5
$\begingroup$

Let $G$ be a subgroup of the symmetric group $S_n = \operatorname{Sym}(X)$.

Let $k$ be a field, and let $V$ be the permutation module corresponding to $X$.

Then $V$ is not irreducible, it has a $1$-dimensional submodule $W$ generated by $\sum_{x \in X} x$, and a submodule $Z$ of codimension $1$ consisting of $\sum_{x \in X} \lambda_x x $ with $\sum_{x \in X} \lambda_x = 0$.

The quotient $Z/W \cap Z$ is an irreducible $k[S_n]$-module. (If $n$ is coprime to the characteristic of the field, then $V = W \oplus Z$.)

Is there a classification of $G \leq S_n$ for which $Z/W \cap Z$ is an irreducible $k[G]$-module? What are the examples with $G$ solvable?

Improve this question
$\endgroup$
3
  • $\begingroup$ At least in coprime characteristic, you're asking for permutation groups which have rank 2: these are precisely the 2-transitive groups. These are classified, for example in "Permutation Groups" by Dixon and Mortimer. An infinite family of solvable examples are the groups $AGL_{1}(q)$, and a theorem of Burnside shows that any solvable example has prime power degree. $\endgroup$
    – Padraig Ó Catháin
    Commented May 21, 2022 at 12:22
  • $\begingroup$ @PadraigÓCatháin I think you also need to assume that the field is algebraically closed, otherwise there are other example, such as a cyclic group of prime order with $k = {\mathbb Q}$. $\endgroup$
    – Derek Holt
    Commented May 21, 2022 at 12:30
  • 1
    $\begingroup$ Yes - I had $\mathbb{C}$ in mind, I should have been more careful! $\endgroup$
    – Padraig Ó Catháin
    Commented May 21, 2022 at 12:36

1 Answer 1

Reset to default
7
$\begingroup$

As Padraig Ó Catháin explained in a comment, in coprime characteristic, this module is irreducible whenever $G$ is doubly transitive, and this is a sufficient condition when the field $k$ is algebraically closed.

For doubly transitive groups in the modular case, the module is usually irreducible, but there are some exceptions, and these are classified completely in the paper

The Modular Permutation Representations of the Known Doubly Transitive Groups Brian Mortimer Proceedings of the London Mathematical Society, Volume s3-41, Issue 1, July 1980, Pages 1–20, https://doi.org/10.1112/plms/s3-41.1.1/

There is an old classification by Huppert of the solvable doubly transitive finite groups.

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ Thanks! After some more searching I find a recent paper which finishes off a more general problem: "Kleshchev, Alexander; Morotti, Lucia; Tiep, Pham Huu: Irreducible restrictions of representations of symmetric and alternating groups in small characteristics. Adv. Math. 369 (2020), 107184, 66 pp." In particular the statement there contains an answer to my question in the case where the field is algebraically closed. (The new results of that paper are in characteristic $p = 2$ and $p = 3$, I think.) $\endgroup$
    – spin
    Commented May 22, 2022 at 3:43
  • 1
    $\begingroup$ By the way, it seems the paper by Mortimer on 2-transitive groups left a few cases unsettled. The paper by Kleshchev/Morotti/Tiep notes that these can be handled with later results, referring to a work of G. Hiss for Ree groups and a GAP calculation for $Co_3$. In Table II, page 5, they give an updated version of the table by Mortimer. $\endgroup$
    – spin
    Commented May 22, 2022 at 3:48

You must log in to answer this question.

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