| layout | title | arXiv | comments | coauthors | categories | published | journal-ref | journal-web | image | image-alt | show-date | source | post-source | post-pdf | pages | cv-number | ||||
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Refined Cauchy identity for spin Hall-Littlewood symmetric rational functions |
2007.10886 [math.CO] |
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blog math paper |
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Journal of Combinatorial Theory Ser. A, vol. 184 (2021), 105519 |
__STORAGE_URL__/img/papers/refined_sHL.png |
Two lattices with equal partition functions. This leads to a refined Cauchy identity |
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37-publ-petrov-post-tenure-published.pdf |
37source.txt |
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true |
40 |
37 |
Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $$\mathsf{F}\lambda$$ arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two $\mathsf{F}\lambda$'s as a determinant. The determinant is of Izergin-Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang-Baxter equation.
We rewrite our Izergin-Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall-Littlewood rational functions with (the Hall-Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin-Korepin type determinant yields a deformation of Schur symmetric polynomials.
In the spin-$\frac12$ specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar-Parisi-Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.