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Li Yutong
Li Yutong
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Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$$\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have non-negative entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset SL(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.

Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset SL(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.

Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have non-negative entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset SL(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.

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Li Yutong
Li Yutong
  • 3.5k
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  • 34

Subgroup of $GL$SL(n,\mathbb{R})$ with positive entries

Let $GL(n, \mathbb{R})$$SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=G(n, \mathbb{R}_{\geq 0})$$G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in G(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$$\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset G(n, \mathbb{R}_{\geq 0})$$S_n \subset SL(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.

Subgroup of $GL(n,\mathbb{R})$ with positive entries

Let $GL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices in real numbers. Let $G:=G(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in G(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset G(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.

Subgroup of $SL(n,\mathbb{R})$ with positive entries

Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset SL(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.

Source Link
Li Yutong
Li Yutong
  • 3.5k
  • 16
  • 34

Subgroup of $GL(n,\mathbb{R})$ with positive entries

Let $GL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices in real numbers. Let $G:=G(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in G(n,\mathbb{R}) \mid M, M^{-1} \text{ both have positive entries}\}$. Is there any known results on this group? I am particular interest in how big (roughly) this group could be. I know $S_n \subset G(n, \mathbb{R}_{\geq 0})$ (considering the permutation of a basis of a $n$- dimensional vector space), but I wish it could be much bigger than that.