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The Character Table of a Subgroup of the Symmetric Group of Degree 8*

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The Character Table of a Subgroup of the Symmetric Group of Degree 8* THE CHARACTER TABLE OF A SUBGROUP OF THE SYMMETRIC GROUP OF DEGREE 8* By T. VENKATARAYUDU, M.A., PH.D., F.A.Sc. From the Department of Physics, Andhra University, Guntur) Received January 5, 1943 Introduction :—POINT groups anL formed by taking the symmetry opera- tions (1) rotations about symmetry axes, (2) reflections in symmetry planes and (3) rotation reflections. It is possible to extend point groups by in- troducing other kinds of symmetry operations. The application of such extended groups to some physical problems will be reserved for a later communication. The aim of the present paper is to extend the point group S8 by introducing a twist about the symmetry axis and to give the character table of this extended group. The abstract form of this group is P8 =Q 2 =R4 =E ; PQ =QP -1 ; RP —P 3R 3 ; RQ =QR 3 . It may be considered to be generated by P,Q,R where P,Q,R are respectively the permutations (15263748), (24) (57), (1234). A general method of calculating the characters of any group based on Young Tableaux is given in Littlewood's recent book on group characters. Burnside has also given another general method in his book on the theory of groups. But both these methods are only of theoretical importance and they are most unsuitable for use in any particular case. Recently Richardson and D.E. Littlewood have calculated the tables of characters of the symmetric groups up to degree 13. Later Littlewood showed how to calculate the characters of subgroups of symmetric groups. Since any group may be regarded as a subgroup of a suitable symmetry group, his methods are somewhat more practicable. They are particularly useful for subgroups of high order. The group characters can be more systemetically obtained if the following theorem could be proved as suggested by Littlewood. "If G is a subgroup of H and G has two linear invariants in an irreducible representation of H, then there is a subgroup G of H containing G which has only one linear invariant." ' One of the papers submitted for the Rannanujam Memorial Prize, 1941. 79 80 T. Venkatarayudu Orthogonal relations :—The following very important orthogonal relations between the characters of a group are well known and they are constantly used in obtaining the character table:— (1) There are as many irreducible representations of a group as there are conjugate classes of elements in it. (2) The character of the identity element is equal to the dimension of the irreducible representation and is always a divisor of the order of the group. (3) If x Prepresents the character of any element S belonging to a class p in the irreducible representation A and if h p 'is the number of elements in the class p (a) E XP(X) XP.(X) __ 0 P P (b) hP 2\ XP( ^` )2 = h (the order of the group) (4) (a) Ep hp Xp XP(µ)== 0 A p. (b) hp XP(X1 2 •v = h. The character table for the extended group Class .. ( Cl I Ca Ca I CA (Ce Cc I Cv Ca Ca Cio Cn I C12 I C13 I Cu CU Cu Order 11 I 4 I 2 I 4 I 8 I 8 I 8 I 2 I 4 I 2 I 4 I 4 I 1 4 1 4( 4 A l 1 1 1 1 1 1 1 1 1 lI 1 1 1 1 1 1 1 A a 1 —1 1 —1 1 1 —1 1 —1 1 1 —1 1 1 —1 —1 A$ 1 1 1 —1 —1 1 --1 1 1 1 —1 1 1 —1 1 —1 A, 1 —1 1 1 —1 1 1 1 —1 1 —1 —1 1 —1 —1 1 B1 1 1 1 1 1 —1 —1 1 —1 1 —1 1 1 —1 —1 1 B 2 1 —1 1 —1 1 —1 1 1 1 1 —1 —1 1 —1 1 —1 Bs 1 1 1 —1 —1 —1 1 1 —1 1 1 1 1 1 —1 --1 B4 1 —1 1 1 —1 —1 —1 1 1 1 1-1 1 1 1 1 E l 2 0 —2 0 0 0 0 2 2 —2 0 0 2 0 —2 0 E g 2 0 —2 0 0 0 0 2 —2 —2 0 0 2 0 2 0 E3 2 0 —2 0 0 0 0 —2 0 2 2 0 2 —2 0 0 E4 2 0 —2 0 0 0 0 —2 0 2 —2 0 2 2 0 0 E5 2 2 2 0 0 0 0 —2 0 —2 0 —2 2 0 0 0 E B 2 —2 2 0 0 0 0 —2 0 —2 0 2 2 0 0 0 G2 4 0 0 —2 0 0 0 0 0 0 0 0 —4 ( 0 0 —2 Extension of the point group S8 :—An example of a point system coming under the group S 8 is a set of 8 particles of the same kind which occupy the corners of a puckered octagon made up of two equal squares one turned through 45° with respect to the other. The planes of the squares are parellel but not coincident. If the particles at the corners of the squares be represented by the numbers 1, 2, 3, 4 in the lower square and 5, 6, 7, 8 in the upper square, the point group which is of order 16 is easily seen to Character Table of Subgroup of Symmetric Group of Degree 8 81 be generated by the two permutations (15263748) and (24) (57) (68). A new symmetry operation, namely a twist about the symmetry axis, may be perform- ed as follows.- Keep one of the squares, say the square containing the particles 5, 6, 7, 8 stationary and rotate the other square containing the particles 1, 2, 3, 4 about the symmetry axis through a right angle. The operation may be represented by the permutation R =_ (1234). The group generated by the three operations P, Q and R is of order 64 and the elements of this group fall into 16 conjugate classes. They are CI =E C2 = (R, RPs, R 3P 2, R3) C3 = (R2, R 2P4) C 4 = (RQP 2, RQP6, R3QP 2, R3QP6) C5 __ (Q, QP 2, QP4, QP6, R 2Q, R 2QP 2, R 2QP4, R 2QP6) C6 :._ (P, P 3, P5, P 7, R2 P, R 2P 3, R 2P, R2P7) 7 5 C = (RQP, RQl' 3, RQP5, RQP', R 3QP, R 3QP 3, R3QP , R 3QP , ) C8 = (P 2, P6) C9 = ( RP, RP5, R3P 3, R 3P7) C10 = (R2P 2, R2P6) C11 = (R2QP, R 2QP 3, R2QP5, R 2QP7) C12 = (RP 2, RP4, R3P4, R3P6) C13 P4 C14 _ (QP, QP 3, QP5, QP') C15 -- (RP 3, RP 7, R3P, R3P5) C16 == (RQ, R 3Q, RQP4, R3QP4) Since there are 16 conjugate classes of elements in the group, there will be 16 irreducible representations for this group. The following eight corres- pondences obviously define eight one dimensional irreducible representations P-±1,Q-±landR-±1. Since all the possible combinations are taken, there can be no other one dimensional irreducible representation. Now the only way of expressing 64 which is the order of the group as the sum of 16 squares of integers of which eight are equal to unity is the following one 64=8.1 2 +6.2 2 +2.4 2 . t In the practical case of sulphur, which conforms to the above model, this new operation does not exist because there are chemical bonds connecting 15, 25, 26, etc. 82 T. Venkatarayudu This relation shows that the group has 8 one dimensional and 6 two dimensional and 2 four dimensional irreducible representations. Since the character of the identity element is equal to the dimension of the corres- ponding irreducible representation, the first column in the character table is at once written. The correspondence 1 0 0--i P-. and Q.-- 0 1 1 0 defines a two dimensional irreducible representation of S 8 . Now if we make R to correspond to the two matrices. 1 0 —1 0 and 0 1 0 —1 we obtain the irreducible representations E l and E. and the characters in the two irreducible representations are now written. From the orthogonality relations it follows that the remaining elements in the columns 2, 3, 5, 6, 7 are all zero. Following Littlewood's method, the other elements in the character table are obtained after a few trials from the compound characters derived from the characters of the symmetric group of degree 8. It may be mentioned that Littlewood has given the character tables of the subgroups of order 1344, 1152, 192, 384 and 336 of the symmetric group of degree 8. The character table of the group of order 64 considered in this paper has not been previously published. An effective check for the correctness or otherwise of the character table is finally made with the help of the orthogonality relations of the group characters. My grateful thanks are due to Prof. S. Bhagavantam for the constant help and encouragement he has been giving me..
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