nLab Brauer induction theorem

Idea

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

(1)$$[V] \;=\; \underset{ \mathclap{ { H_i \hookrightarrow G } \atop { {W_i \in Rep(H_i)\,,} \atop { dim(W_i) = 1 } } } }{\sum} \, n_i \, \Big[ ind_{H_i}^G \big(W_i\big) \Big] \,, \phantom{AAA} a_i \in \mathbb{Z} \,.$$

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

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 $G$ a finite group and $H \subset G$ a subgroup, there is a linear map

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

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

$$\mathcal{N}_H^G\big( \alpha \smile \beta\big) \;=\; \mathcal{N}_H^G\big( \alpha\big) \smile \mathcal{N}_H^G\big( \beta\big) \,.$$

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 $G \in$ FinGrp there is a linear map (homomorphism of abelian groups)

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

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 $H \subset G$,

such that

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

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

2. $L$ is a section of the natural transformation

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

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

   $$\big( \sum ind \big) \circ L(V) \coloneqq \underset{ [H \subset G] }{ \sum } ind_H^G\left[ L(V)_H \right] \;=\; V$$

3. $L$ is compatible with the total Chern classes of linear representations

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

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

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

   hence in that the following diagram commutes:

   $$\array{ R_{\mathbb{C}}\big( G\big) &\overset{L}{\longrightarrow}& \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right] \\ {}^{\mathllap{c}}\Big\downarrow && \Big\downarrow {}^{ \underset{ [H \subset G] }{\prod} \left( c \circ \mathcal{N}_H^G \right) } \\ \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big) & \underset{ \underset{ [H \subset G] }{\smile} }{\longleftarrow} & \underset{ [H \subset G] }{\prod} \underset{ k \in \mathbb{Z} } {\Prod} H^{2k}\big( B G, \mathbb{Z}\big) }$$

4. a 1-dimensional representation $W \in 1dRep\big(G\big)_{/\sim} \subset R_{\mathbb{C}}\big(G\big)$ is sent to the tuple $L(W) = (W,0,0, \cdots)$ whose component over $G \subset G$ is $V$ itself, and all whose other components vanish;

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

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

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References

Due to

• Richard Brauer, … (1946)

See also

• Wikipedia, Brauer’s theorem on induced characters

In the context of characteristic classes of linear representations:

• Peter Symonds, A splitting principle for group representations, Comment. Math. Helv. (1991) 66: 169 (dml:140229, doi:10.1007/BF02566643)

based on

• Leonard Evens, A Generalization of the Transfer Map in the Cohomology of Groups, Transactions of the American Mathematical Society Vol. 108, No. 1 (Jul., 1963), pp. 54-65 (doi:10.1090/S0002-9947-1963-0153725-1, jstor:1993825)

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)