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In mathematics, 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.
A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K). Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class.
Since the orthogonal group is a subgroup of the general linear group, representations of () can be decomposed into representations of (). The decomposition of a tensor representation is given in terms of Littlewood-Richardson coefficients c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} by the Littlewood restriction rule [ 12 ]
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods ...
The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1. In the simple case n = 1 , the group U(1) corresponds to the circle group , isomorphic to the set of all complex numbers that have absolute value 1, under multiplication.
The general linear group GL n (R) is the group of all R-linear automorphisms of R n. There is a subgroup: the special linear group SL n (R), and their quotients: the projective general linear group PGL n (R) = GL n (R)/Z(GL n (R)) and the projective special linear group PSL n (R) = SL n (R)/Z(SL n (R)).
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group. PGL(V) = GL(V) / Z(V)
The mirabolic subgroup is used to define the Kirillov model of a representation of the general linear group. As an example, the group of all matrices of the form ( a b 0 1 ) {\displaystyle {\begin{pmatrix}a&b\\0&1\end{pmatrix}}} where a is a nonzero element of the field k and b is any element of k is a mirabolic subgroup of the 2-dimensional ...