# General theory for cyclic modules of the group algebra of the symmetric group

Let $$\mathbb{C}[\mathfrak{S}_n]$$ be the group algebra of the symmetric group. An element of this algebra is of the form

$$v = \displaystyle \sum_{g \in \mathfrak{S}_n} a_g g,$$

where $$a_g \in C$$. For a fixed $$v$$, we consider the cyclic module $$\mathbb{C}[\mathfrak{S}_n] \cdot v$$, where $$h \cdot v = \displaystyle \sum_{g \in \mathfrak{S}_n} a_g (hg)$$ (extended linearly).

For example, if $$H$$ is a subgroup of $$\mathfrak{S}_n$$ and $$a_g = \begin{cases} 1 & \textrm{if g \in H} \\ 0 & \text{else} \end{cases}$$, then $$\mathbb{C}[\mathfrak{S}_n] \cdot v$$ is a permutation module, isomorphic to the representation generated by the cosets $$\mathfrak{S}_n/H$$.

A more complicated example is the Specht modules, which are the simple $$\mathbb{C}[\mathfrak{S}_n]$$-modules. (See the first part of this question : Are idempotents in the group algebra of $S_n$ equivalent to Specht modules? for a fast description of Specht modules from this point of view).

Constructing other examples seems hard to do naively (at least for me). For example, if $$n=3$$, then one can think of $$v = 2 \textrm{Id} + (12)$$, but then $$\mathbb{C}[\mathfrak{S}_n] \cdot v \cong \mathbb{C}[\mathfrak{S}_3]$$.

Whenever I try to find new reprensentations, I end up finding modules that are either isomorphic to $$\mathbb{C}[\mathfrak{S}_n]$$, a permutation module, a permutation module tensored with the sign representation, or a Specht module.

So my question is : Is there a general theory of cyclic modules the form $$\mathbb{C}[\mathfrak{S}_n] \cdot v$$? Or at least, is there other interesting examples that are not in the above list?

• Every module is a direct sum of Specht modules, so are you sure you want that "constructed from representations on this list" on there? Commented Jun 15 at 21:25
• True, I have remove this condition. Commented Jun 15 at 21:43

## 1 Answer

Equivalently, the question is to classify the principal left ideals of $$\mathbb{C}[G]$$, as modules (representations). We have the usual Artin-Wedderburn decomposition

$$\mathbb{C}[G] \cong \prod_i \text{End}(V_i) \cong \prod_i M_{n_i}(\mathbb{C})$$

where $$V_i$$ are the irreducible representations of $$G$$ and $$n_i = \dim V_i$$. A principal left ideal of a finite product of rings is a tuple of principal left ideals in each ring, so the problem reduces to classifying the principal left ideals of a matrix algebra $$M_n(\mathbb{C})$$.

This is standard: the classification as ideals is that the principal left ideal $$M_n(\mathbb{C}) X$$ generated by a matrix $$X$$ is uniquely and freely determined by the kernel $$\ker X \subseteq \mathbb{C}^n$$ (it consists of all matrices whose kernel contains $$\ker X$$), producing a natural bijection between principal left ideals and subspaces of $$\mathbb{C}^n$$. The classification up to module isomorphism is that only the dimension of the subspace (equivalently, the rank of $$X$$) matters: the principal left ideals, as modules, have the form $$(\mathbb{C}^n)^k$$ where $$0 \le k \le n$$, and representatives can be generated by any collection of matrices of each rank $$k$$. Explicitly if we consider diagonal matrices with $$k$$ diagonal entries equal to $$1$$, then the principal left ideal they generate is given by matrices with the corresponding $$k$$ columns arbitrary and other columns zero.

The conclusion is that the principal left ideals of $$\mathbb{C}[G]$$, as representations, are exactly the representations in which each irreducible $$V_i$$ appears with multiplicity $$\le n_i$$.

Edit: Here is an alternative simpler argument which doesn't require this explicit detour into matrix algebras. Equivalently, the question is to classify the cyclic submodules of the regular representation.

Generally, the cyclic (left) modules over a ring $$R$$ are exactly the quotients of $$R$$ as a (left) module over itself. We know that the regular representation of $$\mathbb{C}[G]$$ decomposes as a direct sum of $$n_i$$ copies of the irreducible $$V_i$$. So the conclusion is that the cyclic modules are exactly the quotients of the regular representation, hence are the representations in which each irreducible $$V_i$$ appears with multiplicity $$\le n_i$$ as above. These are also exactly the subrepresentations of the regular representation, so every such cyclic module can also be embedded into the regular representation $$\mathbb{C}[G]$$, and hence every cyclic module is in fact isomorphic to a cyclic submodule of the regular representation.

• Thnak you very much! It helps me formulate the real question I had (I think). Let $k_i \leq n_i$ and $\underline{k} = (k_i)_i$. Let $E_{\underline{k}}$ be the set of all $v \in \mathcal{C}[\mathfrak{S}_n]$ such that $\mathcal{C}[\mathfrak{S}_n] \cdot v$ is a representation such that $V_i$ appears $k_i$ times. Is there some research along these lines? I am pretty sure the whole case is unknown (as there exists representations for which we don't know the decomposition), but maybe results about the sets $E_{\underline{k}}$ are known? Commented Jun 17 at 1:50
• I tried to do some research about that, but couldn't find any result... Commented Jun 17 at 1:51
• @eti902: in principle, for fixed $n$ you can extract descriptions of such $v$ from the argument I gave above about matrices, together with the character table of $S_n$, but I don't know anything about how to do it uniformly in $n$. Commented Jun 17 at 7:23