Fundamental representation

Fundamental representation

In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie groupor Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.

Examples

* In the case of the general linear group, all fundamental representations are exterior powers of the defining module.
* In the case of the special unitary group SU(n), the n-1 fundamental representations are the wedge products Alt^k {mathbb C}^n consisting of the alternating tensors, for k=1,2,...,n-1.
* The spin representation of the twofold cover of an odd orthogonal group, the odd spin group, and the two half-spin representations of the twofold cover of an even orthogonal group, the even spinor group, are fundamental representations that cannot be realized in the space of tensors.
* The adjoint representation of the simple Lie group of type E8 is a fundamental representation.

Explanation

The irreducible representations of a simply-connected compact Lie group are indexed by their highest weights. These weights are the lattice points in an orthant "Q"+ in the weight lattice of the Lie group consisting of the dominant integral weights. It can be provedthat there exists a set of "fundamental weights", indexed by the vertices of the Dynkin diagram, such that any dominant integral weight is a non-negative integer linear combinations of the fundamental weights. The corresponding irreducible representations are the fundamental representations of the Lie group. 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.

Other uses

Outside of Lie theory, the term "fundamental representation" is sometimes loosely used to refer to a smallest-dimensional faithful representation, though this is also often called the "standard" or "defining" representation (a term referring more to the history, rather than having a well-defined mathematical meaning).

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Fundamental Rights, Directive Principles and Fundamental Duties of India — The Fundamental Rights, Directive Principles of State Policy and Fundamental Duties are sections of the Constitution of India that prescribe the fundamental obligations of the Statefn|° to its citizens and the duties of the citizens to the State …   Wikipedia

  • Fundamental Rights in India — The Fundamental Rights in India enshrined in the Part III of the Constitution of India guarantee civil liberties such that all Indians can lead their lives in peace and harmony as citizens of India. These include individual rights common to most… …   Wikipedia

  • representation — rep·re·sen·ta·tion n 1: one that represents: as a: a statement or account made to influence opinion or action compare warranty 3 b: an incidental or collateral statement of fact on the faith of which a contract is entered into the contract of… …   Law dictionary

  • Representation (arts) — Representation describes the signs that stand in for and take the place of something else.Mitchell, W. 1995, Representation , in F Lentricchia T McLaughlin (eds), Critical Terms for Literary Study , 2nd edn, University of Chicago Press, Chicago]… …   Wikipedia

  • Fundamental Law of Vatican City State — Vatican City State This article is part of the series: Politics and government of Vatican City State Fundamental Law Pope …   Wikipedia

  • Fundamental theorem of arithmetic — In number theory, the fundamental theorem of arithmetic (or unique prime factorization theorem) states that every natural number greater than 1 can be written as a unique product of prime numbers. For instance, : 6936 = 2^3 imes 3 imes 17^2 , ,! …   Wikipedia

  • Representation of a Lie group — In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the… …   Wikipedia

  • Fundamental Laws of England — In the 1760s William Blackstone described the Fundamental Laws of England in Commentaries on the Laws of England, Book the First Chapter the First : Of the Absolute Rights of Individuals [http://www.yale.edu/lawweb/avalon/blackstone/bk1ch1.htm]… …   Wikipedia

  • Representation theory of finite groups — In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction. This article discusses the representation theory of… …   Wikipedia

  • Representation theory of SU(2) — In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”