The Classical Groups
Encyclopedia
In mathematics, The Classical Groups: Their Invariants and Representations is a book by , which describes classical invariant theory
in terms of representation theory
. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by Hilbert's solution of its main problems in the 1890s.
gave an informal talk about the topic of his book.
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.
Chapter 4 discusses Schur-Weyl duality between representations of the symmetric and general linear groups.
Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal
and symplectic group
s, showing that the ring of invariants is generated by the obvious ones.
Chapter VII describes the Weyl character formula
for the characters of representations of the classical groups.
Chapter VIII on invariant theory proves Hilbert's theorem that invariants of the special linear group are finitely generated.
Chapter IX and X give some supplements to the previous chapters.
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
in terms of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by Hilbert's solution of its main problems in the 1890s.
gave an informal talk about the topic of his book.
Contents
Chapter I defines invariants and other basic ideas and describes the relation to Klein's Erlanger program in geometry.Chapter II describes the invariants of the special and general linear group
General linear group
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...
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.
Chapter 4 discusses Schur-Weyl duality between representations of the symmetric and general linear groups.
Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
and symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
s, showing that the ring of invariants is generated by the obvious ones.
Chapter VII describes the Weyl character formula
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
for the characters of representations of the classical groups.
Chapter VIII on invariant theory proves Hilbert's theorem that invariants of the special linear group are finitely generated.
Chapter IX and X give some supplements to the previous chapters.