Problems related to Invariant theory of
torus and finite groups
Santosha Kumar Pattanayak
under the guidance of Prof. S. S. Kannan
Ph.D. Defense
June 24, 2011
Outline of the Thesis
Motivation
Projective Normality of finite group quotients
Torus action on homogeneous spaces
Bibliography
Motivation
K - an algebraically closed field.
G = SLn - the special linear group.
T - a maximal torus of G .
B- a Borel subgroup of G containing T .
N- the normalizer of T in G .
W = N/T - the Weyl group.
M2×n - vector space of 2 × n matrices.
Motivation
P(M2×n )
G2,n = SL2 \\P(M2×n )




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P(M2×n )//N = P(Vn )
G2,n //N = SL2 \\P(Vn )
where Vn is the vector space of homogeneous polynomials of degree n in
two variables.
Motivation
More generally one has the following isomorphism;
T \\(G /P)ss (Lr ) = T \\(Gr ,n )ss (Lr ) ' SLr \\(Pr −1 )n ,
where G = SLn (K ),
P - the maximal parabolic subgroup associated to the simple root αr ,
Gr ,n - the Grassmannian of r -dimensional subspaces of an n dimensional
vector space,
Lr - the line bundle on G /P = Gr ,n associated to $r .
Motivation
Now N\\Gr ,n = W \T \\Gr ,n ← T \\Gr ,n = T \\(G /P).
Projective Normality of finite group quotients
Projective Normality of finite group quotients
Projective Normality of finite group quotients
Projective Normality
A projective variety X is said to be projectively normal if the affine cone
X̂ over X is normal at its vertex, i.e., the stalk at the vertex is a normal
domain.
Projective Normality of finite group quotients
Projective Normality
A projective variety X is said to be projectively normal if the affine cone
X̂ over X is normal at its vertex, i.e., the stalk at the vertex is a normal
domain.
Projective normality depends on the particular projective embedding of
the variety.
Projective Normality of finite group quotients
Projective Normality
A projective variety X is said to be projectively normal if the affine cone
X̂ over X is normal at its vertex, i.e., the stalk at the vertex is a normal
domain.
Projective normality depends on the particular projective embedding of
the variety.
Example: The projective line P1 is obviously projectively normal since its
cone is the affine plane K 2 (which is non-singular). However, it can be
also embedded in P3 as the quartic curve, namely,
V+ = {(a4 , a3 b, ab 3 , b 4 ) ∈ P3 : (a, b) ∈ P1 },
Then it is normal but not projectively normal [Hartshorne, Ch. I, Ex.
3.18].
Introduction to the problem
Remark: Let L be a very ample line bundle on a projective variety X .
Then the polarized variety (X , L) is projectively normal if the natural map
Symm H 0 (X , L) → H 0 (X , Lm ) is surjective for all m ≥ 0 [Hartshorne, Ch.
II, Ex. 5.14].
Introduction to the problem
Remark: Let L be a very ample line bundle on a projective variety X .
Then the polarized variety (X , L) is projectively normal if the natural map
Symm H 0 (X , L) → H 0 (X , Lm ) is surjective for all m ≥ 0 [Hartshorne, Ch.
II, Ex. 5.14].
Introduction to the problem :
Introduction to the problem
Remark: Let L be a very ample line bundle on a projective variety X .
Then the polarized variety (X , L) is projectively normal if the natural map
Symm H 0 (X , L) → H 0 (X , Lm ) is surjective for all m ≥ 0 [Hartshorne, Ch.
II, Ex. 5.14].
Introduction to the problem :
G - a finite group.
V - a finite dimensional representation of G over a field K .
Then the K -algebra of invariants K [V ]G is finitely generated. (Noether:
1916, 1926).
Introduction to the problem
When K is algebraically closed, it is an interesting problem to study the
quotient varieties V /G = Spec(K [V ]G ) and P(V )/G .
Introduction to the problem
When K is algebraically closed, it is an interesting problem to study the
quotient varieties V /G = Spec(K [V ]G ) and P(V )/G .
Let O(1) denotes the ample generator of the Picard group of P(V ).
Introduction to the problem
When K is algebraically closed, it is an interesting problem to study the
quotient varieties V /G = Spec(K [V ]G ) and P(V )/G .
Let O(1) denotes the ample generator of the Picard group of P(V ).
Then ∀x ∈ P(V ), the isotropy Gx acts trivially on the fiber of the line
bundle O(1)⊗|G | at x.
Introduction to the problem
When K is algebraically closed, it is an interesting problem to study the
quotient varieties V /G = Spec(K [V ]G ) and P(V )/G .
Let O(1) denotes the ample generator of the Picard group of P(V ).
Then ∀x ∈ P(V ), the isotropy Gx acts trivially on the fiber of the line
bundle O(1)⊗|G | at x.
Hence, by a descent lemma of Kempf, when |G | is a unit in K , the line
bundle O(1)⊗|G | descends to the quotient P(V )/G .
Let us denote it by L.
Introduction to the problem
When K is algebraically closed, it is an interesting problem to study the
quotient varieties V /G = Spec(K [V ]G ) and P(V )/G .
Let O(1) denotes the ample generator of the Picard group of P(V ).
Then ∀x ∈ P(V ), the isotropy Gx acts trivially on the fiber of the line
bundle O(1)⊗|G | at x.
Hence, by a descent lemma of Kempf, when |G | is a unit in K , the line
bundle O(1)⊗|G | descends to the quotient P(V )/G .
Let us denote it by L.
On the other hand, V /G = Spec(K [V ]G ) is normal.
Introduction to the problem
When K is algebraically closed, it is an interesting problem to study the
quotient varieties V /G = Spec(K [V ]G ) and P(V )/G .
Let O(1) denotes the ample generator of the Picard group of P(V ).
Then ∀x ∈ P(V ), the isotropy Gx acts trivially on the fiber of the line
bundle O(1)⊗|G | at x.
Hence, by a descent lemma of Kempf, when |G | is a unit in K , the line
bundle O(1)⊗|G | descends to the quotient P(V )/G .
Let us denote it by L.
On the other hand, V /G = Spec(K [V ]G ) is normal.
So, it is a natural question to ask if P(V )/G is projectively normal with
respect to the line bundle L.
Introduction to the problem
More generally, an interesting question is the following:-
Introduction to the problem
More generally, an interesting question is the following:Question:
Let G be a finite group acting on a projectively normal polarized variety
(X , M), where M is G -linearized very ample line bundle making X ⊂ P(V )
projectively normal and L ∈ Pic(X /G ) is the descent of M⊗|G | on X . Is
the polarized variety (X /G , L) projectively normal ?
Introduction to the problem
More generally, an interesting question is the following:Question:
Let G be a finite group acting on a projectively normal polarized variety
(X , M), where M is G -linearized very ample line bundle making X ⊂ P(V )
projectively normal and L ∈ Pic(X /G ) is the descent of M⊗|G | on X . Is
the polarized variety (X /G , L) projectively normal ?
(P(V )/G , L) = Proj(⊕d∈Z≥0 (H 0 (P(V ), O(1)⊗d|G | )G )
=Proj(⊕d∈Z≥0 (Symd|G | V ∗ )G ).
Introduction to the problem
More generally, an interesting question is the following:Question:
Let G be a finite group acting on a projectively normal polarized variety
(X , M), where M is G -linearized very ample line bundle making X ⊂ P(V )
projectively normal and L ∈ Pic(X /G ) is the descent of M⊗|G | on X . Is
the polarized variety (X /G , L) projectively normal ?
(P(V )/G , L) = Proj(⊕d∈Z≥0 (H 0 (P(V ), O(1)⊗d|G | )G )
=Proj(⊕d∈Z≥0 (Symd|G | V ∗ )G ).
Let R := ⊕d≥0 Rd ; Rd := (Symdn V ∗ )G .
Introduction to the problem
More generally, an interesting question is the following:Question:
Let G be a finite group acting on a projectively normal polarized variety
(X , M), where M is G -linearized very ample line bundle making X ⊂ P(V )
projectively normal and L ∈ Pic(X /G ) is the descent of M⊗|G | on X . Is
the polarized variety (X /G , L) projectively normal ?
(P(V )/G , L) = Proj(⊕d∈Z≥0 (H 0 (P(V ), O(1)⊗d|G | )G )
=Proj(⊕d∈Z≥0 (Symd|G | V ∗ )G ).
Let R := ⊕d≥0 Rd ; Rd := (Symdn V ∗ )G .
Question:
When is the map Symd R1 → Rd surjective, for all d ∈ N ?
Introduction to the problem
More generally, an interesting question is the following:Question:
Let G be a finite group acting on a projectively normal polarized variety
(X , M), where M is G -linearized very ample line bundle making X ⊂ P(V )
projectively normal and L ∈ Pic(X /G ) is the descent of M⊗|G | on X . Is
the polarized variety (X /G , L) projectively normal ?
(P(V )/G , L) = Proj(⊕d∈Z≥0 (H 0 (P(V ), O(1)⊗d|G | )G )
=Proj(⊕d∈Z≥0 (Symd|G | V ∗ )G ).
Let R := ⊕d≥0 Rd ; Rd := (Symdn V ∗ )G .
Question:
When is the map Symd R1 → Rd surjective, for all d ∈ N ?
Here we give an affirmative answer to this question in many cases.
Solvable Case
Theorem (Kannan, -, Sardar, Proc. AMS, 2009)
Let G be a finite solvable group, and let V be a finite dimensional
faithful representation of G over a field K of characteristic not dividing
|G |. Then, the polarized variety (P(V )/G , L) is projectively normal.
Solvable Case
Theorem (Kannan, -, Sardar, Proc. AMS, 2009)
Let G be a finite solvable group, and let V be a finite dimensional
faithful representation of G over a field K of characteristic not dividing
|G |. Then, the polarized variety (P(V )/G , L) is projectively normal.
The proof uses necessarily the celebrated theorem in additive number
theory due to Erdös, Ginzburg and Ziv.
Solvable Case
Theorem (Kannan, -, Sardar, Proc. AMS, 2009)
Let G be a finite solvable group, and let V be a finite dimensional
faithful representation of G over a field K of characteristic not dividing
|G |. Then, the polarized variety (P(V )/G , L) is projectively normal.
The proof uses necessarily the celebrated theorem in additive number
theory due to Erdös, Ginzburg and Ziv.
Erdös-Ginzburg-Ziv Theorem :
Let n ≥ 1 and a1 , . . . , a2n−1 be a finite sequence of integers of length
atleast 2n − 1. Then there exists a subsequence ai1 , . . . , ain of length
exactly n such that ai1 + · · · + ain ≡ 0 mod n.
Pseudo-reflection groups
Pseudo-reflection groups:
Let V be a vector space of dimension n over a field K . A linear transformation σ : V → V is called a pseudo-reflection, if it fixes pointwise a
subspace of co-dimension one. Let G ≤ GL(V ) be a finite group acting
linearly on V . We say that G is a pseudo-reflection group if G is generated
by pseudo-reflections.
Pseudo-reflection groups
Pseudo-reflection groups:
Let V be a vector space of dimension n over a field K . A linear transformation σ : V → V is called a pseudo-reflection, if it fixes pointwise a
subspace of co-dimension one. Let G ≤ GL(V ) be a finite group acting
linearly on V . We say that G is a pseudo-reflection group if G is generated
by pseudo-reflections.
Theorem
(Chevalley-Serre-Shephard-Todd). Let V be a finite dimensional
representation of a finite group G over a field K . Assume that the order
of G is relatively prime to the characteristic of K . Then G is generated
G
by pseudo-reflections if and
Qn only if K [V ] is a polynomial algebra. In this
case one also has |G | = i=1 deg (fi ), where f1 , f2 , · · · , fn are the
generators of K [V ]G .
Pseudo-reflection groups
Theorem (Kannan, -, Sardar, Proc AMS, 2009)
Let V be a finite dimensional vector space over a field K . Let G be a
finite subgroup of GL(V ) which is generated by pseudo reflections.
Further assume that characteristic of K does not divide |G |, then the
polarized variety (P(V )/G , L) is projectively normal.
Pseudo-reflection groups
Theorem (Kannan, -, Sardar, Proc AMS, 2009)
Let V be a finite dimensional vector space over a field K . Let G be a
finite subgroup of GL(V ) which is generated by pseudo reflections.
Further assume that characteristic of K does not divide |G |, then the
polarized variety (P(V )/G , L) is projectively normal.
Proof uses the Chevalley-Serre-Shephard-Todd theorem and the following
combinatorial lemma:-
Pseudo-reflection groups
Theorem (Kannan, -, Sardar, Proc AMS, 2009)
Let V be a finite dimensional vector space over a field K . Let G be a
finite subgroup of GL(V ) which is generated by pseudo reflections.
Further assume that characteristic of K does not divide |G |, then the
polarized variety (P(V )/G , L) is projectively normal.
Proof uses the Chevalley-Serre-Shephard-Todd theorem and the following
combinatorial lemma:Qr
Let a=(a1 , a2 , · · · ar ) ∈ Nr and NP
a =
i=1 ai . Consider the semigroup
r
r
Ma = {(m1 , m2 , · · · mr ) ∈ Z≥0 :P i=1 mi ai ≡ 0 mod Na } and the set
Sa = {(m1 , m2 , · · · mr ) ∈ Zr≥0 : ri=1 mi ai = Na }.
Lemma
Ma is generated by Sa for a ∈ Nr .
Pseudo-reflection groups
Corollary
Let G be a semi-simple algebraic group over C. Let T be a maximal
torus of G and let W = NG (T )/T be the Weyl group of G with respect
to T . Let h = Lie(T ) be the standard representation of W . Let L be the
descent of O(1)⊗|W | . Then the polarized variety (P(h)/W , L) is
projectively normal.
Vector Invariants and Projective Normality
Let V be the standard representation of W .
Let V m := V ⊕ V ⊕ · · · ⊕ V .
Then W acts diagonally on V m .
Let L be the descent of O(1)⊗|W | .
Vector Invariants and Projective Normality
Let V be the standard representation of W .
Let V m := V ⊕ V ⊕ · · · ⊕ V .
Then W acts diagonally on V m .
Let L be the descent of O(1)⊗|W | .
Question:
Does projective normality hold for (P(V m )/W , L) ?
Vector Invariants and Projective Normality
Let V be the standard representation of W .
Let V m := V ⊕ V ⊕ · · · ⊕ V .
Then W acts diagonally on V m .
Let L be the descent of O(1)⊗|W | .
Question:
Does projective normality hold for (P(V m )/W , L) ?
Theorem (Kannan, -, Proc. IAS, 2011)
Let G be a semi-simple algebraic group of type An , Bn , Cn , Dn , F4 or G2 .
Let W denote the corresponding Weyl group. Let V be the standard
representation of W . Then P(V m )/W is projectively normal with respect
to the line bundle O(1)⊗|W | .
Vector Invariants and Projective Normality
For type An , Bn , Cn and Dn the proof uses the classical technique of
polarization.
Vector Invariants and Projective Normality
For type An , Bn , Cn and Dn the proof uses the classical technique of
polarization.
The Weyl groups of type F4 and G2 are solvable. So, infact projective
normality holds for any representation of these groups (not necessarily
the standard representation).
Vector Invariants and Projective Normality
For type An , Bn , Cn and Dn the proof uses the classical technique of
polarization.
The Weyl groups of type F4 and G2 are solvable. So, infact projective
normality holds for any representation of these groups (not necessarily
the standard representation).
Corollary
Let G be a finite group of order n and U be any finite dimensional
representation of G over C. Let L denote the descent of O(1)⊗n! . Then
P(U)/G is projectively normal with respect to L.
Proof uses the Cayley embedding G ,→ Sn , g 7→ (gj := ggi ) and the
surjectivity of the restriction map
η̃ : Sym(V k )Sn → Sym(U)G
where V is the natural representation of Sn .
Normality, Projective Normality and EGZ theorem
Normality of a Semigroup
An affine semigroup M is a finitely generated sub-semigroup of Zn containing 0 for some n. Let N be the subgroup of Zn generated by M. Then,
M is called normal if it satisfies the following condition: if kx ∈ M for
some x ∈ N and k ∈ N, then x ∈ M.
Normality, Projective Normality and EGZ theorem
Normality of a Semigroup
An affine semigroup M is a finitely generated sub-semigroup of Zn containing 0 for some n. Let N be the subgroup of Zn generated by M. Then,
M is called normal if it satisfies the following condition: if kx ∈ M for
some x ∈ N and k ∈ N, then x ∈ M.
For an affine semigroup M and a field K we can form the affine
semigroup algebra K [M] in the following way: as a K vector space K [M]
has a basis consisting of the symbols X a , a ∈ M, and the multiplication
on K [M] is defined by the K -bilinear extension of X a .X b = X a+b .
Normality, Projective Normality and EGZ theorem
Theorem
Let M be an affine semigroup, and K be a field. Then M is normal if and
only if K [M] is normal, i.e., it is integrally closed in its field of fractions.
Normality, Projective Normality and EGZ theorem
Theorem (Kannan, -, INTEGERS-JCNT, 2011)
The following are equivalent
1. Erdös-Ginzburg-Ziv theorem: Let (a1 , a2 , · · · , am ), m ≥ 2n − 1 be a
sequence of elements of Z/nZ. Then there exists a subsequence
(ai1 , ai2 , · · · , ain ) of length n whose sum is zero.
2. Let G be a cyclic group of order n and V be any finite dimensional
representation of G over C. Let L be the descent of O(1)⊗n . Then
(P(V )/G , L) is projectively normal.
20 . Let G be a cyclic group of order n and V be the regular
representation of G over C. Let L be the descent of O(1)⊗n . Then
(P(V )/G , L) is projectively normal.
3. The sub-semigroup M of Zn generated by the set
S = {(m0 , m
P1 , · · · , mn−1 )
Pn−1
∈ (Z≥0 )n : n−1
i=0 mi = n and
i=0 imi ≡ 0 mod n} is normal.
Projective Normality of finite group quotients
Question:
Let G be a reductive group (not necessarily finite) acting morphically on a
projective variety X . Let M be a G -linearized very ample line bundle on
X such that M descends to the quotient X //G . Let L be the descent. Is
the polarized variety (X //G , L) projectively normal ?
Torus action on homogeneous spaces
Torus action on homogeneous spaces
Torus action on homogeneous spaces
G - a simply connected semi-simple algebraic group over an algebraically
closed field K
T - a maximal torus of G
B - a Borel subgroup of G containing T
P - a parabolic subgroup of G containing B
NG (T ) - the normalizer of T in G
W = NG (T )/T - the Weyl group of G with respect to T
Φ - set of roots with respect to T
Φ+ - positive roots with respect to B
∆ = {α1 , · · · , αl } ⊆ Φ+ - a set of simple roots
Torus action on homogeneous spaces
G - a simply connected semi-simple algebraic group over an algebraically
closed field K
T - a maximal torus of G
B - a Borel subgroup of G containing T
P - a parabolic subgroup of G containing B
NG (T ) - the normalizer of T in G
W = NG (T )/T - the Weyl group of G with respect to T
Φ - set of roots with respect to T
Φ+ - positive roots with respect to B
∆ = {α1 , · · · , αl } ⊆ Φ+ - a set of simple roots
For I ⊆ ∆ denote W I = {w ∈ W |w (α) > 0, α ∈ I }
WI = hsα : α ∈ I i.
Then for w ∈ W , w = w I .wI , with w I ∈ W I and wI ∈ WI .
Torus action on homogeneous spaces
Then T acts on the generalized flag variety G /P by left translation.
There are only finitely many fixed points {ew := wWP : w ∈ W /WP }.
For w ∈ W /WP , the B-orbit CP (w ) := Bew = BwP/P in G /P is a
locally closed subset of G /P, called the Schubert cell.
Torus action on homogeneous spaces
Then T acts on the generalized flag variety G /P by left translation.
There are only finitely many fixed points {ew := wWP : w ∈ W /WP }.
For w ∈ W /WP , the B-orbit CP (w ) := Bew = BwP/P in G /P is a
locally closed subset of G /P, called the Schubert cell.
CP (w ) is called the Schubert variety associated to w , and is denoted by
XP (w ).
Thus Schubert varieties in G /P are indexed by W P .
Torus action on homogeneous spaces
Then T acts on the generalized flag variety G /P by left translation.
There are only finitely many fixed points {ew := wWP : w ∈ W /WP }.
For w ∈ W /WP , the B-orbit CP (w ) := Bew = BwP/P in G /P is a
locally closed subset of G /P, called the Schubert cell.
CP (w ) is called the Schubert variety associated to w , and is denoted by
XP (w ).
Thus Schubert varieties in G /P are indexed by W P .
Note that if P = B, then WP = {id}, and the Schubert varieties in G /B
are indexed by the elements of W .
We denote the Schubert variety corresponding to w ∈ W by X (w ).
Torus action on homogeneous spaces
Let X be a projective variety with an action of reductive group G
Let L be an ample line bundle on G /P.
Torus action on homogeneous spaces
Let X be a projective variety with an action of reductive group G
Let L be an ample line bundle on G /P.
Semi-stable points
A point x ∈ X is said to be semi-stable with respect to a G -linearized line
bundle L if there is a positive integer m ∈ N, and a G -invariant section
s ∈ H 0 (X , Lm ) with s(x) 6= 0.
Torus action on homogeneous spaces
Let X be a projective variety with an action of reductive group G
Let L be an ample line bundle on G /P.
Semi-stable points
A point x ∈ X is said to be semi-stable with respect to a G -linearized line
bundle L if there is a positive integer m ∈ N, and a G -invariant section
s ∈ H 0 (X , Lm ) with s(x) 6= 0.
All parabolic subgroups P of G containing B for which there exists an
s
ample line bundle L on G /P such that (G /P)ss
T (L) = (G /P)T (L) has
been described. [Kannan, Proc. IAS, 1998, 1999].
Torus action on homogeneous spaces
Let X be a projective variety with an action of reductive group G
Let L be an ample line bundle on G /P.
Semi-stable points
A point x ∈ X is said to be semi-stable with respect to a G -linearized line
bundle L if there is a positive integer m ∈ N, and a G -invariant section
s ∈ H 0 (X , Lm ) with s(x) 6= 0.
All parabolic subgroups P of G containing B for which there exists an
s
ample line bundle L on G /P such that (G /P)ss
T (L) = (G /P)T (L) has
been described. [Kannan, Proc. IAS, 1998, 1999].
Now T acts on all Schubert varieties.
Torus action on homogeneous spaces
Let X be a projective variety with an action of reductive group G
Let L be an ample line bundle on G /P.
Semi-stable points
A point x ∈ X is said to be semi-stable with respect to a G -linearized line
bundle L if there is a positive integer m ∈ N, and a G -invariant section
s ∈ H 0 (X , Lm ) with s(x) 6= 0.
All parabolic subgroups P of G containing B for which there exists an
s
ample line bundle L on G /P such that (G /P)ss
T (L) = (G /P)T (L) has
been described. [Kannan, Proc. IAS, 1998, 1999].
Now T acts on all Schubert varieties.
Question:
Describe all minimal Schubert varieties in G /P admitting semi-stable
points with respect to L for the action of a maximal torus T .
Schubert varieties admitting semi-stable points
Here is a criterion for a Schubert variety to admit semi-stable points.
Proposition (Kannan, -, Proc. IAS, 2009)
P
Let χ = α∈∆ aα $α be a dominant character of T which is in the root
c
lattice. Let I = Supp(χ) = {α ∈ ∆ : aα 6= 0} and let w ∈ W I , where
I c = ∆ \ I . Then X (w )ss
T (Lχ ) 6= ∅ if and only if w χ ≤ 0.
Schubert varieties admitting semi-stable points
Here is a criterion for a Schubert variety to admit semi-stable points.
Proposition (Kannan, -, Proc. IAS, 2009)
P
Let χ = α∈∆ aα $α be a dominant character of T which is in the root
c
lattice. Let I = Supp(χ) = {α ∈ ∆ : aα 6= 0} and let w ∈ W I , where
I c = ∆ \ I . Then X (w )ss
T (Lχ ) 6= ∅ if and only if w χ ≤ 0.
When Q is a maximal parabolic subgroup of G and L = L$ , where $ is
a minuscule dominant weight, there exists unique minimal Schubert
variety X (w ) admitting semi-stable points with respect to L.
[Kannan-Sardar, Proc. IAS, 2009].
Schubert varieties admitting semi-stable points
Here is a criterion for a Schubert variety to admit semi-stable points.
Proposition (Kannan, -, Proc. IAS, 2009)
P
Let χ = α∈∆ aα $α be a dominant character of T which is in the root
c
lattice. Let I = Supp(χ) = {α ∈ ∆ : aα 6= 0} and let w ∈ W I , where
I c = ∆ \ I . Then X (w )ss
T (Lχ ) 6= ∅ if and only if w χ ≤ 0.
When Q is a maximal parabolic subgroup of G and L = L$ , where $ is
a minuscule dominant weight, there exists unique minimal Schubert
variety X (w ) admitting semi-stable points with respect to L.
[Kannan-Sardar, Proc. IAS, 2009].
Note that this includes type A.
Schubert varieties admitting semi-stable points
Let L$r denote the line bundle corresponding to the fundamental weight
$r .
Schubert varieties admitting semi-stable points
Let L$r denote the line bundle corresponding to the fundamental weight
$r .
Type-A
Theorem (Kannan, Sardar, Proc. IAS, 2009)
Let rank(G ) = n = qr + t, with 1 ≤ t ≤ r and let w ∈ W Ir . Then
X (w )ss
T (L$r ) 6= ∅ if and only if w = (sa1 · · · s1 ) · · · (sar · · · sr ), where
{ai : i = 1, 2 · · · r } is an increasing sequence of positive integers such that
ai ≥ i(q + 1) ∀ i ≤ t − 1 and ai = iq + (t + 1) ∀ t ≤ i ≤ r .
Schubert varieties admitting semi-stable points
Type- B,C,D
Notation:
Jp,q = {(i1 , i2 , · · · , ip ) : ik ∈ {1, 2, · · · , q} ∀ k and ik+1 − ik ≥ 2}.
Schubert varieties admitting semi-stable points
Type- B,C,D
Notation:
Jp,q = {(i1 , i2 , · · · , ip ) : ik ∈ {1, 2, · · · , q} ∀ k and ik+1 − ik ≥ 2}.
Ir
Let Wmin
= Minimal elements of the set of all τ ∈ W Ir such that
ss
X (τ )T (L$r ) 6= ∅.
Schubert varieties admitting semi-stable points
Type- Bn
Theorem (Kannan, -, Proc. IAS, 2009)
I1
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
= {w }.
Schubert varieties admitting semi-stable points
Type- Bn
Theorem (Kannan, -, Proc. IAS, 2009)
I1
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
= {w }.
(ii) Let r be an even integer in {2, 3, · · · , n − 1}. For any
Ir
such that
i = (i1 , i2 , · · · , i 2r ) ∈ J 2r ,n−1 , there exists unique wi ∈ Wmin
r
P2
Ir
wi ($r ) = −( k=1 αik ). Further, Wmin
= {wi : i ∈ J 2r ,n−1 }.
Schubert varieties admitting semi-stable points
Type- Bn
Theorem (Kannan, -, Proc. IAS, 2009)
I1
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
= {w }.
(ii) Let r be an even integer in {2, 3, · · · , n − 1}. For any
Ir
such that
i = (i1 , i2 , · · · , i 2r ) ∈ J 2r ,n−1 , there exists unique wi ∈ Wmin
r
P2
Ir
wi ($r ) = −( k=1 αik ). Further, Wmin
= {wi : i ∈ J 2r ,n−1 }.
(iii) Let r be an odd integer in {2, 3, · · · , n − 1}. For any
Ir
i = (i1 , i2 , · · · , i r −1 ) ∈ J r −1 ,n−2 , there exists unique wi ∈ Wmin
such that
2
2
P r −1
Ir
2
= {wi : i ∈ J r −1 ,n−2 }.
wi ($r ) = −( k=1
αik + αn ). Further, Wmin
2
Schubert varieties admitting semi-stable points
Type- Bn
Theorem (Kannan, -, Proc. IAS, 2009)
I1
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
= {w }.
(ii) Let r be an even integer in {2, 3, · · · , n − 1}. For any
Ir
such that
i = (i1 , i2 , · · · , i 2r ) ∈ J 2r ,n−1 , there exists unique wi ∈ Wmin
r
P2
Ir
wi ($r ) = −( k=1 αik ). Further, Wmin
= {wi : i ∈ J 2r ,n−1 }.
(iii) Let r be an odd integer in {2, 3, · · · , n − 1}. For any
Ir
i = (i1 , i2 , · · · , i r −1 ) ∈ J r −1 ,n−2 , there exists unique wi ∈ Wmin
such that
2
2
P r −1
Ir
2
= {wi : i ∈ J r −1 ,n−2 }.
wi ($r ) = −( k=1
αik + αn ). Further, Wmin
2
(iv) Let r = n. If n is even, then, w = w n2 · · · w1 , where,
wi = s2i−1 . . . sn , i = 1, 2, · · · n2 and if n is odd, then, w = w[ n2 ]+1 · · · w1 ,
In
where, wi = s2i−1 . . . sn , i = 1, 2, · · · [ n2 ] + 1. Further, Wmin
= {w }.
Schubert varieties admitting semi-stable points
(2) Type Cn :
Theorem
I1
= {w }.
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
Schubert varieties admitting semi-stable points
(2) Type Cn :
Theorem
I1
= {w }.
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
(ii) Let r be an even integer in {2, 3, · · · , n}. For any
Ir
i = (i1 , i2 , · · · , i 2r ) ∈ J 2r ,n−1 , there exists unique wi ∈ Wmin
such that
r
P2
Ir
= {wi : i ∈ J 2r ,n−1 }.
wi ($r ) = −( k=1 αik ). Further, Wmin
Schubert varieties admitting semi-stable points
(2) Type Cn :
Theorem
I1
= {w }.
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
(ii) Let r be an even integer in {2, 3, · · · , n}. For any
Ir
i = (i1 , i2 , · · · , i 2r ) ∈ J 2r ,n−1 , there exists unique wi ∈ Wmin
such that
r
P2
Ir
= {wi : i ∈ J 2r ,n−1 }.
wi ($r ) = −( k=1 αik ). Further, Wmin
(iii) Let r be an odd integer in {2, 3, · · · , n}. For any
Ir
i = (i1 , i2 , · · · , i r −1 ) ∈ J r −1 ,n−2 , there exists unique wi ∈ Wmin
such that
2
2
P r −1
Ir
2
wi ($r ) = −( k=1
αik + 21 αn ). Further, Wmin
= {wi : i ∈ J r −1 ,n−2 }.
2
Schubert varieties admitting semi-stable points
(3)Type Dn :
Theorem
I1
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
= {w }.
Schubert varieties admitting semi-stable points
(3)Type Dn :
Theorem
I1
(i) Let r = 1. Then w = sn sn−1 . . . s1 . Further, Wmin
= {w }.
(ii) Let r be an even integer in {2, 3, · · · , n − 2}. For any
Ir
such that
i = (i1 , i2 , · · · , i 2r ) ∈ J 2r ,n \ Z , there exists unique wi ∈ Wmin
P 2r
wi ($r ) = −( k=1 αik ), where Z = {(i1 , i2 , · · · , i 2r −2 , n − 2, n) : ik ∈
{1, 2, · · · , n − 4} and ik+1 − ik ≥ 2 ∀ k}. Further,
Ir
= {wi : i ∈ J 2r ,n \ Z }.
Wmin
Schubert varieties admitting semi-stable points
(iii) Let r be an odd integer in {2, 3, · · · , n − 2}. For any
Ir
i = (i1 , i2 , · · · , i r −1 ) ∈ J r −1 ,n−3 , there exists unique wi ∈ Wmin
such that
2
2
P r −1
2
wi ($r ) = −( k=1
αik + 21 αn−1 + 12 αn ). Also, for any
Ir
i = (i1 , i2 , · · · , i r −1 ) ∈ J r −1 ,n−2 , there exists unique wi,1 ∈ Wmin
such
2
2
P r −1
2
αik + 21 αn−1 + 32 αn ) and there exists unique
that wi,1 ($r ) = −( k=1
P r −1
Ir
2
such that wi,2 ($r ) = −( k=1
αik + 32 αn−1 + 12 αn ). Further,
wi,2 ∈ Wmin
S
Ir
= {wi : i ∈ J r −1 ,n−3 } {wi,j : i ∈ J r −1 ,n−2 and j = 1, 2}.
Wmin
2
2
Schubert varieties admitting semi-stable points
(iii) Let r be an odd integer in {2, 3, · · · , n − 2}. For any
Ir
i = (i1 , i2 , · · · , i r −1 ) ∈ J r −1 ,n−3 , there exists unique wi ∈ Wmin
such that
2
2
P r −1
2
wi ($r ) = −( k=1
αik + 21 αn−1 + 12 αn ). Also, for any
Ir
i = (i1 , i2 , · · · , i r −1 ) ∈ J r −1 ,n−2 , there exists unique wi,1 ∈ Wmin
such
2
2
P r −1
2
αik + 21 αn−1 + 32 αn ) and there exists unique
that wi,1 ($r ) = −( k=1
P r −1
Ir
2
such that wi,2 ($r ) = −( k=1
αik + 32 αn−1 + 12 αn ). Further,
wi,2 ∈ Wmin
S
Ir
= {wi : i ∈ J r −1 ,n−3 } {wi,j : i ∈ J r −1 ,n−2 and j = 1, 2}.
Wmin
2
2
Q[ n−1
]
2
(iv) Let r = n − 1 or n. Then, w = i=1 wi , where,
τi sn if i is odd.
wi =
τi sn−1 if i is even.
Ir
with, τi = s2i−1 . . . sn−2 , i = 1, 2, · · · [ n−1
2 ]. Further, Wmin = {w }.
Coxeter elements admitting semi-stable points
Coxeter elements in the Weyl group:
Let W be the Weyl group of a semi-simple algebraic group of rank n.
An element w ∈ W is said to be a Coxeter element if it is of the form
w = si1 si2 . . . sin , with sij 6= sik unless j = k.
Coxeter elements admitting semi-stable points
Coxeter elements in the Weyl group:
Let W be the Weyl group of a semi-simple algebraic group of rank n.
An element w ∈ W is said to be a Coxeter element if it is of the form
w = si1 si2 . . . sin , with sij 6= sik unless j = k.
A Schubert variety X (w ) in G /B contains a (rank G )-dimensional
T -orbit if and only if w ≥ τ for some Coxeter element τ .
Coxeter elements admitting semi-stable points
Coxeter elements in the Weyl group:
Let W be the Weyl group of a semi-simple algebraic group of rank n.
An element w ∈ W is said to be a Coxeter element if it is of the form
w = si1 si2 . . . sin , with sij 6= sik unless j = k.
A Schubert variety X (w ) in G /B contains a (rank G )-dimensional
T -orbit if and only if w ≥ τ for some Coxeter element τ .
Question
Given a coxeter element τ , does there exist a non-trivial line bundle L on
G /B such that X (τ )ss
T (L) 6= ∅.
Coxeter elements admitting semi-stable points
Theorem (Kannan, -, Proc. IAS, 2009)
(A) Type An : (1) A3 : For any Coxeter element w , X (w )ss
T (Lχ ) 6= ∅ for
some non-zero dominant weight χ.
Coxeter elements admitting semi-stable points
Theorem (Kannan, -, Proc. IAS, 2009)
(A) Type An : (1) A3 : For any Coxeter element w , X (w )ss
T (Lχ ) 6= ∅ for
some non-zero dominant weight χ.
(2) An , n ≥ 4: If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w must be either sn sn−1 . . . s1 or
si . . . s1 si+1 . . . sn for some 1 ≤ i ≤ n − 1.
Coxeter elements admitting semi-stable points
Theorem (Kannan, -, Proc. IAS, 2009)
(A) Type An : (1) A3 : For any Coxeter element w , X (w )ss
T (Lχ ) 6= ∅ for
some non-zero dominant weight χ.
(2) An , n ≥ 4: If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w must be either sn sn−1 . . . s1 or
si . . . s1 si+1 . . . sn for some 1 ≤ i ≤ n − 1.
(B) Type Bn : (1) B2 : For any Coxeter element w , X (w )ss
T (Lχ ) 6= ∅ for
some non-zero dominant weight χ.
Coxeter elements admitting semi-stable points
Theorem (Kannan, -, Proc. IAS, 2009)
(A) Type An : (1) A3 : For any Coxeter element w , X (w )ss
T (Lχ ) 6= ∅ for
some non-zero dominant weight χ.
(2) An , n ≥ 4: If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w must be either sn sn−1 . . . s1 or
si . . . s1 si+1 . . . sn for some 1 ≤ i ≤ n − 1.
(B) Type Bn : (1) B2 : For any Coxeter element w , X (w )ss
T (Lχ ) 6= ∅ for
some non-zero dominant weight χ.
(2) Bn , n ≥ 3: If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w = sn sn−1 . . . s1 .
Coxeter elements admitting semi-stable points
(C) Type Cn : If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w = sn sn−1 . . . s1 .
Coxeter elements admitting semi-stable points
(C) Type Cn : If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w = sn sn−1 . . . s1 .
(D) Type Dn : (1) D4 : If w is a Coxeter element, then X (w )ss
T (Lχ ) 6= ∅
for some non-zero dominant weight χ if and only if l(ws2 ) = l(w ) + 1
and l(wsi ) = l(w ) − 1 for exactly one i 6= 2.
Coxeter elements admitting semi-stable points
(C) Type Cn : If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w = sn sn−1 . . . s1 .
(D) Type Dn : (1) D4 : If w is a Coxeter element, then X (w )ss
T (Lχ ) 6= ∅
for some non-zero dominant weight χ if and only if l(ws2 ) = l(w ) + 1
and l(wsi ) = l(w ) − 1 for exactly one i 6= 2.
(2) Dn , n ≥ 5: If X (w )ss
T (Lχ ) 6= ∅ for some non-zero dominant weight χ
and w is a Coxeter element, then w = sn sn−1 . . . s1 .
Coxeter elements admitting semi-stable points
(E) E6 , E7 , E8 : There is no Coxeter element w for which there exist a
non-zero dominant weight χ such that X (w )ss
T (Lχ ) 6= ∅.
Coxeter elements admitting semi-stable points
(E) E6 , E7 , E8 : There is no Coxeter element w for which there exist a
non-zero dominant weight χ such that X (w )ss
T (Lχ ) 6= ∅.
(F) F4 : There is no Coxeter element w for which there exist a non-zero
dominant weight χ such that X (w )ss
T (Lχ ) 6= ∅.
Coxeter elements admitting semi-stable points
(E) E6 , E7 , E8 : There is no Coxeter element w for which there exist a
non-zero dominant weight χ such that X (w )ss
T (Lχ ) 6= ∅.
(F) F4 : There is no Coxeter element w for which there exist a non-zero
dominant weight χ such that X (w )ss
T (Lχ ) 6= ∅.
(G) G2 : There is no Coxeter element w for which there exist a non-zero
dominant weight χ such that X (w )ss
T (Lχ ) 6= ∅.
THANK YOU
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