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Lie GroupsRepresentations
GeometryMore Symmetry
Results
Computation of Invariants for Harish-ChandraModules of SU(p, q) by Combining Algebraic and
Geometric Methods.
Matthew [email protected]
November 6th, 2010
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
Slides and notes are available atwww.math.utah.edu/~housley → talks.
2 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Definition: Lie Group
I G is a differentiable manifold with a group operation.
I Smooth multiplication.
I Smooth inverse.
I Complex and real flavors.
3 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Definition: Lie Group
I G is a differentiable manifold with a group operation.
I Smooth multiplication.
I Smooth inverse.
I Complex and real flavors.
4 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Definition: Lie Group
I G is a differentiable manifold with a group operation.
I Smooth multiplication.
I Smooth inverse.
I Complex and real flavors.
5 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Definition: Lie Group
I G is a differentiable manifold with a group operation.
I Smooth multiplication.
I Smooth inverse.
I Complex and real flavors.
6 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Lie groups: smooth symmetries.
7 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Examples:
I GL(n,C) is the set of all invertible complex lineartransformations on Cn. (Complex)
I GL(n,R) is the set of all invertible real linear transformationson Rn. (Real)
I SO(n) is the set of orientation preserving isometries of the(n − 1)-sphere. (Real)
I E.g. SO(3) is the set of rotations of the 2-sphere.I Has applications to quantum mechanics.
8 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Examples:
I GL(n,C) is the set of all invertible complex lineartransformations on Cn. (Complex)
I GL(n,R) is the set of all invertible real linear transformationson Rn. (Real)
I SO(n) is the set of orientation preserving isometries of the(n − 1)-sphere. (Real)
I E.g. SO(3) is the set of rotations of the 2-sphere.I Has applications to quantum mechanics.
9 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Examples:
I GL(n,C) is the set of all invertible complex lineartransformations on Cn. (Complex)
I GL(n,R) is the set of all invertible real linear transformationson Rn. (Real)
I SO(n) is the set of orientation preserving isometries of the(n − 1)-sphere. (Real)
I E.g. SO(3) is the set of rotations of the 2-sphere.I Has applications to quantum mechanics.
10 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Examples:
I GL(n,C) is the set of all invertible complex lineartransformations on Cn. (Complex)
I GL(n,R) is the set of all invertible real linear transformationson Rn. (Real)
I SO(n) is the set of orientation preserving isometries of the(n − 1)-sphere. (Real)
I E.g. SO(3) is the set of rotations of the 2-sphere.I Has applications to quantum mechanics.
11 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Examples:
I GL(n,C) is the set of all invertible complex lineartransformations on Cn. (Complex)
I GL(n,R) is the set of all invertible real linear transformationson Rn. (Real)
I SO(n) is the set of orientation preserving isometries of the(n − 1)-sphere. (Real)
I E.g. SO(3) is the set of rotations of the 2-sphere.
I Has applications to quantum mechanics.
12 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Examples:
I GL(n,C) is the set of all invertible complex lineartransformations on Cn. (Complex)
I GL(n,R) is the set of all invertible real linear transformationson Rn. (Real)
I SO(n) is the set of orientation preserving isometries of the(n − 1)-sphere. (Real)
I E.g. SO(3) is the set of rotations of the 2-sphere.I Has applications to quantum mechanics.
13 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Symmetry
Examples:
I SU(p, q) is the set of invertible complex linear transformationsof Cp+q that preserve the hermitian form given by
〈v ,w〉 = v∗(Ip 00 −Iq
)w .
(real)
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Reductive Groups
I The above examples are reductive.
I Roughly speaking: no interesting normal subgroups.
I We’ll assume this from now on.
15 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Reductive Groups
I The above examples are reductive.
I Roughly speaking: no interesting normal subgroups.
I We’ll assume this from now on.
16 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Reductive Groups
I The above examples are reductive.
I Roughly speaking: no interesting normal subgroups.
I We’ll assume this from now on.
17 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
Start with a Lie group G . Define g to be the tangent space to Gat id .
18 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.
I Roughly speaking: g approximates G near the identity.
I [−,−] is bilinear.
I [x , y ] = −[y , x ].
I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.
I Can be derived from the Lie bracket of differential geometry.
I The structures of G and g are closely related.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.
I Roughly speaking: g approximates G near the identity.
I [−,−] is bilinear.
I [x , y ] = −[y , x ].
I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.
I Can be derived from the Lie bracket of differential geometry.
I The structures of G and g are closely related.
20 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.
I Roughly speaking: g approximates G near the identity.
I [−,−] is bilinear.
I [x , y ] = −[y , x ].
I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.
I Can be derived from the Lie bracket of differential geometry.
I The structures of G and g are closely related.
21 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.
I Roughly speaking: g approximates G near the identity.
I [−,−] is bilinear.
I [x , y ] = −[y , x ].
I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.
I Can be derived from the Lie bracket of differential geometry.
I The structures of G and g are closely related.
22 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.
I Roughly speaking: g approximates G near the identity.
I [−,−] is bilinear.
I [x , y ] = −[y , x ].
I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.
I Can be derived from the Lie bracket of differential geometry.
I The structures of G and g are closely related.
23 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.
I Roughly speaking: g approximates G near the identity.
I [−,−] is bilinear.
I [x , y ] = −[y , x ].
I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.
I Can be derived from the Lie bracket of differential geometry.
I The structures of G and g are closely related.
24 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
DefinitionSymmetryReductive GroupsLie Algebras
Lie Algebras
I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.
I Roughly speaking: g approximates G near the identity.
I [−,−] is bilinear.
I [x , y ] = −[y , x ].
I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.
I Can be derived from the Lie bracket of differential geometry.
I The structures of G and g are closely related.
25 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.
I Given a smooth group homomorphism from G to GL(V ), wecall V a representation of G .
I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
26 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
27 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
28 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.
I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
29 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.
I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
30 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.
I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
31 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.
I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
32 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.
I Generating new representations. (Tensor products, exteriorpowers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
33 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
34 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Representations
I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .I Reductive: more or less the whole group acts on V .
I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior
powers, etc.)
I Finite dimensional representations of real and complex Liegroups are well understood.
35 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Irreducibility
I V is an irreducible representation of G if it contains no propernonzero subrepresentations of G .
I Analogous to prime numbers, finite simple groups, etc.
I Roughly: irreducible representations are simplest actions ofthe symmetries in G .
36 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Irreducibility
I V is an irreducible representation of G if it contains no propernonzero subrepresentations of G .
I Analogous to prime numbers, finite simple groups, etc.
I Roughly: irreducible representations are simplest actions ofthe symmetries in G .
37 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Irreducibility
I V is an irreducible representation of G if it contains no propernonzero subrepresentations of G .
I Analogous to prime numbers, finite simple groups, etc.
I Roughly: irreducible representations are simplest actions ofthe symmetries in G .
38 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Lie Algebra Representations
I If V is a G representation, it becomes a g representation viadifferentiation.
I We complexify g to gC.
I Irreducible representations of gC are important in classifyingirreducible representations of G .
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Lie Algebra Representations
I If V is a G representation, it becomes a g representation viadifferentiation.
I We complexify g to gC.
I Irreducible representations of gC are important in classifyingirreducible representations of G .
40 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Lie Algebra Representations
I If V is a G representation, it becomes a g representation viadifferentiation.
I We complexify g to gC.
I Irreducible representations of gC are important in classifyingirreducible representations of G .
41 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Infinite Dimensional Motivations
I Start with a manifold M with a measure.
I Find a real Lie group G that acts on M and preserves themeasure.
I Irreducible representations of complex Lie groups are finitedimensional.
I Consider the action of G on L2(M).
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Infinite Dimensional Motivations
I Start with a manifold M with a measure.I Find a real Lie group G that acts on M and preserves the
measure.
I Irreducible representations of complex Lie groups are finitedimensional.
I Consider the action of G on L2(M).
43 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Infinite Dimensional Motivations
I Start with a manifold M with a measure.I Find a real Lie group G that acts on M and preserves the
measure.I Irreducible representations of complex Lie groups are finite
dimensional.
I Consider the action of G on L2(M).
44 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Infinite Dimensional Motivations
I Start with a manifold M with a measure.I Find a real Lie group G that acts on M and preserves the
measure.I Irreducible representations of complex Lie groups are finite
dimensional.
I Consider the action of G on L2(M).
45 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Harish-Chandra Modules
I Harish-Chandra modules: algebraizations of infinitedimensional representations.
I We’ll ignore this distinction.
I We can study infinite dimensional representations usingalgebraic and algebro-geometric techniques.
46 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Harish-Chandra Modules
I Harish-Chandra modules: algebraizations of infinitedimensional representations.
I We’ll ignore this distinction.
I We can study infinite dimensional representations usingalgebraic and algebro-geometric techniques.
47 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case
Harish-Chandra Modules
I Harish-Chandra modules: algebraizations of infinitedimensional representations.
I We’ll ignore this distinction.
I We can study infinite dimensional representations usingalgebraic and algebro-geometric techniques.
48 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
Geometric InvariantsSU(p, q)
Geometric Invariants
Two geometric invariants to consider:
I Associated variety AV(X ): a variety contained in gC.
I Associated cycle AC(X ): finer invariant that attaches aninteger (multiplicity) to each component of AV(X ).
I We’d like to calculate the multiplicities.
49 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
Geometric InvariantsSU(p, q)
Geometric Invariants
Two geometric invariants to consider:
I Associated variety AV(X ): a variety contained in gC.
I Associated cycle AC(X ): finer invariant that attaches aninteger (multiplicity) to each component of AV(X ).
I We’d like to calculate the multiplicities.
50 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
Geometric InvariantsSU(p, q)
Geometric Invariants
Two geometric invariants to consider:
I Associated variety AV(X ): a variety contained in gC.
I Associated cycle AC(X ): finer invariant that attaches aninteger (multiplicity) to each component of AV(X ).
I We’d like to calculate the multiplicities.
51 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
Geometric InvariantsSU(p, q)
Geometric Invariants
Two geometric invariants to consider:
I Associated variety AV(X ): a variety contained in gC.
I Associated cycle AC(X ): finer invariant that attaches aninteger (multiplicity) to each component of AV(X ).
I We’d like to calculate the multiplicities.
52 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
Geometric InvariantsSU(p, q)
SU(p, q)
I For this group, associated variety for irreducible X is anirreducible variety.
I We only need to compute one multiplicity to get AC(X ).
53 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
Geometric InvariantsSU(p, q)
SU(p, q)
I For this group, associated variety for irreducible X is anirreducible variety.
I We only need to compute one multiplicity to get AC(X ).
54 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
More Symmetry: the Weyl Group
I The Weyl group W is a finite set of internal symmetries of G .
I W = NG (T )/ZG (T ).
I The Weyl group for SU(p, q) is the symmetric group Sp+q.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
More Symmetry: the Weyl Group
I The Weyl group W is a finite set of internal symmetries of G .
I W = NG (T )/ZG (T ).
I The Weyl group for SU(p, q) is the symmetric group Sp+q.
56 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
More Symmetry: the Weyl Group
I The Weyl group W is a finite set of internal symmetries of G .
I W = NG (T )/ZG (T ).
I The Weyl group for SU(p, q) is the symmetric group Sp+q.
57 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Cells
I Let X be an infinite dimensional representation of SU(p, q).
I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.
I Take the formal Z-span of the elements of C.
I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.
I Let mXidenote the multiplicity in the associated variety of Xi .
I The representation relates the multiplicities mXifor the
various Xi in C.
I If we know mXjfor some Xj we can calculate mXi
for theother Xi in the cell C.
58 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Cells
I Let X be an infinite dimensional representation of SU(p, q).
I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.
I Take the formal Z-span of the elements of C.
I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.
I Let mXidenote the multiplicity in the associated variety of Xi .
I The representation relates the multiplicities mXifor the
various Xi in C.
I If we know mXjfor some Xj we can calculate mXi
for theother Xi in the cell C.
59 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Cells
I Let X be an infinite dimensional representation of SU(p, q).
I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.
I Take the formal Z-span of the elements of C.
I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.
I Let mXidenote the multiplicity in the associated variety of Xi .
I The representation relates the multiplicities mXifor the
various Xi in C.
I If we know mXjfor some Xj we can calculate mXi
for theother Xi in the cell C.
60 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Cells
I Let X be an infinite dimensional representation of SU(p, q).
I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.
I Take the formal Z-span of the elements of C.
I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.
I Let mXidenote the multiplicity in the associated variety of Xi .
I The representation relates the multiplicities mXifor the
various Xi in C.
I If we know mXjfor some Xj we can calculate mXi
for theother Xi in the cell C.
61 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Cells
I Let X be an infinite dimensional representation of SU(p, q).
I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.
I Take the formal Z-span of the elements of C.
I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.
I Let mXidenote the multiplicity in the associated variety of Xi .
I The representation relates the multiplicities mXifor the
various Xi in C.
I If we know mXjfor some Xj we can calculate mXi
for theother Xi in the cell C.
62 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Cells
I Let X be an infinite dimensional representation of SU(p, q).
I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.
I Take the formal Z-span of the elements of C.
I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.
I Let mXidenote the multiplicity in the associated variety of Xi .
I The representation relates the multiplicities mXifor the
various Xi in C.
I If we know mXjfor some Xj we can calculate mXi
for theother Xi in the cell C.
63 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Cells
I Let X be an infinite dimensional representation of SU(p, q).
I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.
I Take the formal Z-span of the elements of C.
I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.
I Let mXidenote the multiplicity in the associated variety of Xi .
I The representation relates the multiplicities mXifor the
various Xi in C.
I If we know mXjfor some Xj we can calculate mXi
for theother Xi in the cell C.
64 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Strategy
I For SU(p, q) and any cell C of representations, we can alwaysfind an Xi ∈ C so that mXi
can be computed by geometricmeans.
(Springer fiber.)
I Compute the Sp+q representation on spanZ C.
I Compute mXifor other Xi in C.
65 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Strategy
I For SU(p, q) and any cell C of representations, we can alwaysfind an Xi ∈ C so that mXi
can be computed by geometricmeans. (Springer fiber.)
I Compute the Sp+q representation on spanZ C.
I Compute mXifor other Xi in C.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Strategy
I For SU(p, q) and any cell C of representations, we can alwaysfind an Xi ∈ C so that mXi
can be computed by geometricmeans. (Springer fiber.)
I Compute the Sp+q representation on spanZ C.
I Compute mXifor other Xi in C.
67 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Strategy
I For SU(p, q) and any cell C of representations, we can alwaysfind an Xi ∈ C so that mXi
can be computed by geometricmeans. (Springer fiber.)
I Compute the Sp+q representation on spanZ C.
I Compute mXifor other Xi in C.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Problem
It can be difficult to compute the Sp+q representation on spanZ C.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Symmetric Group Representations
I Let n = p + q.
I Basis: {e1 − e2, e2 − e3, . . . , en−1 − en}.I Let Sn act in the “obvious” way.
For example, acting by (12):e1 − e2 → e2 − e1 and e3 − e1 → e3 − e2.This is the standard representation V of Sn.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Symmetric Group Representations
I Let n = p + q.
I Basis: {e1 − e2, e2 − e3, . . . , en−1 − en}.
I Let Sn act in the “obvious” way.
For example, acting by (12):e1 − e2 → e2 − e1 and e3 − e1 → e3 − e2.This is the standard representation V of Sn.
71 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Symmetric Group Representations
I Let n = p + q.
I Basis: {e1 − e2, e2 − e3, . . . , en−1 − en}.I Let Sn act in the “obvious” way.
For example, acting by (12):e1 − e2 → e2 − e1 and e3 − e1 → e3 − e2.This is the standard representation V of Sn.
72 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Symmetric Group Representations
I Let n = p + q.
I Basis: {e1 − e2, e2 − e3, . . . , en−1 − en}.I Let Sn act in the “obvious” way.
For example, acting by (12):e1 − e2 → e2 − e1 and e3 − e1 → e3 − e2.
This is the standard representation V of Sn.
73 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Symmetric Group Representations
I Let n = p + q.
I Basis: {e1 − e2, e2 − e3, . . . , en−1 − en}.I Let Sn act in the “obvious” way.
For example, acting by (12):e1 − e2 → e2 − e1 and e3 − e1 → e3 − e2.This is the standard representation V of Sn.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
More Representations of Sn
Irreducible representations of Sn are parametrized by Youngdiagrams with n boxes.
Construction: take subspaces of tensor powers of the standardrepresentation.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
More Representations of Sn
Irreducible representations of Sn are parametrized by Youngdiagrams with n boxes.
Construction: take subspaces of tensor powers of the standardrepresentation.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
More Representations of Sn
Irreducible representations of Sn are parametrized by Youngdiagrams with n boxes.
Construction: take subspaces of tensor powers of the standardrepresentation.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
Hook type diagram: upside down L.
Two rows with one box on the bottom row: standardrepresentation.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
Hook type diagram: upside down L.
Two rows with one box on the bottom row: standardrepresentation.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
General hook type with m + 1 rows:∧m V , where V is the
standard representations.
Example: (12) action on (e1 − e2) ∧ (e2 − e3) for∧2 V :
(e2 − e1) ∧ (e1 − e3)
= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)
= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)
= −(e1 − e2) ∧ (e2 − e3).
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
General hook type with m + 1 rows:∧m V , where V is the
standard representations.Example: (12) action on (e1 − e2) ∧ (e2 − e3) for
∧2 V :
(e2 − e1) ∧ (e1 − e3)
= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)
= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)
= −(e1 − e2) ∧ (e2 − e3).
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
General hook type with m + 1 rows:∧m V , where V is the
standard representations.Example: (12) action on (e1 − e2) ∧ (e2 − e3) for
∧2 V :
(e2 − e1) ∧ (e1 − e3)
= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)
= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)
= −(e1 − e2) ∧ (e2 − e3).
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
General hook type with m + 1 rows:∧m V , where V is the
standard representations.Example: (12) action on (e1 − e2) ∧ (e2 − e3) for
∧2 V :
(e2 − e1) ∧ (e1 − e3)
= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)
= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)
= −(e1 − e2) ∧ (e2 − e3).
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
General hook type with m + 1 rows:∧m V , where V is the
standard representations.Example: (12) action on (e1 − e2) ∧ (e2 − e3) for
∧2 V :
(e2 − e1) ∧ (e1 − e3)
= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)
= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)
= −(e1 − e2) ∧ (e2 − e3).
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
The Weyl GroupCellsSymmetric Group Representations
Hook Type Representations
General hook type with m + 1 rows:∧m V , where V is the
standard representations.Example: (12) action on (e1 − e2) ∧ (e2 − e3) for
∧2 V :
(e2 − e1) ∧ (e1 − e3)
= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)
= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)
= −(e1 − e2) ∧ (e2 − e3).
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
Strategy
I Find a hook type cell C.
I Find one Xj in the cell such that computation of mXjis easy.
I Find mXifor other Xi in C by using the Sn representation on
spanZ C.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
Strategy
I Find a hook type cell C.
I Find one Xj in the cell such that computation of mXjis easy.
I Find mXifor other Xi in C by using the Sn representation on
spanZ C.
87 / 91
Lie GroupsRepresentations
GeometryMore Symmetry
Results
Strategy
I Find a hook type cell C.
I Find one Xj in the cell such that computation of mXjis easy.
I Find mXifor other Xi in C by using the Sn representation on
spanZ C.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
Result
We get a formula for mXiwhere Xi is any infinite dimensional
representation in a hook type cell:
mXi= Am
1∏|τk |!
∑σ∈Sτ
sgn(σ)σ·
∑σ′∈Sm
sgn(σ′)σ′ ·
( ∏i=1...m
xm−i+1τ(i)
)where
Am =1
m! · (m − 1)! · · · 1.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
Result
We get a formula for mXiwhere Xi is any infinite dimensional
representation in a hook type cell:
mXi= Am
1∏|τk |!
∑σ∈Sτ
sgn(σ)σ·
∑σ′∈Sm
sgn(σ′)σ′ ·
( ∏i=1...m
xm−i+1τ(i)
)where
Am =1
m! · (m − 1)! · · · 1.
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Lie GroupsRepresentations
GeometryMore Symmetry
Results
Details
Details are available at www.math.utah.edu/~housley →research.
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