Regular representation

We will first consider an abelian group, namely the cyclic group Zra. The pi s forming a regular representation commute with each other and... 

Figure 6. Formfactor K(r) and nearest neighbour spacing distribution P(s) for (a) the pi s form the regular representation of the cyclic group Z24 (b) pi s represent the symmetric group. S, (c) a random set of pi s without symmetries. The dashed curve in (b) labeled red. Poisson corresponds to a distribution of degenerate levels being Poisson distributed otherwise.

A particular representation which we shall need is the regular representation of dimension g, which symbolically uses the group elements themselves as basis vectors of a permutation representation . 

The representation so defined is the regular representation jT. It has dimension g, and each row and each column of any DlR)(s) has exactly one element "1 , with the rest being zero. Only the unit element, 1, has diagonal matrix elements. Thus, (l) =g, with the other characters being zero. 

The proof that the sum of the squares of the dimensions of the irreducible representations is equal to the order of the group has three parts (1) intro, duction of the regular representation, (2) the Celebrated Theorem, (3) the final steps. 

The regular representation is a reducible representation composed of matrices constructed as follows first write down the group multiplication table in such a way that the order of the rows corresponds to the inverses of the operations heading the columns in this way will appear only along the diagonal of the table. For example, from Table 3 4.2 we would have... 

Next, the matrix of the regular representation of the operation R is formed from the resulting table by replacing R by unity and all the other operations by zero. For example, in the above case we have... 

This theorem states that the number of times each irreducible representation T " occurs in the regular representation F is equal to the dimension of T (n ). This is easily proved by using eqn (7 4.2). We find, since jre (E) = g and if R E, ]f (R) = 0, that... 

Example 4.4-5 Find the regular representation for the group C3. C3 = E C3+ C3. Interchanging the second and third columns of Table 4.4(b) gives Table 4.4(c). 

For a set of equivalent nuclei in general site the matrices 11(G) are identical with the right regular representation matrices2 lf the nuclear position vectors of all K nuclei of a SRM are included in the basis Xkd), 11(G) denotes a K by K permutation matrix. In addition to the matrix groups (2.49) and (2.49 ) the set... 

Lemma. Let G be an affine group scheme over a field. Every finite-dimensional representation of G embeds in a finite sum of copies of the regular representation. 

Suppose the regular representation of G is a sum of irreducibles. Show that every representation is a sum of irreducibles. 

D. T. Edmonds, N. K. Rogers, and M. J. E. Sternberg, Mol. Phys., 52, 1487 (1984). Regular Representation of Irregular Charge Distributions. Application to Electrostatic Potentials of Globular Proteins. 

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