Consider the category $\mathrm{Rep}_{\mathbb{C}}(GL_n)$ of representations of GL_n over complex numbers. Then a theorem of Rajan(See https://doi.org/10.4007/annals.2004.160.683) says that if $V_1, V_2, \dots, V_n$ and $W_1, W_2, \dots, W_m$ are irreducible representations of GL_n of nonzero highest weights such that $V_1 \otimes \dots \otimes V_n \cong W_1 \otimes \dots \otimes W_m$, then $m=n$ and $V_i \simeq W_{\sigma(i)} \otimes \det^{\alpha_i}$, where $\alpha_i$ are integers and $\sigma$ is a permutation of $n$.
Recall that Deligne had constructed the semisimple rigid abelian tensor categories(See https://publications.ias.edu/deligne/paper/438). Can we say something similar in the context of $\mathrm{Rep}(GL_t)$ where $t \in \mathbb{C} \setminus \mathbb{N}$? More precisely I have the following question
$\it{Question}$ : Let $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_m$ be irreducible objects of the category $\mathrm{Rep}(GL_t)$(for $t \in \mathbb{C} \setminus \mathbb{Z}$) over some algebraically closed field of characteristic zero. Suppose $X_1 \otimes \dots \otimes X_n \cong Y_1 \otimes \dots \otimes Y_m$. Is $m=n$? Are there any relation between individual objects $X_i$ and $Y_j$ as in the above mentioned result?
More generally, tet $(\mathcal{C}, \otimes)$ be a rigid abelian category. You may even assume semisimple. I wonder if we impose the condition of "unique factorization" as above then would it be possible to say that there cannot be too many such categories. Perhaps something in terms of growth of lengths of tensor power of objects?
I have found following related query but with no answers : Has the notion of a unique factorization category been defined and studied?.
Apologies if the question is too naive. Any comments, references and suggestions are welcome. Thank you!
