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In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant
Special groups include the general linear group, the special linear group, and the symplectic group. Special groups are necessarily connected. Products of special groups are special. The projective linear group is not special because there exist Azumaya algebras, which are trivial over a finite separable extension, but not over the base field.
Among the examples above the additive, multiplicative groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its coordinate ring it can be shown that every affine algebraic group is a linear (or matrix group), meaning that it is isomorphic to an algebraic subgroup of the general linear group.
The group GL(n, F) and its subgroups are often called linear groups or matrix groups (the automorphism group GL(V) 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 ...
The group GL n (K) itself; The special linear group SL n (K) (the subgroup of matrices with determinant 1); The group of invertible upper (or lower) triangular matrices; If g i is a collection of elements in GL n (K) indexed by a set I, then the subgroup generated by the g i is a linear group.
Chapter II describes the invariants of the special and general linear group of a vector space V on the polynomials over a sum of copies of V and its dual. It uses the Capelli identity to find an explicit set of generators for the invariants. Chapter III studies the group ring of a finite group and its decomposition into a sum of matrix algebras.
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane.
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 ]