TITLE: Decomposition of tensor power of symmetric square
QUESTION [7 upvotes]: Studying some representation theory I came up with the following problem.
We work over a field of characteristic $0$. Let $V$ be the standard representation of $\mathrm{GL}_n$ and let $W$ be the representation $(\mathrm{Sym}^2(V))^{\otimes n}$.
Is it possible to describe the weights of the irreducible components of $W$ (after choosing the usual Borel and so on)? In particular I would like to know if the representation $\det(V)^{\otimes 2}$ appears in $W$.

REPLY [9 votes]: By Pieri's formula, a partition with $2n$ elements in $n$ rows, corresponding to a representation of $GL_n$, occurs in this representation with multiplicity equal to the number of ways of obtaining that partition by starting with the empty partition and $n$ times adding two elements, no two in the same column.
For the determinant squared, which corresponds to a partition with $2$ columns of length $n$, this occurs with multiplicity exactly one.