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Brauer induction theorem

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Contents

Idea

The Brauer induction theorem (Brauer 46) states that, for ground field the complex numbers, finite-dimensional linear representations VV of a finite group GG all arise as virtual combinations of induced representations ind H G(W)ind_{H}^G \big(W\big) of just 1-dimensional representations WW, dim (W)=1dim_{\mathbb{C}}(W) =1:

(1)[V]=H iGW iRep(H i),dim(W i)=1n i[ind H i G(W i)],AAAa i. [V] \;=\; \underset{ \mathclap{ { H_i \hookrightarrow G } \atop { {W_i \in Rep(H_i)\,,} \atop { dim(W_i) = 1 } } } }{\sum} \, n_i \, \Big[ ind_{H_i}^G \big(W_i\big) \Big] \,, \phantom{AAA} a_i \in \mathbb{Z} \,.

In other words, this says that the representation ring R (G)R_{\mathbb{C}}(G) is generated from isomorphism classes [ind H big(W)]\left[ind_{H}^big(W\big)\right] of induced representations ind H G(W)ind_{H}^G \big(W\big) of 1-dimensional representations WW of subgroups HGH \subset G.

This may be thought of as (implying) a splitting principle for linear representations (Symonds 91), for more on this see at characteristic classes of linear representations the section splitting principle.

Brauer induction generalizes the immediate statement that finite-dimensional permutation representations are all direct sums of induced representations of the trivial 1-dimensional representation; see at induced representation of the trivial representation.

The analogous statement holds true also for ground ring the quaternions, while for ground field the real numbers one has to induce not just from 1-dimensional but also from 2-dimensional representations.

Of course, the expansions (1) are not unique. But one may find functorial choices that satisfy good extra properties, see below Snaith’s explicit Brauer induction and Symond’s explicit Brauer induction.


Properties

Snaith’s explicit Brauer induction

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Symonds’ explicit Brauer induction

We state an explicit and natural choice of Brauer induction due to Symonds 91 (Prop. ) below. Its main property is good compatibility with the total Chern classes of linear representations via a certain multiplicative transfer map on integral cohomology (the latter recalled as Lemma below).

Lemma

(Evens’ multiplicative transfer)

For GG a finite group and HGH \subset G a subgroup, there is a linear map

𝒩 H G:kH 2k(BH,)kH 2k(BG,) \mathcal{N}_H^G \;\colon\; \underset{k \in \mathbb{N}}{\prod} H^{2k}\big( B H, \mathbb{Z}\big) \longrightarrow \underset{k \in \mathbb{N}}{\prod} H^{2k}\big( B G, \mathbb{Z}\big)

from the cohomology ring of the classifying space of HH to that of GG which is multiplicative in that its respects the product structure, hence the cup product, on both sides

𝒩 H G(αβ)=𝒩 H G(α)𝒩 H G(β). \mathcal{N}_H^G\big( \alpha \smile \beta\big) \;=\; \mathcal{N}_H^G\big( \alpha\big) \smile \mathcal{N}_H^G\big( \beta\big) \,.

This is due to Evens 63. There, the maps themselves are introduced on the bottom of p. 7, while their multiplicativity is stated as Prop. 4 on p. 10.

Proposition

(Symonds’ explicit Brauer induction)

For GG \in FinGrp there is a linear map (homomorphism of abelian groups)

R (G)L[HG][1dRep (H) /] R_{\mathbb{C}}\big( G \big) \overset {L} {\longrightarrow} \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right]

from the underlying abelian group of the representation ring to the product group of the free abelian groups that are spanned by the isomorphism classes of 1-dimensional representations over all conjugacy classes of subgroup HGH \subset G,

such that

  1. LL is a natural transformation of functors FinGrp opAbFinGrp^{op} \to Ab,

    hence L(f *V)=f *(L(V))L\big( f^\ast V\big) = f^\ast( L(V) );

  2. LL is a section of the natural transformation

    [HG]R 1d(H)indR (G) \underset{ [H \subset G] }{\prod} R^{1d}_{\mathbb{Z}}\big( H\big) \overset {\sum ind} {\longrightarrow} R_{\mathbb{C}}\big( G \big)

    which applies induction and then sums everything up, in that the composition (ind)L\big( \sum ind \big) \circ L is the identity:

    (ind)L(V)[HG]ind H G[L(V) H]=V \big( \sum ind \big) \circ L(V) \coloneqq \underset{ [H \subset G] }{ \sum } ind_H^G\left[ L(V)_H \right] \;=\; V
  3. LL is compatible with the total Chern classes of linear representations

    R (G)ckH 2k(BG,) R_{\mathbb{C}}\big( G \big) \overset{c}{\longrightarrow} \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big)

    via their multiplicative transfer 𝒩 H G\mathcal{N}_H^G (Lemma ) in that

    c(V)=[HG]𝒩 H G(c(L(V) H)), c \big( V \big) \;=\; \underset{ [H \subset G] }{\smile} \mathcal{N}_H^G \Big( c \big( L(V)_H \big) \Big) \,,

    hence in that the following diagram commutes:

    R (G) L [HG][1dRep (H) /] c [HG](c𝒩 H G) kH 2k(BG,) [HG] [HG]ProdkH 2k(BG,) \array{ R_{\mathbb{C}}\big( G\big) &\overset{L}{\longrightarrow}& \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right] \\ {}^{\mathllap{c}}\Big\downarrow && \Big\downarrow {}^{ \underset{ [H \subset G] }{\prod} \left( c \circ \mathcal{N}_H^G \right) } \\ \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big) & \underset{ \underset{ [H \subset G] }{\smile} }{\longleftarrow} & \underset{ [H \subset G] }{\prod} \underset{ k \in \mathbb{Z} } {\Prod} H^{2k}\big( B G, \mathbb{Z}\big) }
  4. a 1-dimensional representation W1dRep(G) /R (G)W \in 1dRep\big(G\big)_{/\sim} \subset R_{\mathbb{C}}\big(G\big) is sent to the tuple L(W)=(W,0,0,)L(W) = (W,0,0, \cdots) whose component over GGG \subset G is VV itself, and all whose other components vanish;

  5. in contrast, if VRep (G) /V \in Rep_{\mathbb{C}}\big( G \big)_{/\sim} has no 1-dimensional direct summand, then the GG-compnents of L(V)L(V) is zero;

(Symonds 91, Prop. 2.1, Theorem 2.2, and Theorem 2.4)

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References

Due to

See also

In the context of characteristic classes of linear representations:

based on

and

  • Ove Kroll, An Algebraic Characterisation of Chern Classes of Finite Group Representations, Bulletin of the LMS, Volume19, Issue3 May 1987 Pages 245-248 (doi:10.1112/blms/19.3.245)

Last revised on January 29, 2019 at 12:20:49. See the history of this page for a list of all contributions to it.