Powers of Representations of the Rotation Group (Their Symmetric, Alternating, and Other Parts)

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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 R3 contains the representations D(2j),D(2j - 1),...,D(1),D(0) of R3, each occurring once. I give in this paper a method, involving no more than the expansion of (1 + x +... + x2j)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 R3, 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 (12) 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, the part associated with the l-part partition (n) of n being termed the symmetric part of (D(j))n and the part associated with the n-part partition (1n) of n, where n ≤ 2j + 1, being termed the alternating part of (D(j))n. If n > 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 ≤ j, the same as the symmetric part of (D(j - m))n and that the alternating part of (D(j))n is the same as the alternating part of (D(j))n′, where n + n′ = 2j + 1. This implies that, when n is even and <2j + 1, the alternating part of (D(j))n is the same as the symmetric part of (D(n/2))2j+1-n.

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