Question

As part of the result of solving the problem 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.

ADDED:

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.

Answer 1

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.

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.