Let be a finite group. The model character for is . A nice short paper by Inglis, Richardson and Saxl gives a self-contained inductive proof that if is the permutation character of acting by conjugacy on its set of fixed-point-free involutions then
is the model character for . Assuming Pieri’s rule, that if is a partition of then , where the sum is over all partitions obtained from by adding boxes, no two in the same row, this follows from the well-known fact (proved inductively in the paper) that .
Note that is the induction of a linear character from the centralizer of the involution . (When we count the identity as an involution as an honorary involution.) Up to conjugacy, each involution is used exactly once to define the model character.
In an interesting paper, Baddeley generalizes the Inglis–Richardson–Saxl result to a larger class of groups. He makes the following definition.
Definition. An involution model for a finite group is a collection such that is a set of conjugacy-class representatives for the involutions of and is a linear character for each , chosen so that
For example, if an abelian group has an involution model then, since each centralizer is itself, comparing degrees shows that , and so is an elementary abelian 2-group. Conversely, any such group clearly has an involution model. By the Frobenius–Schur count of involutions, a necessary condition for a group to have an involution model is that all its irreducible representations are defined over the reals.
Baddeley’s main theorem is as follows.
Theorem. [Baddeley] If a finite group has an involution model then so does .
The aim of this post is to sketch my version of Baddeley’s proof of his theorem in the special case when . Some familiarity with the theory of conjugacy classes and representations of wreath products in Chapter 4 of James & Kerber, Representation theory of the symmetric group is assumed. The characters defined below differ from Baddeley’s by a factor of ; this is done to make (defined below) a permutation character, in analogy with the Inglis–Richardson–Saxl character .
Aside: the Hyperoctahedral group
The group of all matrices with entries that become permutation matrices when all entries are changed to is isomorphic to . Thus is the hyperoctahedral group of symmetries of the -hypercube. It is a nice exercise to identify , the rotational symmetry group of the cube, as an explicit index subgroup of .
From now on let . The group acts on by
This is a place permutation: the element , in position on the left-hand side, occupies position on the right-hand side. Let
We write elements of as where each .
Imprimitive action of
For each introduce a formal symbol . (This could be thought of as , but I find that bar makes for a more convenient notation.) Let . Given , we define by for all and for all . Then is isomorphic to the subgroup defined by where
So acts imprimitively on with blocks , , .
Irreducible representations of
Let be the faithful character of and let denote the linear character of on which each of the factors of in the base group factor acts as . Given a bipartition with and , we define
Basic Clifford theory shows that the characters for form a complete irredundant set of irreducible characters of . For example, the -dimensional representation of as the symmetry group of the -hypercube has character labelled by the bipartition .
Conjugacy classes of involutions in
Since , any involution in is of the form where is an involution. Moreover, as the calculation suggests, the place permutation action of on permutes amongst themselves the indices such that . By applying a suitable place permutation we may assume that and , for some . Now using that is conjugate, by , to , we see that a set of conjugacy class representatives for the involutions in is
for and such that . The generalized cycle-type invariant defined in James–Kerber can be used to show no two of these representatives are conjugate.
Centralizers of involutions in
As an element of , the involution defined above is
where, by definition, . If commutes with then, passing to the quotient, commutes with . Therefore the non-singleton orbits
of are permuted by , as are the remaining non-singleton orbits .
where acts on and acts on . Clearly commutes with
if and only if . Therefore the first factor is permutation isomorphic to
Set . Note that is permutation isomorphic to , acting with one orbit on and another on . (One has to get used to the two different ways in which the group arises; in this post I’ve used when the comes from the base group.)
For example, if then
and the centralizer of is generated by , , , , in the base group and , and in the top group . The first two top group generators generate .
Definition of the linear representations and reduction
The second and third factors of the centralizer are both complete wreath product, so, by analogy with the Inglis–Richardson–Saxl paper, it is natural guess to define so that restricts to:
- The trivial character on ;
- on ;
- on .
That is (omitting the details of the inflations for brevity),
Since restricts to the trivial character of , we have . The definition of above is therefore symmetric with respect to and . Moreover, if
then, by Pieri’s rule for the hyperoctahedral group (this follows from Pieri’s rule for the symmetric group in the same way as the hyperoctahedral branching rule follows from the branching rule for the symmetric group — for the latter see Lemma 4.2 in this paper),
where the second sum is over all bipartitions of such that is obtained by adding boxes, no two in the same row, to and is obtained by adding boxes, again no two in the same row, to . Therefore Baddeley’s theorem holds if and only if is multiplicity-free, with precisely the right constituents for the Pieri inductions as varies to give us every character of exactly once. This is the content of the following proposition.
Proof of the proposition
To avoid some messy notation I offer a ‘proof by example’. I believe it shows all the essential ideas of the general case.
Proof by example. Take . We have
and so is induced from the trivial character of
(Recall that where and .) To apply Clifford theory, it would be much more convenient if we induced from a subgroup of containing the full base group . We arrange this by first inducing up to . (For the action of , it is best to think of as .) The calculation
shows that, on restriction to , the induced character
is the sum of all products where each is one of the irreducible characters or on the right-hand side above. The centralizer acts transitively on the 3 factors: glueing together the products in the same orbits into induced characters we get that has the following irreducible constituents:
Note that the ’tilde-construction’ enters in two ways: once to combine characters of each two -factors in the same orbit of , and then again to combine the characters obtained in this way. As a small check, observe that the sum of degrees is , which is the index of in .
Reflecting the isomorphisms
we rewrite these characters as follows:
It is now routine to induce ‘in the top group’ up to
using the decomposition of into characters labelled by even partitions. For the second summand we use transitivity of induction, starting at the subgroup and going via . The third summand is dealt with similarly. Thus is the sum of the for the following bipartitions :