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]((https://books.google.com/books?id=AdKnOpCsH3gC&pg=PA52&lpg=PA52&dq=Shahshahani++Gradient++gradient+vector++field&source=bl&ots=huTRLFy49f&sig=LKs-EYCuLjnlK5t9KtOO18R2_fA&hl=en&sa=X&ved=0ahUKEwjZkOu_8bbbAhW0yqYKHUltC-g4ChDoAQgrMAI#v=onepage&q=Shahshahani%20%20Gradient%20%20gradient%20vector%20%20field&f=false)) $g$ 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  $(M,g)$?