By Louis Auslander
Court cases of the yank Mathematical Society
Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236
Published via: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2035904
Page count number: 7
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Extra resources for An Account of the Theory of Crystallographic Groups
Thus 37 GROUP REPRESENTATIONS Φ'ν Φ'* 8 P a n ar* invariant subspace, and it may easily be verified that they transform according to the irreducible representation Γ (Table 3). Similarly ^'3, φ\ transform according to Γ, φ\ aooording to J% and φ\ according to J'. The whole set φ\> . . 21) » - (φ'ν φ\) + ( f 3, φ'4) + ( f 5) + (φ\). of Γ We have so far discussed some of the properties of reducible and irreducible representations and vector spaces, without indicating how in practioe one might reduce a given representation or prove it to be irreduoible as the case may be.
7). e. φ ='c$j = c'$'j. 13) The question is, how are the coefficients c'j related to the ty? 14) where [D(T)]'1 is the reciprocal of the matrix D(T). 14) is expressed by saying that the φι transform as base vectors and the c% as coefficients according to the representation Dij(T). Equivalence We have seen that the functions #exp(—r), yexp(—r) form a basis for the representation Γ (Table 3) of the group 32. If now in the vector space [x exp(—r), y exp(—r)] we choose different base vectors, say (x ± iy) exp(—r), then we can use the matrices of Γ to 32 GROUP THEORY IN QUANTUM MECHANICS determine the representation according to which the new base vectors transform.
Hint: the functions xtxjnt - WA a n d —V(lß)(xix&* + X\V&% — 2 2/ι^β) transform according to Γ. 6. 19). Show that it is m>t true to say that any function of H is equal to some function of H (1) or some function of K<2) or . . etc. 20 Show that the permutation group p 2 has two one-dimensional representations, the symmetric one in which both elements are represented by + 1 , and the antisymmetric in which the elements (12) and (21) are represented by + 1 and —1 respectively. Write down some functions that transform according to these representations, and prove that no other irreducible representations of p 2 exist.