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Consider a spin $ j $ system. A spin $ j $ spherical tensor $ T^k_q(j) $ of rank $ k $ is a $ (2j+1) \times (2j+1) $ matrix.

Given two spherical tensors of spin $ j $, say $ T^{k_1}_{q_1}(j) $ and $ T^{k_2}_{q_2}(j) $, we can multiply them, since they are just matrices.

It is an interesting fact that the product $ T^{k_1}_{q_1}(j) T^{k_2}_{q_2}(j) $ is always a (real) linear combination of $ T^k_q $ for $ k $ in the range $ |k_1-k_2| \leq k \leq min \{ 2j,k_1+k_2 \} $.

A reference for this fact is equation 16 of 2.4.4. page 45 in Quantum Theory of Angular Momentum by D A Varshalovich, A N Moskalev and V K Khersonskii. However no derivation, explanation or proof is provided in this reference.

Does anyone have a derivation, explanation or proof for the fact $$ T^{k_1}_{q_1}(j) T^{k_2}_{q_2}(j)=\sum_{k=|k_1-k_2|}^{min \{ 2j,k_1+k_2 \}} C(k_1,q_1,k_2,q_2,k,q,j) T^k_q(j) $$ here $ C(k_1,q_1,k_2,q_2,k,q,j) $ is a coefficient with an explicit form in terms of the Clebsch-Gordan coefficients and 6j symbols, for details see the reference.

It is known that the $ \{T_q^k(j): 0 \leq k \leq 2j, -k \leq q \leq k \} $ are an orthonormal basis for the space of $ (2j+1) \times (2j+1) $ matrices. In particular, the $ T^k_q(j) $ form a spanning set. Thus we expect the product $ T^{k_1}_{q_1}(j) T^{k_2}_{q_2}(j) $ to be in the span of some $ T^k_q(j) $ for $ 0 \leq k \leq 2j $. My question is really about the bounds on $ k $ given in the product formula. Why is $ |k_1-k_2|\leq k \leq k_1+k_2 $?

I'm especially interested in understand the upper bound $ k \leq k_1+k_2 $

For example, for $ j=\tfrac{3}{2} $ and $ k_1=1=k_2 $ it is already not obvious/ not intuitive to me why the product $ T^{1}_{q_1}(\tfrac{3}{2}) T^{1}_{q_2}(\tfrac{3}{2}) $ can be expressed as a linear combination of spherical tensors without using any of the seven rank $ 3 $ spherical tensors $ T^3_q(\tfrac{3}{2}) $.

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