Irreducibility criterion

Equation (27) is known as Johnston s irreducibility criterion (Johnston (I960)). 

Finally we establish a criterion for deciding whether two e-operators belong to the same irreducible representation. To do this, we simply note that, because of (14), e jc z ki 4m =4m) on the other hand, if e% and e m belong to two different irreducible representations, then, because of (14) and the fact that every x in U can be expanded in the e s, e jk xe m = 0 for all x e A. special case of this, the only one we will need, is... 

Given the IRs T of H, all the irreducible co-representations F of G can be determined from eqs. (40)-(42). Although the equivalence of T, T and the sign of c(Z) provide a criterion for the classification of the co-representations of point groups with antiunitary operators, this will be more useftd in the form of a character test. 

Even if a system is irreducibly complex (and thus cannot have been produced directly), however, one can not definitively rule out the possibility of an indirect, circuitous route. As the complexity of an interacting system increases, though, the likelihood of such an indirect route drops precipitously. And as the number of unexplained, irreducibly complex biological systems increases, our confidence that Darwin s criterion of failure has been met skyrockets toward the maximum that science allows. 

The smoothness of algebraic matrix groups is a property not shared by all closed sets in /c". To see what it means, take fc = fc and let 5 fc" be an arbitrary irreducible closed set. Let s be a point in S corresponding to the maximal ideal J in k[S]. If S is smooth, n si k = O si /J us) has fc-dimension equal to the dimension of S. (This would in general be called smoothness at s.) If S is defined by equations fj = 0, the generators and relations for OUS] show that S is smooth at s iff the matrix of partial derivatives (dfj/dXi)(s) has rank n — dim V. Over the real or complex field this is the standard Jacobian criterion for the solutions of the system (f = 0) to form a C or analytic submanifold near s. For S to be smooth means then that it has no cusps or self-crossings or other singularities . 

As generally used in stability theory, the diagonal element is negative, and Theorem A.l is used in attempts to show that the radius of the disk (called a Gerschgorin disk) is smaller in absolute value. There are many generalizations that yield finer results at the expense of a more complicated criterion. One of these, useful for our work, involves the concept of an irreducible matrix. 

A criterion for reducibility or irreducibility of a matrix representation is required. It is desired to impose a simple test, which will indicate whether dr not a representation is reducible. It will be proved subsequently that a matrix that commutes with every matrix of an irreducible representation is a constant matrix, and conversely, if there exists a nonconstant matrix—that is, one that is not a simple multiple of the unit matrix—which commutes with all the matrices of a representation then the representation is reducible. This is the central theorem of group representations and we will use this result in other proofs later in the chapter. 

A criterion based on a Mulliken analysis is adopted again to subdivide atomic LWFs associated to the same irreducible atom A into shell-subsets (without loss of generality, A can be assumed to be in reference 0 cell). The bond LWFs are the representatives of double and triple bonds, and are assigned to the (AO —BM)-subset. Thus, part of the LWFs, classihed as SALWFs, only require numerical rehnement, or the subset including all the atypical ones. All others are grouped in subsets S, either of a shell or of a bond type, each one comprising ns members. 

Determination of eigenvalues and eigenfunctions. The irreducible representations for the symmetry points, axes and planes of the Brillouin zone are classified into three types according to the Herring criterion (Herring 1937). These three types... 

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