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48-533: GLN may refer to: General linear group Gide Loyrette Nouel , a French law firm Glenview Railroad Station , in Illinois, United States Global Location Number Gln (or Q), abbreviation for the amino acid glutamine Guelmim Airport , in Morocco Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

96-399: A free R -module M of rank n . One can also define GL( M ) for any R -module, but in general this is not isomorphic to GL( n , R ) (for any n ). Over a field F , a matrix is invertible if and only if its determinant is nonzero. Therefore, an alternative definition of GL( n , F ) is as the group of matrices with nonzero determinant. Over a commutative ring R , more care

144-410: A group G on a set X is given by a function ρ : G → X , the set of functions from X to X , such that for all g 1 , g 2 in G and all x in X : where 1 {\displaystyle 1} is the identity element of G . This condition and the axioms for a group imply that ρ( g ) is a bijection (or permutation ) for all g in G . Thus we may equivalently define

192-410: A group G on a vector space V over a field K is a group homomorphism from G to GL( V ), the general linear group on V . That is, a representation is a map such that Here V is called the representation space and the dimension of V is called the dimension or degree of the representation. It is common practice to refer to V itself as the representation when the homomorphism

240-490: A linear complex structure — concretely, that commute with a matrix J such that J = − I , where J corresponds to multiplying by the imaginary unit i . The Lie algebra corresponding to GL( n , C ) consists of all n × n complex matrices with the commutator serving as the Lie bracket. Unlike the real case, GL( n , C ) is connected . This follows, in part, since the multiplicative group of complex numbers C

288-572: A semidirect product : The special linear group is also the derived group (also known as commutator subgroup) of the GL( n , F ) (for a field or a division ring F ) provided that n ≠ 2 {\displaystyle n\neq 2} or k is not the field with two elements . When F is R or C , SL( n , F ) is a Lie subgroup of GL( n , F ) of dimension n − 1 . The Lie algebra of SL( n , F ) consists of all n × n matrices over F with vanishing trace . The Lie bracket

336-426: A choice of basis in V . Given a basis ( e 1 , ..., e n ) of V and an automorphism T in GL( V ), we have then for every basis vector e i that for some constants a ij in F ; the matrix corresponding to T is then just the matrix with entries given by the a ji . In a similar way, for a commutative ring R the group GL( n , R ) may be interpreted as the group of automorphisms of

384-422: A functor selects an object X in C and a group homomorphism from G to Aut( X ), the automorphism group of X . In the case where C is Vect K , the category of vector spaces over a field K , this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of G in the category of sets . When C is Ab , the category of abelian groups ,

432-410: A permutation representation to be a group homomorphism from G to the symmetric group S X of X . For more information on this topic see the article on group action . Every group G can be viewed as a category with a single object; morphisms in this category are just the elements of G . Given an arbitrary category C , a representation of G in C is a functor from G to C . Such

480-577: A representation ρ on C 2 {\displaystyle \mathbb {C} ^{2}} given by: This representation is faithful because ρ is a one-to-one map . Another representation for C 3 on C 2 {\displaystyle \mathbb {C} ^{2}} , isomorphic to the previous one, is σ given by: The group C 3 may also be faithfully represented on R 2 {\displaystyle \mathbb {R} ^{2}} by τ given by: where Another example: Let V {\displaystyle V} be

528-410: A subgroup of GL( n , F ) isomorphic to ( F ) . In fields like R and C , these correspond to rescaling the space; the so-called dilations and contractions. A scalar matrix is a diagonal matrix which is a constant times the identity matrix . The set of all nonzero scalar matrices forms a subgroup of GL( n , F ) isomorphic to F . This group is the center of GL( n , F ) . In particular, it

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576-516: Is GL n ( F ) or GL( n , F ) , or simply GL( n ) if the field is understood. More generally still, the general linear group of a vector space GL( V ) is the automorphism group , not necessarily written as matrices. The special linear group , written SL( n , F ) or SL n ( F ), is the subgroup of GL( n , F ) consisting of matrices with a determinant of 1. The group GL( n , F ) and its subgroups are often called linear groups or matrix groups (the automorphism group GL( V )

624-405: Is a vector space over the field F , the general linear group of V , written GL( V ) or Aut( V ), is the group of all automorphisms of V , i.e. the set of all bijective linear transformations V → V , together with functional composition as group operation. If V has finite dimension n , then GL( V ) and GL( n , F ) are isomorphic . The isomorphism is not canonical; it depends on

672-412: Is a linear group but not a matrix group). These groups are important in the theory of group representations , and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials . The modular group may be realised as a quotient of the special linear group SL(2, Z ) . If n ≥ 2 , then the group GL( n , F ) is not abelian . If V

720-500: Is a normal, abelian subgroup. The center of SL( n , F ) is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of n th roots of unity in the field F . The so-called classical groups are subgroups of GL( V ) which preserve some sort of bilinear form on a vector space V . These include the These groups provide important examples of Lie groups. The projective linear group PGL( n , F ) and

768-601: Is a vector space we have a linear representation . Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations. The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: Representation theory also depends heavily on

816-451: Is clear from the context. In the case where V is of finite dimension n it is common to choose a basis for V and identify GL( V ) with GL( n , K ) , the group of n × n {\displaystyle n\times n} invertible matrices on the field K . Consider the complex number u = e which has the property u = 1. The set C 3 = {1, u , u } forms a cyclic group under multiplication. This group has

864-403: Is connected. The group manifold GL( n , C ) is not compact; rather its maximal compact subgroup is the unitary group U( n ). As for U( n ), the group manifold GL( n , C ) is not simply connected but has a fundamental group isomorphic to Z . If F is a finite field with q elements, then we sometimes write GL( n , q ) instead of GL( n , F ) . When p is prime, GL( n , p )

912-435: Is given by the commutator . The special linear group SL( n , R ) can be characterized as the group of volume and orientation-preserving linear transformations of R . The group SL( n , C ) is simply connected, while SL( n , R ) is not. SL( n , R ) has the same fundamental group as GL ( n , R ) , that is, Z for n = 2 and Z 2 for n > 2 . The set of all invertible diagonal matrices forms

960-428: Is invariant under the group action is called a subrepresentation . If V has exactly two subrepresentations, namely the zero-dimensional subspace and V itself, then the representation is said to be irreducible ; if it has a proper subrepresentation of nonzero dimension, the representation is said to be reducible . The representation of dimension zero is considered to be neither reducible nor irreducible, just as

1008-463: Is needed: a matrix over R is invertible if and only if its determinant is a unit in R , that is, if its determinant is invertible in R . Therefore, GL( n , R ) may be defined as the group of matrices whose determinants are units. Over a non-commutative ring R , determinants are not at all well behaved. In this case, GL( n , R ) may be defined as the unit group of the matrix ring M( n , R ) . The general linear group GL( n , R ) over

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1056-419: Is the outer automorphism group of the group Z p , and also the automorphism group, because Z p is abelian, so the inner automorphism group is trivial. The order of GL( n , q ) is: This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general,

1104-426: Is the group of n × n invertible matrices of real numbers, and is denoted by GL n ( R ) or GL( n , R ) . More generally, the general linear group of degree n over any field F (such as the complex numbers ), or a ring R (such as the ring of integers ), is the set of n × n invertible matrices with entries from F (or R ), again with matrix multiplication as the group operation. Typical notation

1152-435: Is the product of the determinants of each matrix. SL( n , F ) is a normal subgroup of GL( n , F ) . If we write F for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism that is surjective and its kernel is the special linear group. Therefore, by the first isomorphism theorem , GL( n , F )/SL( n , F ) is isomorphic to F . In fact, GL( n , F ) can be written as

1200-562: The Lorentz group , O(1, 3, F ) ⋉ F . The general semilinear group ΓL( n , F ) is the group of all invertible semilinear transformations , and contains GL. A semilinear transformation is a transformation which is linear “up to a twist”, meaning “up to a field automorphism under scalar multiplication”. It can be written as a semidirect product: where Gal( F ) is the Galois group of F (over its prime field ), which acts on GL( n , F ) by

1248-614: The Schubert decomposition of the Grassmannian, and are q -analogs of the Betti numbers of complex Grassmannians. This was one of the clues leading to the Weil conjectures . Note that in the limit q ↦ 1 the order of GL( n , q ) goes to 0! – but under the correct procedure (dividing by ( q − 1) ) we see that it is the order of the symmetric group (See Lorscheid's article) – in

1296-459: The Zariski topology ), and therefore a smooth manifold of the same dimension. The Lie algebra of GL( n , R ) , denoted g l n , {\displaystyle {\mathfrak {gl}}_{n},} consists of all n × n real matrices with the commutator serving as the Lie bracket. As a manifold, GL( n , R ) is not connected but rather has two connected components :

1344-412: The general linear group of degree n is the set of n × n invertible matrices , together with the operation of ordinary matrix multiplication . This forms a group , because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group. The group is so named because the columns (and also

1392-399: The k th column can be any vector not in the linear span of the first k − 1 columns. In q -analog notation, this is [ n ] q ! ( q − 1 ) n q ( n 2 ) {\displaystyle [n]_{q}!(q-1)^{n}q^{n \choose 2}} . For example, GL(3, 2) has order (8 − 1)(8 − 2)(8 − 4) = 168 . It is

1440-457: The projective special linear group PSL( n , F ) are the quotients of GL( n , F ) and SL( n , F ) by their centers (which consist of the multiples of the identity matrix therein); they are the induced action on the associated projective space . The affine group Aff( n , F ) is an extension of GL( n , F ) by the group of translations in F . It can be written as a semidirect product : where GL( n , F ) acts on F in

1488-400: The symmetry group of a physical system affects the solutions of equations describing that system. The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object

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1536-451: The Cartesian product of O( n ) with the set of positive-definite symmetric matrices. Similarly, it shows that there is a homeomorphism between GL ( n , R ) and the Cartesian product of SO( n ) with the set of positive-definite symmetric matrices. Because the latter is contractible, the fundamental group of GL ( n , R ) is isomorphic to that of SO( n ). The homeomorphism also shows that

1584-664: The Galois action on the entries. The main interest of ΓL( n , F ) is that the associated projective semilinear group PΓL( n , F ) (which contains PGL( n , F )) is the collineation group of projective space , for n > 2 , and thus semilinear maps are of interest in projective geometry . Group representation In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups allow many group-theoretic problems to be reduced to problems in linear algebra . In physics , they describe how

1632-494: The automorphism group of the Fano plane and of the group Z 2 . This group is also isomorphic to PSL(2, 7) . More generally, one can count points of Grassmannian over F : in other words the number of subspaces of a given dimension k . This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem . These formulas are connected to

1680-417: The characteristic of the complex numbers is zero, which never divides the size of a group. In the example above, the first two representations given (ρ and σ) are both decomposable into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}), while the third representation (τ) is irreducible. A set-theoretic representation (also known as a group action or permutation representation ) of

1728-410: The context of studying the Galois group of the general equation of order p . The special linear group, SL( n , F ) , is the group of all matrices with determinant 1. They are special in that they lie on a subvariety – they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices

1776-427: The field of complex numbers , GL( n , C ) , is a complex Lie group of complex dimension n . As a real Lie group (through realification) it has dimension 2 n . The set of all real matrices forms a real Lie subgroup. These correspond to the inclusions which have real dimensions n , 2 n , and 4 n = (2 n ) . Complex n -dimensional matrices can be characterized as real 2 n -dimensional matrices that preserve

1824-430: The field of complex numbers . The other important cases are the field of real numbers , finite fields , and fields of p-adic numbers . In general, algebraically closed fields are easier to handle than non-algebraically closed ones. The characteristic of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the order of the group . A representation of

1872-440: The field of real numbers is a real Lie group of dimension n . To see this, note that the set of all n × n real matrices, M n ( R ), forms a real vector space of dimension n . The subset GL( n , R ) consists of those matrices whose determinant is non-zero. The determinant is a polynomial map, and hence GL( n , R ) is an open affine subvariety of M n ( R ) (a non-empty open subset of M n ( R ) in

1920-451: The group GL( n , R ) is noncompact . “The” maximal compact subgroup of GL( n , R ) is the orthogonal group O( n ), while "the" maximal compact subgroup of GL ( n , R ) is the special orthogonal group SO( n ). As for SO( n ), the group GL ( n , R ) is not simply connected (except when n = 1) , but rather has a fundamental group isomorphic to Z for n = 2 or Z 2 for n > 2 . The general linear group over

1968-423: The matrices with positive determinant and the ones with negative determinant. The identity component , denoted by GL ( n , R ) , consists of the real n × n matrices with positive determinant. This is also a Lie group of dimension n ; it has the same Lie algebra as GL( n , R ) . The polar decomposition , which is unique for invertible matrices, shows that there is a homeomorphism between GL( n , R ) and

GLN - Misplaced Pages Continue

2016-457: The natural manner. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space F . One has analogous constructions for other subgroups of the general linear group: for instance, the special affine group is the subgroup defined by the semidirect product, SL( n , F ) ⋉ F , and the Poincaré group is the affine group associated to

2064-404: The number 1 is considered to be neither composite nor prime . Under the assumption that the characteristic of the field K does not divide the size of the group, representations of finite groups can be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem ). This holds in particular for any representation of a finite group over the complex numbers , since

2112-402: The philosophy of the field with one element , one thus interprets the symmetric group as the general linear group over the field with one element: S n ≅ GL( n , 1) . The general linear group over a prime field, GL( ν , p ) , was constructed and its order computed by Évariste Galois in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in

2160-436: The rows) of an invertible matrix are linearly independent , hence the vectors/points they define are in general linear position , and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers )

2208-640: The space of homogeneous degree-3 polynomials over the complex numbers in variables x 1 , x 2 , x 3 . {\displaystyle x_{1},x_{2},x_{3}.} Then S 3 {\displaystyle S_{3}} acts on V {\displaystyle V} by permutation of the three variables. For instance, ( 12 ) {\displaystyle (12)} sends x 1 3 {\displaystyle x_{1}^{3}} to x 2 3 {\displaystyle x_{2}^{3}} . A subspace W of V that

2256-463: The title GLN . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=GLN&oldid=1139228969 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages General linear group In mathematics ,

2304-401: The type of vector space on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a Hilbert space , Banach space , etc.). One must also consider the type of field over which the vector space is defined. The most important case is

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