7
$\begingroup$

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)?

Improve this question
$\endgroup$
3
  • $\begingroup$ 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". $\endgroup$
    – Emmanuel Briand
    Commented Jun 24, 2011 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"). $\endgroup$
    – Emmanuel Briand
    Commented Jun 24, 2011 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
    Commented Jan 9, 2015 at 5:27

1 Answer 1

Reset to default
8
$\begingroup$

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).

Improve this answer
$\endgroup$

You must log in to answer this question.

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