Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

By Schur–Weyl duality there is an isomorphism of $\mathrm{GL}(V) \times S_m$-representations

$$V^{\otimes m} \cong \bigoplus_\lambda \Delta^\lambda(V) \boxtimes S^\lambda$$

where the sum is over all partitions $\lambda$ of $m$ with at most $n$ parts, $\Delta^\lambda$ is the Schur functor for $\lambda$, $S^\lambda$ is the irreducible $\mathbb{C}S_m$-module canonically labelled by $\lambda$ and $\boxtimes$ denotes the outer tensor product of two representations. Restricting to $S_n \times S_m$ we get

$$V^{\otimes m} \cong \bigoplus_\lambda \bigl( \Delta^\lambda(V) \bigl\downarrow^{\mathrm{GL}(V)}_{S_n} \bigr) \boxtimes S^\lambda.$$

Hence

$$[V^{\otimes m} : S^\nu \boxtimes S^\lambda] = [\Delta^\lambda(V) : S^\nu]$$

for all partitions $\nu$ of $n$ and $\lambda$ of $m$ with at most $n$ parts.

Determining the multiplicities on the right-hand side is, as far as I know, an open problem, equivalent to computing inner plethysms of symmetric functions. Some special cases are known. For instance, if $\lambda = (1^m)$ and $m \le n$ then $\Delta^{(1^m)}(V) = \bigwedge^mV = S^{(n-m,1^m)} \oplus S^{(n-m+1,1^{m-1})}$. This result has been reproved, many, times (see Proposition 5.1) in the symmetric group literature.

Let $\nu$ be a partition of $n$ and $\lambda$ a partition of $m$. Mark Wildon points out, the multiplicity of $S^\nu \otimes S^\lambda$ in $V^{\otimes m}$ is equal to the multiplicity of an irreducible $S_n$ module $S^\nu$ in the irreducible $Gl_n$ module indexed by the partition $\lambda$.

The latter restatement of this multiplicity is sometimes referred to as "the restriction problem" since it is determining the decomposition of the restriction of an irreducible $Gl_n$ module to $S_n$. People say (me included) that there does not yet exists a satisfactory combinatorial method or positive integral formula for computing this multiplicity (and this is what it means when it is said that the problem is "considered open"), but there is more to say about it because formulae for computing it do exist.

This multiplicity can be expressed in terms of inner plethysm of symmetric functions. If we denote $s_\lambda\{s_\nu\}$ as the Frobenius image of the character of the symmetric group module $\Delta^\lambda(S^\nu)$ (the Schur functor indexed by the partition $\lambda$ applied to an irreducible $S_n$ module indexed by the partition $\nu$), then $$\left< s_\lambda\{s_{(n-1,1)} + s_{(n)}\}, s_\nu \right> = \sum_{\gamma \vdash n} \chi^\nu(\gamma)\frac{s_\lambda[\Xi_\gamma]}{z_\gamma}$$ because the Frobenius characteristic of $V$ is $s_{(n-1,1)} + s_{(n)}$. The right hand side of this expression comes from character theory of the symmetric group $S_n \subseteq Gl_n$, where I have used here the notation $s_\lambda[\Xi_\gamma]$ to denote the evaluation of a Schur function at the eigenvalues of a permutation matrix with cycle structure $\gamma$.

There is an additional computational formula due to Littlewood [1] (for more modern treatment see [2]) that can be used to compute the multiplicity of an irreducible $S_n$ module in an irreducible $Gl_n$ module in terms of operation of outer plethysm of symmetric functions.

The formula is (Theorem XI in [1]$^{(*)}$ and Theorem 4.1 in [2]): \begin{equation} [ \Delta^\lambda(V), S^\nu ] = \left< s_\lambda, s_\nu[1+s_1+s_2+s_3+\cdots] \right> \end{equation} where the square bracket $f[g]$ represents the operation of outer plethysm (in notation from Macdonald's book this is denoted $f \circ g$).

This result is proved and reproved in the literature:

[1] D. E. Littlewood, Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric Groups, Canad. J. Math., 10, 1958, 17–32.

[2] T. Scharf, J. Y. Thibon, A Hopf-algebra approach to inner plethysm. Adv. in Math. 104 (1994), pp. 30–58.

(*) Technically Littlewood's result (in modern notation) says $\left< s_\lambda\{ s_{(n-1,1)} \}, s_\nu \right> = \left< s_\lambda, s_\nu[ s_1 + s_2 + \cdots ] \right>$, but as Scharf and Thibon point out, this is equivalent to $\left< s_\lambda\{ s_{(n-1,1)} + s_{(n)} \}, s_\nu \right> = \left< s_\lambda, s_\nu[ 1 + s_1 + s_2 + \cdots ] \right>$.

If you wish to compute any of these multiplicities in Sage for a fixed $\lambda$ and $\nu$ (below $\lambda = (3,2)$ and $\nu = (2,1)$), here are three functions which implement the formulae that I have stated above.

sage: s = SymmetricFunctions(QQ).s()
sage: def eq1LHS(la, nu):
....:     n = sum(nu)
....:     return s(la).inner_plethysm(s[n-1,1]+s[n]).scalar(s(nu))

sage: def eq1RHS(la, nu):
....:     n = sum(nu)
....:     return s(la).character_to_frobenius_image(n).scalar(s(nu))

sage: def eq2RHS(la, nu):
....:     m = sum(la)
....:     return s(nu).plethysm(1+sum(s[r] for r in range(1,m+1))).scalar(s(la))

sage: [eq1LHS([3,2],[2,1]), eq1RHS([3,2],[2,1]), eq2RHS([3,2],[2,1])]
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