## Tensor products of permutation representations of symmetric groups.

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I am looking for a reference for the following fact which must be classical (which makes it harder, for me, to track a reference down). I am interested because there are similar (more complicated) statements about the cohomology of symmetric groups.

If $P$ is a partition, namely $p_{1} + \cdots + p_{k} = n$, we let $\rho_{P}$ denote the permutation representation of $S_{n}$, induced up from the trivial representation of $S_{P}$.

If $P$ and $Q$ are partitions of $n$ then consider any matrix $\hat{A}$ with nonnegative integer entries such that the entries of $i$th row of $A$ add up to $p_{i}$ and those of the $j$th column of $A$ add up to $q_{j}$. Then the entries of $\hat{A}$ form another partition of $n$, which we call $A$ and say that $A$ is a product-refinement of $P$ and $Q$. For example if $P = Q = 1 + 2$ then two possibilities for $\hat{A}$ are $\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 0 & 1 \ 1 & 1 \end{pmatrix}$.

Proposition: If $\rho_{P}$ and $\rho_{Q}$ are permutation representations of $S_n$ then $\rho_{P} \otimes \rho_{Q} \cong \bigoplus_{A} \rho_{A},$ where the sum is over $A$ which are product-refinements of $P$ and $Q$.

Questions: 1) what is the reference for this fact? and 2) what is standard terminology (for product-refinement in particular)?

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The "Frobenius characteristic" is an isomorphism between the space of class functions of $S_n$ and the homogeneous symmetric functions of degree $n$. It sends the character of $\rho_P$ to the product of complete sums $h_P=h_{P_1} \cdot h_{P_2} \cdots$. By transport of structure it gives a new product on the symmetric functions of degree $n$.   So you might find some references for this result by looking for "Kronecker products of complete sum symmetric functions".  – Emmanuel Briand Jun 24 2011 at 0:09
I meant "It gives a new product on the symmetric functions of degree $n$, called the Kronecker product of symmetric functions." (or sometimes "internal product of symmetric functions"). – Emmanuel Briand Jun 24 2011 at 0:12

## 1 Answer

Hi Dev,

It looks to me like a proof of this fact is given in the answer to Exercise 7.84(b) of Richard Stanley's Enumerative Combinatorics, volume 2, along with a reference to Example I.7.23(e), page 131, of I. G. Macdonald's Symmetric Functions and Hall Polynomials (2nd edition).

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