# Relationship between spherical tensors and 3rd tensor power of cartesian tensors

I have some familiarity with these things from a course I took in Sakurai a few years ago.

Cartesian tensors form a 3d irrep of $$\mathrm{SO}(3)$$. The angular momentum operators $$J_+,J_-,J_z$$ form a 3d irrep of $$\mathrm{SO}(3)$$. The spherical tensors $$T^1_q, q=1,0,-1$$ also form a 3d irrep of $$\mathrm{SO}(3)$$. These three 3d irreps of $$\mathrm{SO}(3)$$ are all isomorphic and I have seen some of them described as the same.

Dyadic cartesian tensors (2nd tensor power) are also a well known case. They form a 9d reducible representation of $$\mathrm{SO}(3)$$, see Decomposition of a Cartesian tensor. The dyadic tensor representation decomposes into irreducibles as $$5+3+1$$. The $$5$$ corresponds to $$T^2_q, q=0,\pm 1, \pm 2$$ (traceless symmetric part) the $$3$$ corresponds to $$T^1_q, q=0,\pm 1$$ (antisymmetric part) and the $$1$$ corresponds to scalar part.

What about a tensor product of three cartesian tensors (3rd tensor power)? This should be a reducible 27d representation reducing as $$(3*5)+(3*3)+(3*1)$$ which should split into irreducibles as $$(7+5+3) + (5+3+1) + 3$$. There are many different copies of the 3d irrep in this decomposition. All the different "3"s are in some sense $$T^1_q,q= \pm 1, 0$$, but somehow they are all different?

For example, for the 1st tensor power we would write $$T_1^1=J_+ \\ T^1_0=J_z \\ T^1_{-1}=J_-$$ But for the second tensor power we would write the $$3$$ part of the decomposition into irreducibles (remember this part corresponds to antisymmetric tensors) as $$T_1^1=J_zJ_+-J_+J_z=[J_z,J_+]=J_+ \\ T^1_0=J_+J_- - J_-J_+=[J+,J_-]=2J_z \\ T^1_{-1}=J_zJ_--J_-J_z-=[J_z,J_-]= -J_-$$ so the $$T^1_{q}, q= \pm 1, 0$$ span the same space for the 1st and 2nd tensor power. What is going on here?

In the 3rd tensor power there are three different copies of the irrep $$3$$. Will all of these three copies of the $$3$$ irrep also be "redundant" in the sense of each corresponding to the same space, the span of $$J_+,J_z,J_-$$?

If not, what do these three different copies of the irrep $$T^1_{q}, q= \pm 1, 0$$ inside of the 3rd tensor power actually look like?

This question Quantum mechanics angular momentum spherical tensor components is relevant but only explicitly does the $$7$$ part of the third tensor power (corresponding to $$T^3_q, q= \pm 3, \pm 2, \pm 1, 0$$) not any of the three $$3$$ irreps.

• Do as your link suggests: Add three spin 1 states, Clebsch, and for your three spin 1s in the decomposition, parse out the symmetric and the two mixed-symmetry ones. It is a mere basis change. Commented Oct 15, 2023 at 20:22