Think of a Young diagram as a collection of rows with numbers of elements $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ moves along the column, $j$ along the row; it makes sense also for $s$ outside $R$) $$ l_R(i,j) := \mu_j^t-i, \qquad a_R(i,j) := \mu_i-j $$ where $R^t$ is transpose (namely rotate by 90 degrees).
Then I would like to prove that $$ Z:= \prod_{s\in R_1} \prod_{s'\in R_2} \frac1{1 - b(l_{R_2}(s) +a_{R_1}(s)+1)} \frac1{1 + b(l_{R_1}(s') +a_{R_2}(s')+1)} $$ can be written as (notice the transpose) $$ Z=\prod_k \frac1{(1+b k)^{C_k(R_1,R_2^t)}}$$ if we define $$ f_R(q):= \sum_{i=1}^d \sum_{j=1}^{\mu_i} q^{j-i} $$ and $$ f_{R_1}(q) + f_{R_2}(q) + f_{R_1}(q)f_{R_2}(q) (q+q^{-1}-2) =:\sum_k C_k(R_1,R_2) q^k $$
Origin: the desired equality follows from equality of Nekrasov instanton partition function in supersymmetric gauge theory, as written e.g. in the Nakajima paper and by Nekrasov in the original paper.
There should also be some relation with Jack polynomials and Plancherel measure. The connection with representation theory comes from the fact that the above is computed as a virtual representation
Refs:
http://arxiv.org/abs/math/0306198 thm 2.11
http://arxiv.org/abs/hep-th/0310235 appendix B
http://arxiv.org/abs/hep-th/0206161
http://member.ipmu.jp/yuji.tachikawa/transp/masterthesis.pdf eqs 4.82,83,84
http://arxiv.org/abs/hep-th/0306238 section 3.1
The book Symmetric functions by Macdonald