$\imath$Hall algebra of Jordan quiver and $\imath$Hall-Littlewood functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866917188018896896 |
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| author | Lu, Ming Ruan, Shiquan Wang, Weiqiang |
| author_facet | Lu, Ming Ruan, Shiquan Wang, Weiqiang |
| contents | We show that the $\imath$Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this $\imath$Hall algebra to the ring of symmetric functions in two parameters $t, θ$, which maps the $\imath$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($\imath$HL) functions. The (modified) $\imath$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $\imath$HL functions. The modified $\imath$HL functions specialize at $θ=0$ to the modified HL functions; they specialize at $θ=1$ to the deformed universal characters of type C, which further specialize at $(t=0, θ=1)$ to the universal characters of type C. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_12336 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $\imath$Hall algebra of Jordan quiver and $\imath$Hall-Littlewood functions Lu, Ming Ruan, Shiquan Wang, Weiqiang Representation Theory Combinatorics Quantum Algebra We show that the $\imath$Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this $\imath$Hall algebra to the ring of symmetric functions in two parameters $t, θ$, which maps the $\imath$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($\imath$HL) functions. The (modified) $\imath$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $\imath$HL functions. The modified $\imath$HL functions specialize at $θ=0$ to the modified HL functions; they specialize at $θ=1$ to the deformed universal characters of type C, which further specialize at $(t=0, θ=1)$ to the universal characters of type C. |
| title | $\imath$Hall algebra of Jordan quiver and $\imath$Hall-Littlewood functions |
| topic | Representation Theory Combinatorics Quantum Algebra |
| url | https://arxiv.org/abs/2104.12336 |