 0 0 0 0   0 0 0 ∗∗∗∗ 0 ∼ L(3,4,1) :=   = GL3 × GL4 × GL1. 0 0 0 ∗∗∗∗ 0   0 0 0 ∗∗∗∗ 0   0 0 0 ∗∗∗∗ 0    0 0 0 0 0 0 0 ∗ 

The standard Levi factor of PI.

The standard Levi factor of PI is a maximal reductive subgroup in PI , denoted by LI such that

PI = LI n Ru(PI ),

where Ru(PI ) is the unipotent radical of PI . For example, The standard Levi factor of PI.

The standard Levi factor of PI is a maximal reductive subgroup in PI , denoted by LI such that

PI = LI n Ru(PI ),

where Ru(PI ) is the unipotent radical of PI . For example,

∗∗∗ 0 0 0 0 0   ∗∗∗ 0 0 0 0 0     ∗∗∗ 0 0 0 0 0   0 0 0 ∗∗∗∗ 0 ∼ L(3,4,1) :=   = GL3 × GL4 × GL1. 0 0 0 ∗∗∗∗ 0   0 0 0 ∗∗∗∗ 0   0 0 0 ∗∗∗∗ 0    0 0 0 0 0 0 0 ∗  Theorem Let G be one of the following algebraic groups: GL2,GL3,GL4, or SO5. Then every Schubert variety XwB in G/B, where B is an appropriate Borel subgroup, is a spherical LI(w)-variety.

Some recent results

Theorem Let w ∈ W . If the lower intervals [id,w]R and [id,w]BC have the same underlying set of elements, then XwB is a smooth spherical LI(w)-variety. Some recent results

Theorem Let w ∈ W . If the lower intervals [id,w]R and [id,w]BC have the same underlying set of elements, then XwB is a smooth spherical LI(w)-variety. Theorem Let G be one of the following algebraic groups: GL2,GL3,GL4, or SO5. Then every Schubert variety XwB in G/B, where B is an appropriate Borel subgroup, is a spherical LI(w)-variety. In G2.

s2s1s2s1s2s1 = s1s2s1s2s1s2

s1s2s1s2s1 s2s1s2s1s2

s1s2s1s2 s2s1s2s1

s1s2s1 s2s1s2

s1s2 s2s1

s1 s2

This is the Bruhat-Chevalley order on the Weyl group of G2. The right weak order is indicated by the solid lines. Corollary Every smooth Schubert variety in a partial ﬂag variety of a semisimple algebraic group of rank 2 is spherical.

Some recent results

Theorem Let XwB be a Schubert variety in G/B, where G is of type G2 Then, XwB is a spherical LI(w)-variety if and only if w ∈ {id,s1,s2,s1s2,s2s1,s1s2s1,s2s1s2,s2s1s2s2s1s2}. Some recent results

Theorem Let XwB be a Schubert variety in G/B, where G is of type G2 Then, XwB is a spherical LI(w)-variety if and only if w ∈ {id,s1,s2,s1s2,s2s1,s1s2s1,s2s1s2,s2s1s2s2s1s2}. Corollary Every smooth Schubert variety in a partial ﬂag variety of a semisimple algebraic group of rank 2 is spherical. A conjecture of Hodges and Yong.

Geometric proofs of the above theorems can be found in my paper (with Hodges)

https://arxiv.org/abs/1803.05515

This year, R. Hodges and A. Yong made some conjectures regarding the sphericality of Schubert varieties by introducing the “spherical elements” in Coxeter groups. Their article can be reached from here:

https://arxiv.org/abs/2007.09238 G : (complex) connected reductive group B : a Borel subgroup of G T : a maximal torus s.t. T ⊂ B X(T )+ : the monoid of dominant weights.

X(T )+ parametrizes the irreducible rational representations of G. For G = GLn, we have

X(T )+ ←→ integer partitions.

If λ is a partition, Borel-Weil-Bott theorem gives us a line bundle on G/B, denoted by L(λ), whose space of global sections is an irreducible G-module.