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)?

  • $\begingroup$ 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". $\endgroup$ Jun 24 '11 at 0:09
  • $\begingroup$ 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"). $\endgroup$ Jun 24 '11 at 0:12
  • $\begingroup$ Is the above proposition remain true over positive characteristic? Since the exercise in Richard Stanley's book uses character of permutation module over char 0. $\endgroup$
    – user13315
    Jan 9 '15 at 5:27

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).


This site is temporarily in read only mode and not accepting new answers.

Not the answer you're looking for? Browse other questions tagged .