A conjecture on monomial realizations and polyhedral realizations for crystal bases
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913729798471680 |
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| author | Kanakubo, Yuki |
| author_facet | Kanakubo, Yuki |
| contents | Crystal bases are powerful combinatorial tools in the representation theory of quantum groups $U_q(\mathfrak{g})$ for a symmetrizable Kac-Moody algebras $\mathfrak{g}$. The polyhedral realizations are combinatorial descriptions of the crystal base $B(\infty)$ for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when $\mathfrak{g}$ is finite dimensional simple. It is a fundamental and natural problem to find explicit forms of the polyhedral cone. The monomial realization expresses crystal bases $B(λ)$ of integrable highest weight representations as Laurent monomials with double indexed variables. In this paper, we give a conjecture between explicit forms of the polyhedral cones and monomial realizations. We prove the conjecture is true when $\mathfrak{g}$ is a classical Lie algebra, a rank $2$ Kac-Moody algebra or a classical affine Lie algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A conjecture on monomial realizations and polyhedral realizations for crystal bases Kanakubo, Yuki Quantum Algebra Combinatorics Representation Theory Crystal bases are powerful combinatorial tools in the representation theory of quantum groups $U_q(\mathfrak{g})$ for a symmetrizable Kac-Moody algebras $\mathfrak{g}$. The polyhedral realizations are combinatorial descriptions of the crystal base $B(\infty)$ for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when $\mathfrak{g}$ is finite dimensional simple. It is a fundamental and natural problem to find explicit forms of the polyhedral cone. The monomial realization expresses crystal bases $B(λ)$ of integrable highest weight representations as Laurent monomials with double indexed variables. In this paper, we give a conjecture between explicit forms of the polyhedral cones and monomial realizations. We prove the conjecture is true when $\mathfrak{g}$ is a classical Lie algebra, a rank $2$ Kac-Moody algebra or a classical affine Lie algebra. |
| title | A conjecture on monomial realizations and polyhedral realizations for crystal bases |
| topic | Quantum Algebra Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2503.06417 |