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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
In the theory of algebraic groups, a special group is a linear algebraic group G with the property that every principal G-bundle is locally trivial in the Zariski topology. 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 ...
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 mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special [1] automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. [2]
Reductive groups include the most important linear algebraic groups in practice, such as the classical groups: GL(n), SL(n), the orthogonal groups SO(n) and the symplectic groups Sp(2n). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them.
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.
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.
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 ...