Search results
Results from the WOW.Com Content Network
Irreducibility (mathematics) In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. . Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are kn
The ring [] / is a field if and only if p is an irreducible polynomial. In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p, and Bézout's identity allows computing r and s such that sp + qr = 1; so, r is the multiplicative inverse of q modulo p.
SU (2) is the universal covering group of SO (3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU (2) for the description of non-relativistic spin in theoretical physics; see below for other physical and historical context.
Thus as an irreducible representation of SO(3), H ℓ is isomorphic to the space of traceless symmetric tensors of degree ℓ. More generally, the analogous statements hold in higher dimensions: the space H ℓ of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric ℓ-tensors.
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. Its roots are all n th primitive roots of unity , where k runs over the positive integers less than n and coprime to n (and i is the imaginary unit ...
For avoiding ambiguity, the term irreducible hypersurface is often used. As for algebraic varieties, the coefficients of the defining polynomial may belong to any fixed field k , and the points of the hypersurface are the zeros of p in the affine space K n , {\displaystyle K^{n},} where K is an algebraically closed extension of k .
Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple ...