Michael Larsen, February 20, 2024

Abstract.   It is easy to see that \((\mathbb{C}^2)^{\otimes n}\) decomposes into irreducible pieces as a representation of \(\text{GL}_2(\mathbb{C})\), and the number of these pieces equals the largest coefficients in row n of Pascal's triangle. Moreover, asymptotically, this coefficient grows like \(C n^{-1/2} 2^n\).

The situation in positive characteristic is not yet fully understood. The pieces need no longer be irreducible, just indecomposable. Coulumbier, Ostrik, and Tubbenhauer recently conjectured that for given p, the number of indecomposable factors lies between \(A n^{-\delta} 2^n\) and \(B n^{-\delta} 2^n\), and Etingof gave a conjectural value of \(\delta\), namely \(\log_4 (8/3)\). The same authors have informed me that they know how to prove their conjecture, and in characteristic 2, I can give a more precise result, namely an asymptotic formula for the number of pieces.