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nLab > Latest Changes: Brauer induction theorem

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJan 28th 2019
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Author: Urs
Format: MarkdownItexam starting something here <a href="https://ncatlab.org/nlab/revision/Brauer+induction+theorem/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Brauer+induction+theorem">current</a>

am starting something here

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeJan 29th 2019
- (edited Jan 29th 2019)
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Author: Urs
Format: MarkdownItexI have tried to record the result of [Symonds 91](https://ncatlab.org/nlab/show/Brauer+induction+theorem#Symonds91) ("Symonds' explicit Brauer induction") in somewhat more modern form: *** 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 $L\big( f^\ast V\big) = f^\ast( L(V) )$; 1. $L$ 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 [[induced representation|induction]] and then sums everything up, in that the [[composition]] $\big( \sum ind \big) \circ L$ is the [[identity morphism|identity]]: $$ \big( \sum ind \big) \circ L(V) \coloneqq \underset{ [H \subset G] }{ \sum } ind_H^G\left[ L(V)_H \right] \;=\; V $$ 1. $L$ 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 \ref{TransferEvens}) 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 [[commuting diagram|diagram commutes]]: $$ \array{ R_{\mathbb{C}}\big( G\big) &\overset{L}{\longrightarrow}& \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right] \\ {}^{\mathllap{c}}\Big\downarrow && \Big\downarrow {}^{ \underset{ [H \subset G] }{\prod} \left( c \circ \mathcal{N}_H^G \right) } \\ \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big) & \underset{ \underset{ [H \subset G] }{\smile} }{\longleftarrow} & \underset{ [H \subset G] }{\prod} \underset{ k \in \mathbb{Z} } {\Prod} H^{2k}\big( B G, \mathbb{Z}\big) } $$ 1. 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; 1. in contrast, if $V \in Rep_{\mathbb{C}}\big( G \big)_{/\sim}$ has no 1-dimensional [[direct sum|direct summand]], then the $G$-compnents of $L(V)$ is zero; <a href="https://ncatlab.org/nlab/revision/diff/Brauer+induction+theorem/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Brauer+induction+theorem/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Brauer+induction+theorem">current</a>
I have tried to record the result of Symonds 91 (“Symonds’ explicit Brauer induction”) in somewhat more modern form:

For $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">G \in </annotation></semantics></math>$ FinGrp there is a linear map (homomorphism of abelian groups)
Rℂ(G)L⟶∏[H⊂G]ℤ[1dRepℂ(H)/∼]

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math>$ ,

such that
1. $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>$ is a natural transformation of functors $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>FinGrp</mi> <mi>op</mi></msup><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">FinGrp^{op} \to Ab</annotation></semantics></math>$ ,
  
  hence $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>V</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L\big( f^\ast V\big) = f^\ast( L(V) )</annotation></semantics></math>$ ;
2. $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>$ is a section of the natural transformation
  $<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mo stretchy="false">[</mo><mi>H</mi><mo>⊂</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></munder><msubsup><mi>R</mi> <mi>ℤ</mi> <mrow><mn>1</mn><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>H</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mrow><mo lspace="0.16667em" rspace="0.16667em">∑</mo><mi>ind</mi></mrow></mover><msub><mi>R</mi> <mi>ℂ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \underset{ [H \subset G] }{\prod} R^{1d}_{\mathbb{Z}}\big( H\big) \overset {\sum ind} {\longrightarrow} R_{\mathbb{C}}\big( G \big) </annotation></semantics></math>$
  which applies induction and then sums everything up, in that the composition $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="0.16667em" rspace="0.16667em">∑</mo><mi>ind</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∘</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\big( \sum ind \big) \circ L</annotation></semantics></math>$ is the identity:
  $<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="0.16667em" rspace="0.16667em">∑</mo><mi>ind</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∘</mo><mi>L</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="0.16667em" rspace="0.16667em">∑</mo><mrow><mo stretchy="false">[</mo><mi>H</mi><mo>⊂</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></munder><msubsup><mi>ind</mi> <mi>H</mi> <mi>G</mi></msubsup><mrow><mo>[</mo><mi>L</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mi>H</mi></msub><mo>]</mo></mrow><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mi>V</mi></mrow><annotation encoding="application/x-tex"> \big( \sum ind \big) \circ L(V) \coloneqq \underset{ [H \subset G] }{ \sum } ind_H^G\left[ L(V)_H \right] \;=\; V </annotation></semantics></math>$
3. $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>$ is compatible with the total Chern classes of linear representations
  $<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>R</mi> <mi>ℂ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mi>c</mi></mover><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> R_{\mathbb{C}}\big( G \big) \overset{c}{\longrightarrow} \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big) </annotation></semantics></math>$
  via their multiplicative transfer $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>𝒩</mi> <mi>H</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{N}_H^G</annotation></semantics></math>$ (Lemma \ref{TransferEvens}) in that
  $<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>c</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>V</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><munder><mo>⌣</mo><mrow><mo stretchy="false">[</mo><mi>H</mi><mo>⊂</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></munder><msubsup><mi>𝒩</mi> <mi>H</mi> <mi>G</mi></msubsup><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>c</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mi>H</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="0.16667em"/><mo>,</mo></mrow><annotation encoding="application/x-tex"> c \big( V \big) \;=\; \underset{ [H \subset G] }{\smile} \mathcal{N}_H^G \Big( c \big( L(V)_H \big) \Big) \,, </annotation></semantics></math>$
  hence in that the following diagram commutes:
  $<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>R</mi> <mi>ℂ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>L</mi></mover></mtd> <mtd><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mo stretchy="false">[</mo><mi>H</mi><mo>⊂</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></munder><mi>ℤ</mi><mrow><mo>[</mo><mn>1</mn><msub><mi>dRep</mi> <mi>ℂ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>H</mi><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>]</mo></mrow></mtd></mtr> <mtr><mtd><msup><mrow/> <mpadded width="0px" lspace="-100%width"><mi>c</mi></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd/> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow/> <mrow><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mo stretchy="false">[</mo><mi>H</mi><mo>⊂</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></munder><mrow><mo>(</mo><mi>c</mi><mo>∘</mo><msubsup><mi>𝒩</mi> <mi>H</mi> <mi>G</mi></msubsup><mo>)</mo></mrow></mrow></msup></mtd></mtr> <mtr><mtd><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><munder><mo>⟵</mo><munder><mo>⌣</mo><mrow><mo stretchy="false">[</mo><mi>H</mi><mo>⊂</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></munder></munder></mtd> <mtd><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mo stretchy="false">[</mo><mi>H</mi><mo>⊂</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></munder><munder><mo lspace="0em" rspace="0.16667em">Prod</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℤ</mi></mrow></munder><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ R_{\mathbb{C}}\big( G\big) &amp;\overset{L}{\longrightarrow}&amp; \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right] \\ {}^{\mathllap{c}}\Big\downarrow &amp;&amp; \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) &amp; \underset{ \underset{ [H \subset G] }{\smile} }{\longleftarrow} &amp; \underset{ [H \subset G] }{\prod} \underset{ k \in \mathbb{Z} } {\Prod} H^{2k}\big( B G, \mathbb{Z}\big) } </annotation></semantics></math>$
4. a 1-dimensional representation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>∈</mo><mn>1</mn><mi>dRep</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>⊂</mo><msub><mi>R</mi> <mi>ℂ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">W \in 1dRep\big(G\big)_{/\sim} \subset R_{\mathbb{C}}\big(G\big)</annotation></semantics></math>$ is sent to the tuple $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(W) = (W,0,0, \cdots)</annotation></semantics></math>$ whose component over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G \subset G</annotation></semantics></math>$ is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>$ itself, and all whose other components vanish;
5. in contrast, if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>∈</mo><msub><mi>Rep</mi> <mi>ℂ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex">V \in Rep_{\mathbb{C}}\big( G \big)_{/\sim}</annotation></semantics></math>$ has no 1-dimensional direct summand, then the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ -compnents of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(V)</annotation></semantics></math>$ is zero;
diff, v3, current
- CommentRowNumber3.
- CommentAuthorJohn Baez
- CommentTimeMar 11th 2025
- PermaLink
Author: John Baez
Format: MarkdownItexAdded link to [[Artin's induction theorem]], a page I will now create. <a href="https://ncatlab.org/nlab/revision/diff/Brauer+induction+theorem/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Brauer+induction+theorem/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Brauer+induction+theorem">current</a>

Added link to Artin’s induction theorem, a page I will now create.

diff, v4, current
- CommentRowNumber4.
- CommentAuthorJohn Baez
- CommentTimeMar 12th 2025
- PermaLink
Author: John Baez
Format: MarkdownItexAdded info comparing Brauer's and Artin's theorems. <a href="https://ncatlab.org/nlab/revision/diff/Brauer+induction+theorem/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Brauer+induction+theorem/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Brauer+induction+theorem">current</a>

Added info comparing Brauer’s and Artin’s theorems.

diff, v5, current

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