Skip to main content
MathOverflow

Return to Question

Name of reference
Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

When is $\textrm$\operatorname{Out}(G)$ isomorphic to the symmetries of the character table of $G$?

The$\DeclareMathOperator\Out{Out}\DeclareMathOperator\CTS{CTS}$The outer automorphism group $\textrm{Out}(G)$$\Out(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (representations). As a result, there is an homomorphism $\psi:\textrm{Out}(G)\rightarrow \textrm{CTS}(G)$$\psi:\Out(G)\rightarrow \CTS(G)$, where $\textrm{CTS}(G)$ is the symmetry group of the character table of $G$ (see Proposition 3.8.12 of the pdfSchedler - Group representation theory, lecture notes for reference). When is this homomorphism $\psi$ surjective or injective?

More specifically, given $\sigma\in\textrm{CTS}(G)$$\sigma\in\CTS(G)$, are there conditions involving properties of the conjugacy classes and representations that are being permuted by $\sigma$, that allows one to know that there exists $\alpha\in\textrm{Out}(G)$$\alpha\in\Out(G)$ such that $\psi\cdot\alpha=\sigma$? For example, one can show that the centralizer of the conjugacy classes that are permuted by an outer automorphism are isomorphic,isomorphic; is the converse true? This is consistent with some few cases that I considered, in particular with $Q_8$ and $D_4$, that share the same character table but for which $\textrm{Out}(Q_4)=S_3$$\Out(Q_4)=S_3$ and $\textrm{Out}(D_4)=\mathbb{Z}_2$$\Out(D_4)=\mathbb{Z}_2$.

When is $\textrm{Out}(G)$ isomorphic to the symmetries of the character table of $G$?

The outer automorphism group $\textrm{Out}(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (representations). As a result, there is an homomorphism $\psi:\textrm{Out}(G)\rightarrow \textrm{CTS}(G)$, where $\textrm{CTS}(G)$ is the symmetry group of the character table of $G$ (see Proposition 3.8.12 of the pdf for reference). When is this homomorphism $\psi$ surjective or injective?

More specifically, given $\sigma\in\textrm{CTS}(G)$, are there conditions involving properties of the conjugacy classes and representations that are being permuted by $\sigma$, that allows one to know that exists $\alpha\in\textrm{Out}(G)$ such that $\psi\cdot\alpha=\sigma$? For example, one can show that the centralizer of the conjugacy classes that are permuted by an outer automorphism are isomorphic, is the converse true? This is consistent with some few cases that I considered, in particular with $Q_8$ and $D_4$, that share the same character table but for which $\textrm{Out}(Q_4)=S_3$ and $\textrm{Out}(D_4)=\mathbb{Z}_2$.

When is $\operatorname{Out}(G)$ isomorphic to the symmetries of the character table of $G$?

$\DeclareMathOperator\Out{Out}\DeclareMathOperator\CTS{CTS}$The outer automorphism group $\Out(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (representations). As a result, there is an homomorphism $\psi:\Out(G)\rightarrow \CTS(G)$, where $\textrm{CTS}(G)$ is the symmetry group of the character table of $G$ (see Proposition 3.8.12 of Schedler - Group representation theory, lecture notes for reference). When is this homomorphism $\psi$ surjective or injective?

More specifically, given $\sigma\in\CTS(G)$, are there conditions involving properties of the conjugacy classes and representations that are being permuted by $\sigma$, that allows one to know that there exists $\alpha\in\Out(G)$ such that $\psi\cdot\alpha=\sigma$? For example, one can show that the centralizer of the conjugacy classes that are permuted by an outer automorphism are isomorphic; is the converse true? This is consistent with some few cases that I considered, in particular with $Q_8$ and $D_4$, that share the same character table but for which $\Out(Q_4)=S_3$ and $\Out(D_4)=\mathbb{Z}_2$.

edited tags
Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286
Source Link
Davi Costa
Davi Costa
  • 141
  • 4

When is $\textrm{Out}(G)$ isomorphic to the symmetries of the character table of $G$?

The outer automorphism group $\textrm{Out}(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (representations). As a result, there is an homomorphism $\psi:\textrm{Out}(G)\rightarrow \textrm{CTS}(G)$, where $\textrm{CTS}(G)$ is the symmetry group of the character table of $G$ (see Proposition 3.8.12 of the pdf for reference). When is this homomorphism $\psi$ surjective or injective?

More specifically, given $\sigma\in\textrm{CTS}(G)$, are there conditions involving properties of the conjugacy classes and representations that are being permuted by $\sigma$, that allows one to know that exists $\alpha\in\textrm{Out}(G)$ such that $\psi\cdot\alpha=\sigma$? For example, one can show that the centralizer of the conjugacy classes that are permuted by an outer automorphism are isomorphic, is the converse true? This is consistent with some few cases that I considered, in particular with $Q_8$ and $D_4$, that share the same character table but for which $\textrm{Out}(Q_4)=S_3$ and $\textrm{Out}(D_4)=\mathbb{Z}_2$.