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ADDED: I hadn't heard previously about the recent death of Harsh Pittie. I was somewhat acquainted with him when we were both at NYU-Courant decades ago and recall hearing some of his lectures on topology of Lie groups. His paper from that period was grounded in topology and K-theory, but Steinberg's follow-up (in his typical concise style) rounded out the discussion of representation rings in a more algebraic framework. Moreover, Steinberg exhibits an explicit basis for $R(T)$ as a free $R(G)$-module in the crucial case where $G$ is a semisimple simply connected compact Lie group and $T$ any maximal torus. In particular, the rank here is the order of the Weyl group $W$. (He also observes that the same ideas work for algebraic groups over any algebraically closed field.)

Though I've never worked through the details of Steinberg's paper carefully, the underlying idea can be observed (in an oversimplified way) in the rank 1 case. Denoting the weight lattice (character group of $T$ in additive notation) by $X$, the respective representation rings look like $\mathbb{Z}[X]$ and $\mathbb{Z}[X]^W$. Then Steinberg's basis elements, one for each element $w \in W$, are defined by applying $w^{-1}$ to a product of symbols (in my notation $e^\lambda$) with $\lambda$ running over suitable fundamental weights. In rank 1, the basis just consists of $e^0, e^{-\rho}$.

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To supplement Barry's citations, I'd point out that the journal Topology was at that time managed by a company which eventually gave up on it after editors resigned partly in protest against the high prices charged. While the online rights now belong to the ScienceDirect conglomerate, it's expensive to access. This can be frustrating because each paper discussed here is only 4+ pages long.

On the other hand, Steinberg's paper is reprinted in the moderately priced one volume Collected Papers (AMS 1997). Though Steinberg is long retired from UCLA, he maintains an email link there, and might be able to supply a reprint of his article. Pittie is an Indian mathematician who has taught at one of the colleges of City University of New York but has not published for many years; his entry in the combined membership list CML (www.ams.org) does give a current mailing address in New York City.

Some users of MO including myself do have access to both papers and might be able to answer precisely stated questions about them.