Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\ldots,n\}$$ For $X=(x_1,x_2,\ldots,x_n)\in M$ put $|X|=\sum_{i=1}^n x_i$.

We consider the Shahshahani Riemannian metric $g$ on $M$ with diagonal tensor metric $g_{ii}=\frac{|X|}{x_i}$.

What is the dimension and the precise structure of the group of all isometries of $(M,g)$?

Edit: 28 January 2023 I just realized that this metric is frequently used in this paper https://hal.science/hal-01382281/document

Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\ldots,n\}$$ For $X=(x_1,x_2,\ldots,x_n)\in M$ put $|X|=\sum_{i=1}^n x_i$.

We consider the Shahshahani Riemannian metric $g$ on $M$ with diagonal tensor metric $g_{ii}=\frac{|X|}{x_i}$.

What is the dimension and the precise structure of the group of all isometries of $(M,g)$?

Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\ldots,n\}$$ For $X=(x_1,x_2,\ldots,x_n)\in M$ put $|X|=\sum_{i=1}^n x_i$

We consider the Shahshahani Riemannian metric on $M$ with diagonal tensor metric $g_{ii}=\frac{|X|}{x_i}$.

What is the dimension and the precise structure of the group of all isometries of $(M,g)$?

Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\ldots,n\}$$ For $X=(x_1,x_2,\ldots,x_n)\in M$ put $|X|=\sum_{i=1}^n x_i$.

We consider the Shahshahani Riemannian metric $g$ on $M$ with diagonal tensor metric $g_{ii}=\frac{|X|}{x_i}$.

What is the dimension and the precise structure of the group of all isometries of $(M,g)$?