As part of the result of solving the [problem](https://mathoverflow.net/questions/87285/reference-request-steinbergs-1975-paper-on-a-paper-of-pittieretrieved) I am working on, my advisor and I translated the task of finding a basis for $R(T\_{sl\_{\mathbb{C}}(n)})$ in terms of $R(sl\_{\mathbb{C}}(n))$ into the following problem:


Finding a basis for $\mathbb{Z}[x\_{1},x\_{2},...x\_{n+1}], \text{with condition} \prod x\_{i}=1$ in terms of $$\mathbb{Z}[x\_{1}+x\_{2}..+x\_{n+1}, \sum\_{i\le j} x\_{i}x\_{j}, \sum\_{i\le j\le k} x\_{i}x\_{j}x\_{k},...]$$


The later ring is obviously a ring of elementary symmetric polynomials. We are wondering if anyone from ring theory has already worked out this earlier. My advisor suggest I should look up online resources before I ask in here, but I could not find anything particularly relevant (I do not know if I am being lazy). I already searched sciencedirect, arxiv, wolfram-mathworld, wikipedia, etc. It seems to me that people are working with the basis of the ring of symmetrical polynomials itself instead of this problem. 


I am more willing to work out this myself instead of learning from reading others, but I think I should have an accurate bibilography to acknowledge other's work. The associated question is I do not know what is the usual practice in such situation, for every trivial question in a research project may be something already well-known to experts. For example I do not wish to ask similar questions for type B,C,D, etc's representation ring in here. I wish to apologize in advance if this turned out to be something well-known or mundane.


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For example, the well-known $SU(2)$ case has a torus whose representation ring is isomorphic to Laurent series in one variable, and $R(SU(2))\cong \mathbb{Z}[w+w^{-1}]$, which is the above problem when $n=2$. I am not familiar with ring theory and module theory (I know Hungerford's section in rings and some Hilton&Stammach,etc, but these seem quite unrelated). 


ADDED:


Now voting to close. I might have seen this in Fulton&Harris so it should be well-known.


ADDED: 


I finished the proof that David Speyer's basis is correct. I extended the result by low-dimensional coincidences to a few $SO\_{2n}$ cases. 



Reference Request: Basis in terms of ring of symmetric polynomials
Let $A$ be the ring $\mathbb{Z}[x\_1, \ldots, x\_n]$ and let $\Lambda$ be the subring of symmetric polynomials. Then $A$ is free as a $\Lambda$ module, and there are two fairly standard sets of bases. 


The first is the monomials $x\_1^{a\_1} x\_2^{a\_2} \cdots x\_n^{a\_n}$ with $0 \leq a\_i \leq n-i$. (So $a\_n$ is always $0$, and there is no $x\_n$ term.) 


The second is the [Schubert polynomials](http://en.wikipedia.org/wiki/Schubert_polynomial).


Your question is about $A/(x\_1 \cdots x\_n-1)$ as a module over $\Lambda/(x\_1 \cdots x\_n-1)$. Taking a quotient won't change the properties of being free, or of having a particular basis.


