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The trivial representation is given by () = for all .. A representation of degree of a group is a homomorphism into the multiplicative group: = = {}. As every element of is of finite order, the values of () are roots of unity.
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 ...
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 ...
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 abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules.A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules.
In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is the identity on the subgroup.
There is a simple rule for computing (,) for any Young diagram (Hamermesh 1989): the result is the sum of all Young diagrams that are obtained from by removing one box and then adding one box, where the coefficients are one except for itself, whose coefficient is # {}, i.e., the number of different row lengths minus one.
A cyclic number [1] [2] is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. [3] Any prime number is clearly cyclic. All cyclic numbers are square-free. [4] Let n = p 1 p 2 …
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