Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Group representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.
Irreducible representationIn mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.
Brass instrumentA brass instrument is a musical instrument that produces sound by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips. Brass instruments are also called labrosones or labrophones, from Latin and Greek elements meaning 'lip' and 'sound'. There are several factors involved in producing different pitches on a brass instrument.
Musical instrumentA musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. A person who plays a musical instrument is known as an instrumentalist. The history of musical instruments dates to the beginnings of human culture. Early musical instruments may have been used for rituals, such as a horn to signal success on the hunt, or a drum in a religious ceremony.
Algebra representationIn abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Brass bandA brass band is a musical ensemble generally consisting entirely of brass instruments, most often with a percussion section. Ensembles that include brass and woodwind instruments can in certain traditions also be termed brass bands (particularly in the context of New Orleans and Japan–style brass bands), but may be more correctly termed military bands, concert bands, or "brass and reed" bands. Balkan Brass Band Balkan-style Brass Bands (Труба, trumpet) play a distinctive style of music originating in 19th century Balkans.
Wind instrumentA wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitch of the vibration is determined by the length of the tube and by manual modifications of the effective length of the vibrating column of air. In the case of some wind instruments, sound is produced by blowing through a reed; others require buzzing into a metal mouthpiece, while yet others require the player to blow into a hole at an edge, which splits the air column and creates the sound.
National accountsNational accounts or national account systems (NAS) are the implementation of complete and consistent accounting techniques for measuring the economic activity of a nation. These include detailed underlying measures that rely on double-entry accounting. By design, such accounting makes the totals on both sides of an account equal even though they each measure different characteristics, for example production and the income from it. As a method, the subject is termed national accounting or, more generally, social accounting.
