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Although prehomogeneous vector spaces do not necessarily decompose into direct sums of irreducibles, it is natural to study the irreducible PVS (i.e., when V is an irreducible representation of G). In this case, a theorem of Élie Cartan shows that G ≤ GL(V) is a reductive group, with a centre that is at most one-dimensional. This, together ...
We write () or shortly () for the restriction to the subgroup . It can be proven that the number of irreducible representations of a group G {\displaystyle G} (or correspondingly the number of simple C [ G ] {\displaystyle \mathbb {C} [G]} –modules) equals the number of conjugacy classes of G . {\displaystyle G.}
The vector space of complex-valued class functions of a group has a natural -invariant inner product structure, described in the article Schur orthogonality relations.Maschke's theorem was originally proved for the case of representations over by constructing as the orthogonal complement of under this inner product.
A polynomial decomposition may enable more efficient evaluation of a polynomial. For example, + + + + + + + = () (+ +) can be calculated with 3 multiplications and 3 additions using the decomposition, while Horner's method would require 7 multiplications and 8 additions.
In mathematics, the term cycle decomposition can mean: . Cycle decomposition (graph theory), a partitioning of the vertices of a graph into subsets, such that the vertices in each subset lie on a cycle
The Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements; The Bruhat decomposition = of a semisimple algebraic group into double cosets of a Borel subgroup can be regarded as a generalization of the principle of Gauss–Jordan elimination, which generically writes a matrix as the product of an upper triangular matrix with a lower ...
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The strictly positive values in the Schmidt decomposition of are its Schmidt coefficients, or Schmidt numbers. The total number of Schmidt coefficients of w {\displaystyle w} , counted with multiplicity, is called its Schmidt rank .
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