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Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Oredual version of a theorem of Øystein Ore.

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

remove wrong check
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Sebastien Palcoux
Sebastien Palcoux
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Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Remark: It's true for $[G:H] \le 30$ and $\vert G \vert \le 10^2$ (checked with GAP).

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Remark: It's true for $[G:H] \le 30$ and $\vert G \vert \le 10^2$ (checked with GAP).

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

the program for checking was incorrect, now it's much more long in time, i've improved the check data
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Sebastien Palcoux
Sebastien Palcoux
  • 27k
  • 5
  • 74
  • 186

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Remark: It's true for $[G:H] \le 30$ and $\vert G \vert \le 10^4$$\vert G \vert \le 10^2$ (checked with GAP).

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Remark: It's true for $[G:H] \le 30$ and $\vert G \vert \le 10^4$ (checked with GAP).

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a core-free subgroup, $U$ and $V$ two irred. complex representations of $G$.
Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

Question: Is there an irr. complex rep. $W$ of $G$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?

Remark: It's true for $[G:H] \le 30$ and $\vert G \vert \le 10^2$ (checked with GAP).

Motivation: This would be very helpful for proving the dual version of a theorem of Øystein Ore.

data update
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Sebastien Palcoux
Sebastien Palcoux
  • 27k
  • 5
  • 74
  • 186
check data
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Sebastien Palcoux
Sebastien Palcoux
  • 27k
  • 5
  • 74
  • 186
Source Link
Sebastien Palcoux
Sebastien Palcoux
  • 27k
  • 5
  • 74
  • 186