Let \(T=(\mathbb{C}^\times)^2\) be the algebraic torus acting on \(\mathbb{C}[x,y]\) by
\[(t_1,t_2)\cdot f(x,y)\mapsto f(t_1x,t_2y).\]
We have that the fixed functions are generated by monomials. So fixed points in \((\mathbb{C}^2)^{[n]}\) are given by monomial ideals in \(\mathbb{C}[x,y]\) of codimension \(n\).
Of length \(n\) can be described by a partition of \(n\), there is a correspondence between monomial ideals and partitions of \(n\).
\[I \mapsto \lambda = \{(i,j)\in \mathbb{N}^2 \mid x^iy^j \not\in I\}, \quad \lambda\mapsto I_\lambda = \langle x^iy^j \mid (i,j)\not\in \lambda\rangle.\]
i.e. the ideals are generated by monomials outside the partition. In fact, the set of monomials in \(\lambda\) provides a basis \(B_\lambda\) for the quotient \(\mathbb{C}[x,y]/I_\lambda\).
For a partition \(\lambda\) of \(n\), the affine open set \(U_\lambda\) is the set of monomial ideals of codimension \(n\) that is indeed generated by the images of monomials in \(\lambda\).
\[U_\lambda = \{I \in (\mathbb{C}^2)^{[n]} \mid B_\lambda \text{ is a basis for } \mathbb{C}[x,y]/I\}.\]
So for each point \(I\) in this open set \(I\in U_\lambda\) we can expand
\[ x^ry^s = \sum_{(i,j)\in \lambda} c^{rs}_{ij}x^iy^j + I\]
Example. For \(\lambda = \{(0,0),(1,0)\}\) the basis is \(B_\lambda = \{1,x\}\), \(I_\lambda = (x^2,y)\). The ideal \(I = (x^2,y+\alpha x)\) is in \(U_\lambda\) and we can expand like
\(c^{00}_{00} = c^{10}_{10} = 1\)
\(y = -\alpha x + I, c^{10}_{00} = -\alpha\)
\(xy = -\alpha x^2 + I = 0 + I, c^{11}_{**}=0\)
By multiplying we have interesting relations
\[\begin{align*} x^{r+r'}y^{s+s'} &= \sum_{(i,j),(i',j')\in \lambda} c^{rs}_{ij}c^{r's'}_{i'j'}x^{i+i'}y^{j+j'} + I\\ &= \sum_{(i,j),(i',j')\in \lambda} c^{rs}_{ij}c^{r's'}_{i'j'} \sum_{(k,l)\in \lambda} c^{i+i',j+j'}_{kl}x^ky^l + I\\ &= \sum_{(k,l)} \left(\sum_{(i,j),(i',j')} c^{rs}_{ij}c^{r's'}_{i'j'}c^{i+i',j+j'}_{kl}\right) x^ky^l + I \end{align*}\]
This gives formula
\[c^{r+r',s+s'}_{kl} = \sum_{(i,j),(i',j')\in \lambda} c^{rs}_{ij}c^{r's'}_{i'j'}c^{i+i',j+j'}_{kl}.\]
In particular if we take \((r',s')=(1,0)\in \lambda\)
\[\begin{align*} c^{r+1,s}_{kl} &= \sum_{(i,j),(i',j')\in \lambda} c^{rs}_{ij}c^{10}_{i'j'}c^{i+i',j+j'}_{kl} \\ &= \sum_{(i,j)\in \lambda} c^{rs}_{ij}c^{i+1,j}_{kl} \end{align*}\]
This gives the ring of algebraic functions \(\mathbb{C}[c^{rs}_{ij}]_{(i,j)\in \lambda}\) on \(U_\lambda\), it should have some relations so it is of dimension \(2n\).
Since for an ideal \(I\in U_\lambda\), the coordinates \(c^{rs}_{ij}\) for \((r,s)\in \lambda\) are constants, and the other coordinates are zero then \(I=I_\lambda\), we know that the maximal ideal corresponding to \(I\) is
\[\mathfrak{m}_\lambda := \langle c^{rs}_{ij} \mid (r,s)\not\in \lambda\rangle.\]