# Talk:Restricted representation

## Clifford theory

Probably the Clifford theory of restriction to normal subgroups should also be mentioned. It is pretty similar to the galois theory of fields or basic algebraic number theory, so can probably make for an interesting addition. The "restriction" of the "irreducible representation" pZ is its factorization in a ring of integers R; just like ideals can ramify, split, and remain inert, so too can representations. Restriction to arbitrary subgroups is like factorization in arbitrary rings of integers, but restriction to normal subgroups is like factorization when the extension is galois. In Clifford theory you have the same e,f,g numbers, an inertial subgroup, etc.

A good example for Clifford theory is determining the character table of An from Sn, or SL(2,F) from GL(2,F). Good references are Isaacs's Character Theory and Alperin&Bell's Group Theory.

If one is interested in the modular versions too, then Gorenstein's Finite Groups and Navarro's Characters and Blocks are good.

One of the Curtis and/or Reiner books has the algebraic number theory connection explicit, but the language is basically identical between basic Clifford theory and basic algebraic number theory, so any source should suffice. JackSchmidt (talk) 13:51, 27 August 2008 (UTC)

I learnt this as the "Mackey machine" - this article is definitely not just about finite groups. Normal subgroup analysis is very well explained in Mackey's Chicago lecture notes. Mathsci (talk) 23:01, 28 August 2008 (UTC)
Thanks for the reference. I took a look at some reviews of the book. Does the Mackey machine require any hypothesis on the normal subgroup when the whole group is finite (the reviews indicated it should be abelian, or nearly so)? Clifford theory doesn't have any such requirements. If anything, its utility is usually determined by G/N rather than N (G/N covers the "Galois group", so the simpler it is, the simpler the algebraic number theory becomes).
• Perhaps I could write a draft of Clifford theory, and then you could write the next section on the Mackey machine (viewing it as a generalization to locally compact groups).
Clifford's stuff is a decade earlier, so I don't think it would be a gross misrepresentation. It seems like the theories are fairly dissimilar to me. The Mackey machine seems to be compared to (or a precursor to) Harish-Chandra induction, which is more of the sylow normalizer->perfect jump, rather than the perfect to Aut(perfect) jump which Clifford theory is better at. One of the best parts of Clifford theory is how well it behaves when the algebra is not semi-simple. The restriction of a completely reducible module to a normal subgroup is completely reducible (and the isotypic components are all conjugate), and the restriction of an indecomposable module to a normal subgroup has a similar conjugacy property. If you can arrange that G/N is not too crazy, and N is a p'-group (so for instance in groups of small p-length), then Clifford theory does a very good job of making the p-modular representations seem like ordinary representations. At any rate, we could have a large section:
• Relation to induction
• Frobenius and Nakayama reciprocity
• Clifford theory
• Mackey machine
• Mackey theorem
I'd like to give the simplified Mackey theorem as well, if G=AB, then chi^G_B has a simple form for chi in Irr(A). You could probably talk about his work on semidirect products with Abelian kernel in the Mackey machine section (often the Harish-Chandra thing is a semidirect product too, so this is likely to be pretty interesting). Then the full Mackey theorem is a nice generalization of the G=AB case when A and B do not permute. JackSchmidt (talk) 03:12, 29 August 2008 (UTC)
The Mackey machine works very simply for finite groups and I think is done in Mackey's notes (with the Mackey "little group" and "Mackey obstruction"). Its application to Lie groups and p-adic groups (Stone-von Neumann, Segal-Shale-Weil) is significant in many areas of mathematics and theoretical physics. Chapters 7 and 8 in Serre's book contain much of the material. I'm not sure whether there's a separate treatment of the Mackey machine anywhere for finite groups - the second edition of Volume II of Varadarajan's Geometry of Quantum Theory has some references. The specialization to finite groups of the Chicago lecture notes is quite easy. However this material is more appropriate for articles on induced representations. The restriction theorem for an induced representation (in Serre) could be included here.
The Mackey machine takes a finite group with normal subgroup N. The irreducible representations are described by the orbits of G on the irreducible dual N* of N. Given an irrep σ in N*, let G σ be the stabliser of σ in G. Then G/G σ is the Mackey little group. σ extends uniquely to a projective irrep σ* of G σ, corresponding to a 2-cocycle ωσ in H2(G/G σ, T) and the irreps of G for the orbit are obtained by tensoring σ* with projective irreps of the Mackey little group belonging to the cocycle ωσ–1, the Mackey obstruction. Thus the irreps of G are described by the projective representations of the Mackey little groups with respect to the Mackey obstruction.
I have no idea whether this already appears on WP (I just looked at induced representation - it is content-free). I do think that the classical branching rules could be spelled out more in the current article. I have written the simplest possible case, but the others are all important (and mysterious) for analysis on bundles on the homogeneous space in the case of compact groups for example. They figure in Bott's version of Dirac induction, also reproduced in Wallach's book on Harmonic analysis on homogeneous spaces. They figure in Brauer's thesis and some of the more tantalizing parts of Weyl's "Classical groups". The article on Clifford theory BTW is not what you imagine it to be about! The Frobenius-Mackey theorems should all be added to the article on induced representations. Personally, these are stones that I don't feel like turning over on WP at the moment ... Mathsci (talk) 07:03, 29 August 2008 (UTC)
Definitely two sections then. The Mackey machine looks like it does not apply to the standard situations of Clifford theory, but it sounds like a good second section. Extensions of characters from normal subgroups to the whole group is quite similar to what you describe, but there is quite some machinery for reducing to the case where things like N* make sense. JackSchmidt (talk) 12:23, 29 August 2008 (UTC)
For a finite group N, is there a problem with N*? It's the space of irreducible characters. I perhaps should have written $\hat{N}$ .... There's already an article on system of imprimitivity and the rest is easy. But all of this should be in the article on induced representations, not here. If I have time, I'll try to dig out a version in a text book - I think it's almost in Curtis & Reiner (any way easily deducible from the results there), in Steven Gaal's book and possibly in Pukanszky's Lecons sur les representations des groupes. Cheers, Mathsci (talk) 14:35, 29 August 2008 (UTC)
Just to be clear, for a finite group, you intend N* to be a set with no further structure (not a vector space, just a discrete topological space, not a group or semigroup) other than the action of Aut(N) on it (and the corresponding G-action)?
Another thing I might not have communicated clearly is that Clifford theory has quite a lot to do with restricted representations. One often starts with Irr(G) and seeks to understand Irr(N).
I took a look at Mackey's book and can definitely say that p125-152 has nothing of the Clifford theory at all. It does have a nice presentation of Mackey's theorem, but the only mention of normal subgroups was a corollary of it. Page 184 has a little more mention of normal subgroups, but again barely touches on Clifford theory. Serre's section 8.2 seems to confirm the idea that the normal subgroup should be abelian. Mackey's book appears to use N-hat to mean the discrete, finite set (with no algebraic operations defined) Irr(N), so I'm unsure if you intend N to be arbitrary. In finite groups when discussing extension of characters, one often uses the determinant map to switch to linear characters where the dual is a group, but this only applies when G/N has a nice relationship to the representation (for instance in a Sylow normalizer). Navarro's Th. 8.16 is a version of extension very similar to what you described above (cohomology is taken in the multiplicative group of the field), and attributes it to Clifford as well. A version of Serre's proposition 24 is called Clifford correspondence, and is generalized to Green correspondence in modular representation theory. Perhaps they are intended to be the same thing, but they lose the flavor completely and omit the applications (which cannot be stated simply using the setup in Mackey's or Serre's book, but are obvious in the Isaacs, Gorenstein, Navarro setup). For instance: a character can be extended from a normal subgroup to the group if it can be extended to any Sylow subgroup (so for instance improving the applicability of the linear method from Sylow normalizers to certain local subgroups). JackSchmidt (talk) 15:08, 29 August 2008 (UTC)
I guess the simple summary is, "I am talking about Clifford theory, but you are talking about something only tangentially related. Do you agree they should both be included?" JackSchmidt (talk) 15:25, 29 August 2008 (UTC)
Ahem, Mackey's theory specializes easily to finite groups. This is explained up to page 202 in the book. What you say is incorrect, I'm afraid. The Mackey machine for semidirect products by Abelian groups is in Serre. It is not tangentially related: most people who do the representation theory of Lie groups are quite familiar with the Mackey machine, which is used in Kirillov theory and also in its generalization to solvable Lie groups of Pukanszky. The case of finite groups is a rather easy special case. The normal subgroup does not have to be Abelian. I will add a complete discussion of the Mackey machine for finite groups below so that it's clear what is being discussed. Mathsci (talk) 16:26, 29 August 2008 (UTC)
But what you describe below is not Clifford theory. I am not saying that the Mackey machine is not related to finite groups, I am saying it is not related to Clifford theory. JackSchmidt (talk) 17:05, 29 August 2008 (UTC)
Clifford theory is an immediate consequence of the Mackey machine described below. Certainly that is the case for what's described (not very well) in character theory in the "Clifford theorem" section. Restricting characters or representations in characteristic zero is the same thing. You seem to have some kind of mental block, because what you have just written makes no sense. Sorry, Mathsci (talk) 22:22, 29 August 2008 (UTC)
Please refrain from personal attacks. Clifford theory is certainly not a consequence of the Mackey machine any more than it is a consequence of the group axioms. Perhaps the aspects of it that have filtered into the locally compact group communities are described well by this, but certainly not for finite groups. JackSchmidt (talk) 01:14, 30 August 2008 (UTC)

(unindent) I'm sorry, there was no intention of a personal attack. There still does seem to be some misunderstanding about Clifford theory. Chapter 11 of Isaacs' book on characters of finite groups is on Clifford theory and concerns representations of a finite group G that restrict to a multiple of a given irreducible on a normal subgroup; the last part of Mackey's 1958 "Mackey machine" paper in Acta Mathematica is on representations of a locally compact group G that restrict to a multiple of a given irreducible on a closed normal subgroup (repeated in the Chicago lectures). The latter is a generalization of the former. Both can be phrased in terms of projective representations, factor sets, 2-cocycles, twisted group algebras, central extensions, etc. The confusion probably arises because Clifford was in a direct line from Frobenius and Schur (through Brauer and Weyl), while Mackey was motivated by the Lie group representations arising in quantum mechanics from the work of Stone, von Neumann, Bargmann and Wigner. Although the original idea is the same, there is a different flavour as soon as modular representations are considered in the finite case.

I looked at Clifford's original papers (inspired he says by a suggestion of Hermann Weyl)

Clifford remarks in the Annals paper that, when the group is finite, almost everything was known by Frobenius and is contained in his 1898 paper on the relations between between the characters of a groups and those of its subgroups. Clifford's theory applies to any group, any normal subgroup and any algebraically closed ground field. What he says in the opening paragraph of the Annals paper is not very different from part of what I have written below. Mathsci (talk) 12:43, 30 August 2008 (UTC)

Thanks. I've read over the references, and I agree there is heavy overlap. I am happy saying the flavor is different, both because of the direct line and the modular concerns, and also because of the availability of more arithmetical methods when the normal subgroup has finite index. Clifford theory is a large body of techniques and an area of current research, but I think presenting the Mackey machine as a common feature of both the finite and the locally compact seems wise. There are likely to be some interesting features of the locally compact ("continuous spectrum"?) that make no sense in a purely finite setting, as well as features of the finite case (the arithmetic concerns, the modular concerns) that make no sense for infinite dimensional representations, non-periodic groups, or infinitely generated groups.
One of my main concerns is that in character theory, it is quite often of critical importance that most of the objects encountered are integers, or at least algebraic integers. If you have Isaacs's handy, then things like theorem 6.18, 6.25, exercise 6.3, 6.12, cor 11.29, 11.31 are good examples of some of the arithmetical flavor. The modular Clifford theory is a little weirder since a few critical quantities are no longer integers (of course, this already broke for the dimension of a simple module dividing the order of the group), but it still often depends heavily on these sorts of arithmetic arguments. The second half of chapter 8 of Navarro is basically chapter 11 of Isaacs in the modular case.
• Would it be sensible to have a separate article on the relation of induction and restriction?
It could then be referenced from both of the two articles. Especially if inflation is considered generalized restriction (like in Serre and in this wiki article), then there are the inflation-restriction sequences of group cohomology to worry about (red link in Lyndon–Hochschild–Serre spectral sequence). There are really a ton ideas centered around Frobenius reciprocity, such as Nakayama's formulations, his reformulations for Frobenius algebras and the Nakayama permutation, the Eckmann-Shapiro lemma lifting the inner product from Hom to Ext^i, the Heller operator allowing smooth transition between some of these inner products, and the weirder bilinear pairings used in modular representation theory. There is a neat area of algebra called "torsion theory" and "cotorsion theory" that is basically there to find orthogonal decompositions of a module category in terms of these various bilinear pairings, and often group rings of various sorts are popular examples since the pairing has so many interpretations. JackSchmidt (talk) 17:15, 1 September 2008 (UTC)
Yes, there are several separate articles that can be written here. It's hard to know where to start, but certainly an article on the Mackey machine is not a bad idea. I'm writing on some bits of CFT at the moment, but will be happy to chip in. Mathsci (talk) 22:01, 1 September 2008 (UTC)

#### Mackey machine for finite groups

Let G be a finite group and N a normal subgroup. Let N* denote the finite set of equivalence classes of irreducible representations N. The group G acts by conjugation on N and hence N*. Let (π, V) be a representation of G. The restriction of π to N can be decomposed as a direct sum of irreducible representations (σ, U) of N. On the other hand G acts on this space permuting the isotypic components

Representations Induced in an Invariant Subgroup

1. A. H. Clifford
2. Proceedings of the National Academy of Sciences of the United States of America, Vol. 23, No. 2 (Feb. 15, 1937), pp. 89-90
$W_\sigma=U\otimes \mathrm{Hom}_N(U,V) \subset V$

according to its action on N*. The action must be transitive, since different orbits would give rise to proper subspaces invariant under G, contrary to the irreducibility of π. Let Wσ be an isotypic component. It is left invariant by Gσ, the stabilizer in G of σ, regarded as an element of N*. Thus the G translates of Wσ form a system of imprimitivity of type G/Gσ. So by Mackey's imprimitivity theorem

$V =\mathrm{ind}_{G_\sigma}^G W_\sigma.$

It only remains to analyse Wσ. As a representation of N it has the form U $\otimes$ M, where M is a multiplicity space. The group Gσ act irreducibly on U $\otimes$ M. Since it normalises N and preserves the class of σ, it must act projectively on U, i.e. as a group of automorphisms of End U. But then it acts projectively also on the relative commutant of End U in End U $\otimes$ M, i.e. End M. Thus Gσ acts projectively on M. But N clearly acts trivially on M, so M in fact provides a projective representation of the Mackey little group Gσ/N. The 2-cocycle (or factor set) of the projective representation of the little group on M is inverse to the 2-cocycle of the extension of U to Gσ

Conversely, by applying Mackey's irreducibility criterion, this process can be reversed. Start from an irreducible representation σ of N, determine the Mackey obstruction for Gσ/N, compute the irreducible projective representations of the little group for this cocycle, then induce up to G. Mathsci (talk) 16:26, 29 August 2008 (UTC)

## Mackey theorem

Somewhere one should also mention the decomposition of the restriction of an induced representation, usually called Mackey's theorem. JackSchmidt (talk) 14:04, 27 August 2008 (UTC)

## Restriction to p-regular classes

There are two names for taking an p-integral representation and forming a modular representation, restriction and reduction. For a number field K, with ring of integers R, maximal ideal M lying above pZ, localization S away from M, maximal ideal P of S, residue field F = S/P of S, and finite group G: every KG-module M has a S-free SG-submodule N whose K span is the whole KG-module, and N/PN is an FG-module. The transition from M to N/PN is called "reducing mod p" or the "restriction mod p". The latter term is most clear when viewed as characters: the Brauer character of N/PN, as a function, is the restriction of the character of M, as a function, from the conjugacy classes of G to the conjugacy classes of G whose elements have order relatively prime to p.

While the map from M to N/PN is often not well-defined (there is a choice of N), the map from the character of M to the Brauer character of N/PN is well-defined and just as trivial to calculate as the restriction of a character of a KG-module to a KH-module.

At any rate, the character theoretic version is quite often called restriction. Should it be mentioned here? JackSchmidt (talk) 14:13, 27 August 2008 (UTC)