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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.

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I fixed the typos in your formulas. – Gjergji Zaimi Oct 8 '12 at 4:33
This identity was also found independently in: MR2218820 (2007c:17009) Westbury, Bruce W. Universal characters from the Macdonald identities. Adv. Math. 202 (2006), no. 1, 50--63. – Bruce Westbury Oct 8 '12 at 7:56
Have you looked at Theorem 1 in MR2837629 (2012i:05019) Dehaye, Paul-Olivier ; Han, Guo-Niu . A multiset hook length formula and some applications. Discrete Math. 311 (2011), no. 23-24, 2690--2702. ? – Bruce Westbury Oct 8 '12 at 8:09

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).

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.

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