Mathematical Methods for Physics and Engineering - Matematica.NET

29.7 COUNTING IRREPS USING CHARACTERSClassesIrrep I AB CDEA 1 1 1 1A 2 1 1 −1E 2 −1 0D 3 0 1Table 29.2 The characters of the irreps of the group 3m and of the representationD, which must be a superposition of some of them.29.7.1 Summation rules for irrepsThe first summation rule for irreps is a simple restatement of (29.14), with µ setequal to λ; itthenreads∑ [χ (λ) (X) ] ∗χ (λ) (X) =g.XIn words, the sum of the squares (modulus squared if necessary) of the charactersof an irrep taken over all elements of the group adds up to the order of thegroup. For group 3m (table 29.1), this takes the following explicit forms:for A 1 , 1(1 2 )+2(1 2 )+3(1 2 )=6;for A 2 , 1(1 2 )+2(1 2 )+3(−1) 2 =6;for E, 1(2 2 )+2(−1) 2 +3(0 2 )=6.We next prove a theorem that is concerned not with a summation within an irrepbut with a summation over irreps.Theorem. If n µ is the dimension of the µth irrep of a group G then∑n 2 µ = g,where g is the order of the group.µProof. Define a representation of the group in the following way. Rearrangethe rows of the multiplication table of the group so that whilst the elements ina particular order head the columns, their inverses in the same order head therows. In this arrangement of the g × g table, the leading diagonal is entirelyoccupied by the identity element. Then, for each element X of the group, take asrepresentative matrix the multiplication-table array obtained by replacing X by1 and all other element symbols by 0. The matrices D reg (X) so obtained form theregular representation of G; theyareeachg × g, have a single non-zero entry ‘1’in each row and column and (as will be verified by a little experimentation) have1097