Talk:Induced representation

This page is a bit technical and needs some cleaning up. It would help if there was an introductory paragraph on how induced representations of finite goups works with examples.

Yes, the page's evolution hasn't been particularly kind to it. Various things: the Frobenius character formula isn't given. More obscurely, there is a way of computing with induced representations allied to coset enumeration (it's coset enuneration with some labelled edges) which is John Conway's point of view and a good counterbalance to abstract discussion. Charles Matthews 14:09, 10 November 2005 (UTC)[reply]

See system of imprimitivity, where these are discussed for finite groups first.--CSTAR 15:24, 10 November 2005 (UTC)[reply]

Analytic Definition[edit]

Shouldn't $f$ be in $L^{2}(G)$ rather than $L^{2}(G/H)$ . I'm just learning the material at the moment, but it's not even clear what the latter condition can mean for a function defined on $G$ . 76.93.7.54 (talk) 23:09, 1 February 2011 (UTC)[reply]

I don't know what was here three years ago, but what's on the page under "analytic definition" at the moment is entirely incorrect and mostly nonsensical. To answer the question above: $L^{2}(G/H)$ would have been closer to correct than $L^{2}(G)$ , but it's still incorrect for the reason mentioned above. The functions in the Hilbert space on which the induced representation acts have constant norm on cosets. One wants the square of that norm to be integrable over $G/H$ with respect to a suitable quasi-invariant measure. I don't have time to write this all up in detail in the article, but I've added references to some books and articles that describe the inducing construction and its properties. Maybe in another three years somebody will come along and actually fix the article! 67.171.244.5 (talk) 16:03, 22 April 2014 (UTC)[reply]

The current state of the article[edit]

I had to use this article when writing about principal series representations of semisimple Lie groups for the article Zonal spherical function. The definition of induced representation was wrong for induction from a non-compact subgroup, even when all groups are discrete and countable (eg for Shimura's definition of Hecke operators). It was wrong even in the case of one dimensional characters induced from the integers to the real numbers. Although I don't have time at the moment, it would be nice to have some references, such as the various books and articles by George Mackey, including the books based on his Chicago and Oxford lecure notes, as well as giving the origins of the theory for topological groups, which I have always understood to have been the Stone-von Neumann theorem (see also Varadajan's book, "he Geometry of Quantum Theory"). I don't have time to provide references for this article or to improve it any more at present, but it would be nice if somebody could fully correct the topological part using the references I have cited, or other references by Blattner, Fell or Rieffel. Mathsci (talk) 04:35, 4 June 2008 (UTC)[reply]

Left and right adjoints[edit]

Might remark that the two different inductions in the analytic case are accounted for by taking left or right adjoints to restriction. In particular, one of these inductions is left exact. —Preceding unsigned comment added by 75.72.161.245 (talk) 22:38, 12 March 2009 (UTC)[reply]

Algebraic definition[edit]

There is something weird in the definition:

"Let G be a finite group and H any subgroup of G. Furthermore let (π,V) be a representation of H. The induced representation $\operatorname {Ind} _{H}^{G}\pi$ can be thought of as acting on the following space:

W=\bigoplus _{x\in G/H}xV.

Here each xV is an isomorphic copy of the vector space V. Via the induced representation G acts on W as follows:

g\cdot \sum _{x\in G/H}xv_{x}=\sum _{x\in G/H}gxv_{x}

where $v_{x}\in V$ for each $x$ ."

Namely: The way things are put now the G-action on the induced space only depends on the represenatation (π,V) through its dimension. The particular H-action doesn't play a role. It seems highly unlikely that that is correct. Maybe I interpret the expression gx in the wrong way, but as x is an element of G/H, it only seems natural to assume that so is gx (one checks that it is well defined.)

Probably what was intended was something along the lines of:

"Let n = |G/H| and let x_1, \ldots x_n be a full set of representatives in G of the posets in G/H. The induced representation $\operatorname {Ind} _{H}^{G}\pi$ can be thought of as acting on the following space:

W=\bigoplus _{i=1}^{n}x_{i}V.

Here each x_iV is an isomorphic copy of the vector space V. For each g \in G and each x_i there exists an h_g in H and j(i) \in {1, \ldots, n} such that gx_i = x_jh. (This is just another way of saying that the x_j form a full set of representatives.) Via the induced representation G acts on W as follows:

g\cdot \sum _{i}x_{i}v_{i}=\sum _{i}x_{j(i)}\pi (h_{g})v_{i}

where $v_{x}\in V$ for each $x$ ."

Octonion (talk) 18:26, 13 March 2009 (UTC)[reply]

Oh, sorry. I misunderstood you (though I think in retrospect, what you wrote was clear, Ind(V) ≠ Ind(W)). It was not *you* that forgot an action, but the wording in the first definition in the article that forgot the action. I suspect the writer of the definition intended G/H to mean a set of coset representatives, with the idea that "gxv" means "y (hv)" where "y" is one of the selected coset representatives and gx=yh. I think this exactly what you wrote above, but I didn't check the indices closely.

How should it be corrected?

Isaacs's Character Theory defines it with systems of imprimitivity, which up to notation is just the tensor product definition. James and Liebeck have (characteristic 0) definition that just defines it first for submodules of the regular G module; very concrete and easy to understand, but I think hard to make it work over arbitrary fields. Serre gives more or less the same definition as Isaacs. Alperin's Local rep theory more or less uses the tensor definition. Benson uses the tensor definition. Feit just defines the block matrix of the induced representation directly. Doerk–Hawkes uses the tensor definition. Curtis and Reiner use the tensor definition.

Looks like the tensor definition should be given prominence? The "gV" and "gv" stuff then works out automatically as: g⋅( x ⊗ v ) = (gx) ⊗ v = y ⊗ hv. JackSchmidt (talk) 19:03, 13 March 2009 (UTC)[reply]

Found a source with the start of the bad definition! Fulton–Harris correctly define the induced representation, and they talk about it in several ways. At one point, their notation is reasonably similar to the first definition, but they correctly describe the G-action (unlike our article here). It looks like everything would be fixed by defining what it means for a G-module to be induced from an H-submodule, or by defining the induced module directly from the tensor definition. I think having both (tensor and h-submodule) in the article would be fine. JackSchmidt (talk) 19:30, 13 March 2009 (UTC)[reply]

Hi, thank you for all your investigations. I was not entirely sure that what I typed was correct, but in the mean time I looked up the Fulton and Harris definition and it seems to make sense. Still I agree with you that the tensor definition should be given more prominence, since the coset construction seems a bit artificial. Octonion (talk) —Preceding undated comment added 11:20, 16 March 2009 (UTC).[reply]

Isn't the definition wrong in that "Here each xV is an isomorphic copy of the vector space V " as e.g. annhiliation could appear, which would make the dimension of xV smaller than V?

Wrong formula for normalized induction[edit]

In Kaniuth (p.62) as well as in Folland (p.155) the definition has an inverse for the modular functions (in the definition of the normalized induction of this article).