Irreducible character

From the direct-product characters X(S) belonging to a particular electronic configuration (e.g., ai2a22e2), one must still decompose this list of characters into a sum of irreducible characters. For the example at hand, the direct-product characters X(S)... 

For each element H in Ti, D possesses an irreducible character Th such that, for each irreducible submodule K of //, xk = T h] cf. Lemma 8.3.5 together with Theorem 8.6.1. 

Theorem 8.6.4 A ssume that, for each irreducible character x °f D, the characteristic of C does not divide xio i) Then we have the following. 

Let x be an irreducible character of CS. Since CS is semisimple, we obtain from Theorem 8.6.2(i) together with Theorem 8.6.4(h) that there exists exactly one maximal homogeneous submodule Hx of the OS -module CS such... 

From Theorem 8.5.6(i) we know that there exists, for each irreducible character x of CS, a non-negative integer mx such that... 

Lemma 9.1.6 Let x be an irreducible character of CS. Then the following hold. 

Proof. We are assuming that Irr(CS) = 2. Thus, there exists an irreducible character x °f CS such that lcs>x = Irr(CS ). 

In the following proposition, we completely compute the values of all nonlinear irreducible characters of CS, provided that C is an algebraically closed field of characteristic 0. 

Proposition 12.4.2 Let C be an algebraically closed field of characteristic 0, and let x be an irreducible character of CS of degree 2. Then there exists an element c in Cd such that the following hold. 

In Proposition 12.4.2, we have computed all values of all irreducible characters of CS of degree 2. This allows us to compute, for each of these characters, the left hand side of the equation given in Theorem 9.1.7(h). We shall do this in the following proposition. 

Suppose now that d is even and that 4 < d. Then, according to Lemma 12.4.l(iii), CS possesses at least one irreducible character °f degree 2. 

The normalized irreducible characters Xy express the way in which these specific features propagate when N and S change. Therefore they axe called propagation coefficients [65]. 

Figure 1.23 The sequence of worksheet displays in the calculation of the square symmetric and antisymmetric powers of the IHg irreducible character of Ih- Note that Figure 1.23c is generated from the action of the antisymmetric powers command button in the options menu displayed in Figure 1.6b.

The GT Calculator has been designed to deal with real reducible representations, in which both the components of any given separably degenerate irreducible representations, such as E in C2h appear with equal weight. Data input and output conventions ensure that only such real representations are used. Thus, all the relevant worksheet input cells for these calculations are linked and the second member of each linked pair is set equal to the first. For such calculations, the red-borders marking cells on the worksheets for each type of calculation straddle only one each of these pairs of representations. This is illustrated in Figure 1.28 for the example of the construction of the direct sum character by combination of the irreducible characters of the point group Csh. 

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