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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2312.06138 |
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| _version_ | 1866929544281194496 |
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| author | Garbali, Alexandr Gunna, Ajeeth |
| author_facet | Garbali, Alexandr Gunna, Ajeeth |
| contents | We consider partition functions on the $N\times N$ square lattice with the local Boltzmann weights given by the $R$-matrix of the $U_{t}(\widehat{sl}(n+1|m))$ quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function $Z_N$ on this lattice computes the weighted sum over all possible closed coloured lattice paths with $n+m$ different colours: $n$ ``bosonic'' colours and $m$ ``fermionic'' colours. Each bosonic (fermionic) path of colour $i$ contributes a factor of $z_i$ ($w_i$) to the weight of the configuration. We show the following. (i) $Z_N$ is a symmetric function in the spectral parameters $x_1\dots x_N$ and generates basis elements of the commutative trigonometric Feigin--Odesskii shuffle algebra. The generating function of $Z_N$ admits a shuffle-exponential formula analogous to the Macdonald Cauchy kernel. (ii) $Z_N$ is a symmetric function in two alphabets $(z_1\dots z_n)$ and $(w_1\dots w_m)$. When $x_1\dots x_N$ are set to be equal to the box content of a skew Young diagram $μ/ν$ with $N$ boxes the partition function $Z_N$ reproduces the skew Macdonald function $P_{μ/ν}\left[w-z\right]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06138 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Shuffle algebras, lattice paths and Macdonald functions Garbali, Alexandr Gunna, Ajeeth Mathematical Physics Quantum Algebra Representation Theory We consider partition functions on the $N\times N$ square lattice with the local Boltzmann weights given by the $R$-matrix of the $U_{t}(\widehat{sl}(n+1|m))$ quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function $Z_N$ on this lattice computes the weighted sum over all possible closed coloured lattice paths with $n+m$ different colours: $n$ ``bosonic'' colours and $m$ ``fermionic'' colours. Each bosonic (fermionic) path of colour $i$ contributes a factor of $z_i$ ($w_i$) to the weight of the configuration. We show the following. (i) $Z_N$ is a symmetric function in the spectral parameters $x_1\dots x_N$ and generates basis elements of the commutative trigonometric Feigin--Odesskii shuffle algebra. The generating function of $Z_N$ admits a shuffle-exponential formula analogous to the Macdonald Cauchy kernel. (ii) $Z_N$ is a symmetric function in two alphabets $(z_1\dots z_n)$ and $(w_1\dots w_m)$. When $x_1\dots x_N$ are set to be equal to the box content of a skew Young diagram $μ/ν$ with $N$ boxes the partition function $Z_N$ reproduces the skew Macdonald function $P_{μ/ν}\left[w-z\right]$. |
| title | Shuffle algebras, lattice paths and Macdonald functions |
| topic | Mathematical Physics Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2312.06138 |