The
adjoint representation of a Lie group G is the representation given by the adjoint action of G on the Lie algebra of G (an adjoint action is obtained, roughly, by differentiating a conjugation action.)
admissible
A representation of a real reductive group is called
admissible if (1) a maximal compact subgroup K acts as unitary operators and (2) each irreducible representation of K has finite multiplicity.
2.
Artin's theorem on induced characters states that a character on a finite group is a rational linear combination of characters induced from cyclic subgroups.
Over an algebraically closed field of characteristic zero, the
Borel–Weil–Bott theorem realizes an irreducible representation of a
reductive algebraic group as the space of the global sections of a line bundle on a flag variety. (In the positive characteristic case, the construction only produces
Weyl modules, which may not be irreducible.)
Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups.
A
class functionf on a group G is a function such that ; it is a function on conjugacy classes.
cluster algebra
A
cluster algebra is an integral domain with some combinatorial structure on the generators, introduced in an attempt to systematize the notion of a
dual canonical basis.
“completely reducible" is another term for "semisimple".
complex
1. A
complex representation is a representation of G on a complex vector space. Many authors refer complex representations simply as representations.
2. The
complex-conjugate of a complex representation V is the representation with the same underlying additive group V with the linear action of G but with the action of a complex number through complex conjugation.
3. A complex representation is self-conjugate if it is isomorphic to its complex conjugate.
complementary
A complementary representation to a subrepresentation W of a representation V is a representation W' such that V is the direct sum of W and W'.
Given a field extension , a representation V of a group G over K is said to be
defined over F if for some representation over F such that is induced by ; i.e., . Here, is called an F-form of V (and is not necessarily unique).
The
direct sum of representationsV, W is a representation that is the direct sum of the vector spaces together with the linear group action .
discrete
An irreducible representation of a Lie group G is said to be in the
discrete series if the matrix coefficients of it are all square integrable. For example, if G is compact, then every irreducible representation of it is in the discrete series.
dominant
The irreducible representations of a simply-connected compact Lie group are indexed by their highest weight. These dominant weights form the lattice points in an orthant in the weight lattice of the Lie group.
dual
1. The
dual representation (or the contragredient representation) of a representation V is a representation that is the dual vector space together with the linear group action that preserves the natural pairing
The
Frobenius reciprocity states that for each representation of H and representation of G there is a bijection
that is natural in the sense that is the right
adjoint functor to the restriction functor .
fundamental
Fundamental representation: For the irreducible representations of a simply-connected
compact Lie group there exists a set of fundamental weights, indexed by the vertices of the
Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights.
The corresponding irreducible representations are the fundamental representations of the Lie group. In particular, from the expansion of a dominant weight in terms of the fundamental weights, one can take a corresponding
tensor product of the fundamental representations and extract one copy of the irreducible representation corresponding to that dominant weight.
In the case of the
special unitary groupSU(n), the n − 1 fundamental representations are the wedge products
A
G-linear map between representations is a linear transformation that commutes with the G-actions; i.e., for every g in G.
G-module
Another name for a representation. It allows for the module-theoretic terminology: e.g., trivial G-module, G-submodules, etc.
G-equivariant vector bundle
A
G-equivariant vector bundle is a vector bundle on a G-space X together with a G-action on E (say right) such that is a well-defined linear map.
good
A
good filtration of a representation of a
reductive groupG is a filtration such that the quotients are isomorphic to where are the line bundles on the flag variety .
H
Harish-Chandra
1.
Harish-Chandra (11 October 1923 – 16 October 1983), an Indian American mathematician.
1. Given a complex semisimple Lie algebra , Cartan subalgebra and a choice of a
positive Weyl chamber, the
highest weight of a representation of is the weight of an -weight vector v such that for every positive root (v is called the highest weight vector).
2. The
theorem of the highest weight states (1) two finite-dimensional irreducible representations of are isomorphic if and only if they have the same highest weight and (2) for each dominant integral , there is a finite-dimensional irreducible representation having as its highest weight.
Hom
The
Hom representation of representations V, W is a representation with the group action obtained by the vector space identification .
I
indecomposable
An
indecomposable representation is a representation that is not a direct sum of at least two proper subrepresebtations.
is a representation of G that is induced on the H-linear functions ; cf.
#Frobenius reciprocity.
2. Depending on applications, it is common to impose further conditions on the functions ; for example, if the functions are required to be compactly supported, then the resulting induction is called the
compact induction.
infinitesimally
Two admissible representations of a real reductive group are said to be
infinitesimally equivalent if their associated Lie algebra representations on the space of K-finite vectors are isomorphic.
An
irreducible representation is a representation whose only subrepresentations are zero and itself. The term "irreducible" is synonymous with "simple".
isomorphism
An isomorphism between representations of a group G is an invertible G-linear map between the representations.
isotypic
1. Given a representation V and a simple representation W (subrepresebtation or otherwise), the
isotypic component of V of type W is the direct sum of all subrepresentations of V that are isomorphic to W. For example, let A be a ring and G a group acting on it as automorphisms. If A is
semisimple as a G-module, then the
ring of invariants is the isotypic component of A of trivial type.
2. The
isotypic decomposition of a semisimple representation is the decomposition into the isotypic components.
Maschke's theorem states that a finite-dimensional representation over a field F of a finite group G is a
semisimple representation if the characteristic of F does not divide the order of G.
Mackey theory
The
Mackey theory may be thought of a tool to answer the question: given a representation W of a subgroup H of a group G, when is the induced representation an irreducible representation of G?[1]
A
matrix coefficient of a representation is a linear combination of functions on G of the form for v in V and in the dual space . Note the notion makes sense for any group: if G is a topological group and is continuous, then a matrix matrix coefficient would be a continuous function on G. If G and are algebraic, it would be a
regular function on G.
Given a finite-dimensional complex representation V of a finite group G,
Molien's theorem says that the series , where denotes the space of -invariant homogeneous polynomials on V of degree n, coincides with . The theorem is also valid for a reductive group by replacing by integration over a maximal compact subgroup.
Given a group G, a G-set X and V the vector space of functions from X to a fixed field, a
permutation representation of G on V is a representation given by the induced action of G on V; i.e., . For example, if X is a finite set and V is viewed as a vector space with a basis parameteized by X, then the symmetric group permutates the elements of the basis and its linear extension is precisely the permutation representation.
The term "primitive element" (or a vector) is an old term for a Borel-weight vector.
projective
A
projective representation of a group G is a group homomorphism . Since , a projective representation is precisely a
group action of G on as automorphisms.
proper
A proper subrepresentation of a representation V is a subrepresentstion that is not V.
Q
quotient
Given a representation V and a subrepresentation , the
quotient representation is the representation given by .
Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is (e.g., a cohomology group, tangent space, etc.). As a consequence, many mathematicians other than specialists in the field (or even those who think they might want to be) come in contact with the subject in various ways.
Fulton, William; Harris, Joe, Representation Theory: A First Course
A
linear representation of a group G is a
group homomorphism from G to the
general linear group. Depending on the group G, the homomorphism is often implicitly required to be a morphishm in a category to which G belongs; e.g., if G is a
topological group, then must be continuous. The adjective “linear” is often omitted.
2. Equivalently, a linear representation is a
group action of G on a vector space V that is linear: the action such that for each g in G, is a linear transformation.
3. A
virtual representation is an element of the Grothendieck ring of the category of representations.
2.
Schur's lemma states that a G-linear map between irreducible representations must be either bijective or zero.
3. The
Schur orthogonality relations on a compact group says the characters of non-isomorphic irreducible representations are orthogonal to each other.
4. The
Schur functor constructs representations such as symmetric powers or exterior powers according to a partition . The characters of are
Schur polynomials.
5. The
Schur–Weyl duality computes the irreducible representations occurring in tensor powers of -modules.
A
semisimple representation (also called a completely reducible representation) is a direct sum of simple representations.
simple
Another term for "irreducible".
smooth
1. A
smooth representation of a
locally profinite groupG is a complex representation such that, for each v in V, there is some compact open subgroup K of G that fixes v; i.e., for every g in K.
2. A
smooth vector in a representation space of a Lie group is a vector v such that is a smooth function.
Given a complex semisimple Lie algebra , a Cartan subalgebra and a choice of a
positive Weyl chamber, the
Verma module associated to a linear functional is the quotient of the enveloping algebra by the left ideal generated by for all positive roots as well as for all .[3]
W
weight
1. The term "weight" is another name for a character.
2. The
weight subspace of a representation V of a weight is the subspace that has positive dimension.
3. Similarly, for a linear functional of a complex Lie algebra , is a weight of an -module V if has positive dimension; cf.
#highest weight.
4. weight lattice
5. dominant weight: a weight \lambda is dominant if for some
6. fundamental dominant weight: : Given a set of simple roots , it is a basis of . is a basis of too; the dual basis defined by , is called the fundamental dominant weights.
3. The
Weyl integration formula says: given a compact connected Lie group G with a maximal torus T, there exists a real continuous function u on T such that for every continuous function f on G,
(Explicitly, is 1 over the cardinality of the Weyl group times the product of over the roots .)
2. The
Young symmetrizer is the G-linear endomorphism of a tensor power of a G-module V defined according to a given partition . By definition, the
Schur functor of a representation V assigns to V the image of .
Z
zero
A
zero representation is a zero-dimensional representation. Note: while a zero representation is a trivial representation, a trivial representation need not be zero (since “trivial” mean G acts trivially.)
^Editorial note: this is the definition in (
Humphreys 1972, § 20.3.) as well as (
Gaitsgory 2005, § 1.2.) and differs from the original by half the sum of the positive roots.
References
Adams, J. F. (1969), Lectures on Lie Groups, University of Chicago Press
Theodor Bröcker and Tammo tom Dieck, Representations of compact
Lie groups, Graduate Texts in Mathematics 98, Springer-Verlag, Berlin, 1995.
Knapp, Anthony W. (2001), Representation theory of semisimple groups. An overview based on examples., Princeton Landmarks in Mathematics, Princeton University Press,
ISBN978-0-691-09089-4
N. Wallach, Real Reductive Groups, 2 vols., Academic Press 1988,
Further reading
M. Duflo et M. Vergne, La formule de Plancherel des groupes de Lie semi-simples réels, in “Representations of Lie Groups;” Kyoto, Hiroshima (1986), Advanced Studies in Pure Mathematics 14, 1988.