Powers of representations of the rotation group (their symmetric, alternating, and other parts)

It has been known for more than three quarters of a century that the square of the irreducible representation D(j), of dimension 2j + 1, of the 3-dimensional rotation group R(3) contains the representations D(2j),D(2j - 1),...,D(1),D(0) of R(3), each occurring once. I give in this paper a method, involving no more than the expansion of (1 + x +... + x(2j))(n-1), of analyzing (D(j))(n), the nth power of D(j), n = 3, 4, 5,..., into the sum of the irreducible representations D(nj), D(nj - 1),..., D(1), D(0) of R(3), each occurring a stated number of times. When n = 2, the part D(2j) + D(2j - 2) +... + D(2) + D(0) of (D(j))(2) is associated with the partition (2) of 2 and is termed the symmetric part of (D(j))(2), while the remaining part, D(2j - 1) + D(2j - 3) +... + D(3) + D(1), of (D(j))(2) is associated with the partition (1(2)) of 2 and is termed the alternating part of (D(j))(2). For any value of n, (D(j))(n) contains various parts, each associated with a k-part partition of n, where k 2j + 1, (D(j))(n) has no alternating part. I show how to determine these various parts giving full details when j = 1 and n is arbitary, and when j is arbitrary and n = 3 or 4. I also show that when n = 2m + 1 is odd the alternating part of (D(j))(n) is, when it exists, i.e., when m