Number of irreducible components of tensor powers of the natural representation Is it true for $ GL_n $ that $ V^{\otimes k} $ is the direct sum of $ k! $ irreps?
Here $ (GL_n(\mathbb{C}),V=\mathbb{C}^n) $ is the natural module, and also we specialize to $ n \geq 2 $. Since $ GL_n(\mathbb{C}) $ is a complex reductive group then the $ k $th tensor power $ V^{\otimes k} $ decomposes as the direct sum of some number of irreducible representations.
When $ k=1 $ the number is $ k!=1!=1 $ because $ (GL_n(\mathbb{C}),V=\mathbb{C}^n) $ is irreducible. When $ k=2 $ the number is $ k!=2!=2 $ because $ V \otimes V $ is isomorphic to the adjoint representation of $ GL_n $ which decomposes into irreducibles as a $ 1 $ dimensional trivial representation direct sum with the adjoint representation on $ \mathfrak{sl}_n $ (recall that we assumed earlier $ n \geq 2$ ).
Since $ U_n $ is the compact form of $ GL_n $ then the above should be equivalent to the question: Is it true for $ U_n $ that $ V^{\otimes k} $ is the direct sum of $ k! $ irreps? Here $ (U_n,V=\mathbb{C}^n) $ is the natural module, and also we specialize to $ n \geq 2 $.
Similarly, does this hold for the natural module $ (SL_n(\mathbb{C}),V=\mathbb{C}^n) $ equivalently $ (SU_n(\mathbb{C}),V=\mathbb{C}^n) $? In other words,
Is it true for $ SL_n $ that $ V^{\otimes k} $ is the direct sum of $ k! $ irreps?
 A: The claim cannot hold: For any $G$ the number of $G$-irreps in the decomposition is no larger than $\dim (V^{\otimes k}) = n^k$, but for any fixed $n$, $k! > n^k$ for sufficiently large $k$.
In general the number $a(n, k)$ of $GL_n$-irreps into which $V^{\otimes k}$ decomposes is the number of Young tableaux with $n$ cells of height $\leq k$. See OEIS A182172, which records $a(n, k)$ for small $n, k$. Since each tableau of height $\leq k$ has height $\leq k + 1$, for each fixed $n$, $a(n, k)$ is a nondecreasing sequence in $k$. In fact each Young diagram with $n$ cells corresponds to a $GL_n$-irrep, and the multiplicity of each irrep in $V^{\otimes k}$ is just the number of Young tableaux with the shape of the corresponding diagram.
The height of any Young tableau with $n$ cells is $\leq n$, i.e., the number $a(n, k)$ of irreps into which $V^{\otimes k}$ decomposes is the same as the number $a(n) := a(n, n)$ of irreps into which $V^{\otimes n}$ decomposes for all $k \geq n$, and so for each $n$, $a(n) \geq a(n, k)$ for all $k$. These counts coincide respectively with the involution numbers (a.k.a. telephone numbers) $a(n) := a(n, n)$, i.e., the numbers of permutations of $n$ letters whose square is the identity permutation (OEIS A000085). For $n = 1, \ldots, 10$, these counts are, respectively, $1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496$.
Chowla showed that as $n \to \infty$,
$$a(n) \sim \frac{1}{\sqrt{2} \sqrt[4]{e}} \left(\frac{n}{e}\right)^{n / 2} \exp \sqrt{n} .$$ Using Stirling's approximation yields an asymptotic for $a(n)$ in terms of the factorial function:
$$a(n) \sim \frac{1}{\sqrt[4]{8\pi e}} \frac{\exp \sqrt{n}}{\sqrt[4]{n}} \sqrt{n!} .$$
For fixed $0 \leq k \leq 12$, the sequences $a(n, k)$ can be found in the OEIS, including:

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*$a(n, 1)$: the all $1$'s sequence; OEIS A000012

*$a(n, 2)$: the values ${n\choose{\left\lfloor n / 2\right\rfloor}}$; OEIS A001405

*$a(n, 3)$: the Motzkin numbers (number of ways of drawing any number of nonintersecting chords joining $n$ labeled points on a circle); OEIS A001006

*$a(n, 4)$: OEIS A001998.

