TITLE: A q,t-extension of Plancherel Measure thru Yang-Mills Theory ?
QUESTION [8 upvotes]: Buried in the physics paper by Nekrasov and Okounkov, a strange identity is proven:
$$ \prod_{n > 0} (1 - q^n)^{\mu^2-1} = \sum_{\mathbf{k}} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - \frac{\mu^2}{h(\square)^2}\right) $$
where the left side is a q-series and the right side is the sum over all partitions.  Ihis was proven by physical considerations, evaluating the Yang-Mills partition function in 2 different ways.
The partitions could index representations of the permutation group $S_n$.    We can define measure on partitions, $\mathrm{Irr}(S_n)$ by
$$ \mathbb{P}\_{\mu, t} (\mathbf k) = \prod_{n \geq 1} (1-t^n)^{1-\mu^2} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - \frac{\mu^2}{h(\square)^2}\right) $$
In fact, 3 years later Alexei Borodin explains this formula interpolates between  uniform and Plancherel measures on partitions.

Can this be extended to a q,t-deformation of uniform measure on the permutation group?  Maybe through something similar to Robinson-Schensted correspondence.

REPLY [4 votes]: Recently there is An insertion algorithm associated with q-Whittaker functions 
defining a q-weighted form of the Robinson-Schensted algorithm.  
The MacDonald processes contain many examples of "integrable" combinatorial processes on Gelfand-Tsetlin patterns (see this talk).

q-Whittaker processes
Hall-Littlewood processes
Random matrices
Whittaker processes
Kingman partition structures
schur processes

Many of these feel nature from the point of view of representation theory of symmetric groups.  Macdonald functions are a q,t-deformation of this.