Is there a non-explicit characterization of the Specht modules?

Question

It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$.

My question is, can the association between partitions and irreps be specified without having to explicitly construct the irreps? In other words, is there for each partition $\lambda$ some relatively simple property that is possessed by the Specht module $S^\lambda$ but none of the other irreps of $S_n$?

In particular I was imagining that an answer might come from Jucys-Murphy type theory but I am not familiar enough with the material to know how to (or if it is possible to) use it to characterize $S^\lambda$ without having to give a bunch of background information.

Answer 1

The characterization using Jucys-Murphy elements is this: let $z$ be a formal variable, and $X_i$ the Jucys-Murphy elements. The product $(z-X_1)\cdots (z-X_n)$ is central in the group algebra of the symmetric group, so it acts on any irreducible representation by a polynomial with scalar coefficients. On the Specht module $V_\lambda$, the polynomial it acts by the product of $z+i-j$ where $(i,j)$ are the boxes of the Young diagram. This is one of the consequences of the Okounkov-Vershik theory as mentioned by Steven and Andrew above.

EDIT: By way of normalization, note that in $S_2$ this is $z^2-z\cdot (12)$, so it acts on the trivial rep by $z(z-1)$ and on the sign by $z(z+1)$. The difference comes from the fact that the box $(1,2)$ is in the diagram of the trivial, and $(2,1)$ is in the diagram of the sign. So, if somewhere you flipped this convention (which is very easy), you must change $z+i-j$ to $z-i+j$.

Answer 2

Schur-Weyl duality tells you that under the commuting actions of $GL(V)$ and $S_n$ on $V^{\otimes n}$, it is a direct sum $\bigoplus_{\lambda} S^{\lambda} \otimes V_\lambda$ where $S^\lambda$ is some irreducible representation of $S_n$ and $V_\lambda$ is some irreducible representation of $GL(V)$ and where the $\lambda$ range over some indexing set. Furthermore, the correspondence is a bijection, i.e., $S^\lambda \cong S^\mu$ if and only if $\lambda = \mu$ if and only if $V_\lambda \cong V_\mu$.

So you can transfer the question to $GL(V)$, and for that group there is a more satisfactory answer: $V_\lambda$ is the unique irreducible representation which has a eigenvector for the subgroup of upper-triangular matrices whose generalized eigenvalue is given by $\lambda$, i.e., the eigenvalue is a function $b \mapsto b^\lambda$ where $b^\lambda = \prod_i x_i^{\lambda_i}$ and $x_1, \dots, x_n$ are the diagonal entries of $b$.

This clarifies certain combinatorial operations on partitions (like addition, which corresponds to the fact that the eigenvalue of a tensor product of eigenvectors is a sum) that are used in combinatorial statements like Murnaghan's stability theorem.

Alternatively, you can use Jucys-Murphy elements to use eigenvalues to label representations if you don't want to introduce $GL(V)$.

Answer 3

As Sam told me to do so, I convert my comment to an answer, although it is essentially the same as David Speyer's.

The Specht module $S^\lambda$ is the unique simple module of $\mathfrak S_n$ whose restriction to the $\mathfrak S_\lambda$ contains the trivial and whose restriction to $\mathfrak S_{\lambda^*}$ contains the sign (where the star denotes conjugation of partitions, and with the usual notation for Young subgroups).

To see this, one just needs to know that if $S^\lambda$ occurs in the permutation module $M^\mu$, then $\lambda \geq \mu$ (for the dominance order), that tensoring with sign induces the transposition of partitions, and that the transposition is an anti-automorphism of the poset of partitions. The books by James, or James and Kerber, for example, can be used as a reference for those facts.

Answer 4

I'm not sure what makes a characterization good or bad for you. Here is the fastest way I know to describe $S^{\lambda}$.

Let $\lambda$ be a partition of $n$. Let $M^{\lambda}$ be the permutation representation of $S_n$ on the set partitions of $\{1,2,\ldots, n \}$ into sets of size $\lambda_1$, $\lambda_2$, ..., $\lambda_r$. Let $\epsilon$ be the sign representation of $S_n$. Then $\mathrm{Hom}_{S_n}(M^{\lambda}, M^{\lambda^T} \otimes \epsilon)$ is one dimensional. (Here $\lambda^T$ is the transpose, or conjugate, partition.) Let $\alpha$ be a non-zero $S_n$-equivariant Hom from $M^{\lambda}$ to $M^{\lambda^T} \otimes \epsilon$. The Specht module $S^{\lambda}$ is the image of $\alpha$.

I wrote up some notes constructing $S^{\lambda}$ from this perspective.