Let $G=GL\_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the invariant functions $$\mathbb{C}[G \times G]^T $$ where $T$ acts by simultaneous conjugation?



Invariants of general linear groups under torus action
Call the two matrices ${}\_1A$ and ${}\_2A$.


Your ring can be expressed by taking the $T$-invariants of the free ring in $2n^2$ variables $\mathbb C[{}\_kA\_{ij}]\_{1\leq i,j\leq n, 1\leq k \leq 2}$ and then inverting $\det {}\_1 A$ and $\det {}\_2 A$.


The $T$-invariants of the free ring are the ring of functions on an affine toric variety. It has an explicit combinatorial structure. A basis for it is given by the $T$-invariant monomials.


A generating set is given by the monomials that can't be decomposed into two monomials. These have the following form: For $k \leq n$ a natural number, $i\_1,\dots ,i\_k$ distinct indices from $1$ to $n$, and $\epsilon\_1,\dots, \epsilon\_k$ values from $1$ to $2$, the element $$ \prod\_{j=1}^k {}\_{\epsilon\_j}A\_{i\_j i\_{j+1}}$$ is such a monomial, and it's not too hard to see how they have this form.


To see the ring has this form, simply note that any function on $G \times G$ is a polynomial in the entries divided by some power of the two determinants, and since the determinants are $T$-invariant, any invariant polynomial must be an invariant polynomial in the entries divided by some power of the two determinants.


