Let $G$ be a finite group. A (complex) character $\chi$ of $G$ is said to be a virtual permutation character if it can be expressed as a $\mathbb{Z}$-linear combination of characters induced from the trivial characters of subgroups of $G$, i.e., $\chi = \sum_{H \leq G} n_H \mathrm{ind}^{G}_{H} 1_H$ for some $n_H \in \mathbb{Z}$.

My question is: which finite groups have the property that every character is a virtual permutation character? Is there a classification of such groups? If so, can you provide a reference?

One well-known example is the symmetric group $S_n$. Moreover, it is clear that the class of groups we're looking for is closed under direct products. But are there other examples?

Note that every virtual permutation character is rational-valued, but the converse is false. For example, the every character of the quaternion group of order $8$ is rational-valued, but the unique non-linear character is not a virtual permutation character. Moreover, non-trivial Schur induces are not the only obstacle: even if the representation attached to $\chi$ takes values in matrices over $\mathbb{Q}$, the character $\chi$ need not be a virtual permutation character. These obstacles are the topic of the article "Rational representations and permutation representations of finite groups" Math. Ann. 364, Issue 1 (2016), 539-558 by Alex Bartel and Tim Dokchitser (see http://arxiv.org/abs/1405.6616).

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    $\begingroup$ A quick search by computer algebra found some further examples: the dihedral group of order $8$, and the split extension of $\langle x, y \rangle \cong C_3 \times C_3$ by a fixed-point-free automorphism $t$ of order $2$, acting as $x^t = x^2$, $y^t = y^2$. $\endgroup$ – Mark Wildon Jul 28 '16 at 20:59
  • $\begingroup$ Certainly doesn't answer the question, but seems at least tangentially relevant: all modular representations are a direct summand of a module that admits a finite resolution by permutation modules arxiv.org/abs/2003.04373. $\endgroup$ – Joshua Grochow Apr 11 '20 at 7:14

No classification of such groups is known. As you say, for every character to be a virtual permutation character, necessary conditions are that

  1. all irreducible characters are $\mathbb{Q}$-valued; equivalently, every element $g$ of the group is conjugate to $g^i$ for all $i$ coprime to the order of $g$; and
  2. all Schur indices are trivial.

Also, as you say, these conditions are not sufficient, in general. Incidentally, we also do not have a classification of the finite groups satisfying the first condition.

Regarding examples: all of the Weyl groups satisfy the first condition above. Of these, in addition to the symmetric groups (Weyl groups of type $A_n$), the Weyl groups of types $B_n$ (isomorphic to $C_2\wr S_n$) and $D_n$ have the property that every character is a virtual permutation character. As for the exceptional types: the Weyl group of type $F_4$ also has that property, while the groups of types $E_6$, $E_7$, and $E_8$ do not.

You can find these facts, with proofs, and many more in the book by Dennis Kletzing, Structure and Representations of $\mathbb{Q}$-Groups, Lecture Notes in Mathematics 1084.

Miscellaneous remarks: in the chain of inclusions

{permutation characters} $\subseteq$ {characters of rational representations} $\subseteq$ {$\mathbb{Q}$-valued characters} $\subseteq$ {all complex characters},

many of the obstructions to these inclusions being equalities can be analysed "locally" at the level of normalisers of cyclic subgroups of your group. For example, the "equivalently..." in condition 1 above already does that for the last inclusion in the chain. The obstruction for the middle inclusion is the theory of Schur indices, and these, too, can be analysed locally, although that analysis is fairly involved – see [1,2]. The obstruction for the left-most inclusion is even more intricate, but one can still draw some conclusion from a local analysis. For example the index of that inclusion is $1$ for the given group $G$ if and only if it is $1$ for all the maximal quasi-elementary subgroups of $G$. This easily follows from Solomon's induction theorem, but one can make much finer statements, as is discussed in my paper with Tim that you have mentioned in the question.

[1] U. Riese and P. Schmid, Schur Indices and Schur Groups, II, J. Algebra 182.

[2] P. Schmid, Schur indices and Schur groups, J. Algebra 169.

  • $\begingroup$ I think I remember there's a whole Springer Lecture Notes about these groups but I don't remember better right now. $\endgroup$ – YCor May 9 '19 at 16:11
  • $\begingroup$ @YCor, do you mean a different one than the one I cite just above the horizontal line? Maybe I should have added it to the references at the bottom of the post? $\endgroup$ – Alex B. May 9 '19 at 16:30
  • $\begingroup$ Oh yes, I didn't see it! $\endgroup$ – YCor May 9 '19 at 16:34

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