A conjecture on monomial realizations and polyhedral realizations for crystal bases

Abstract

Crystal bases are powerful combinatorial tools in the representation theory of quantum groups U q ( 𝔤 ) for a symmetrizable Kac-Moody algebras 𝔤 . The polyhedral realizations are combinatorial descriptions of the crystal base B ( ∞ ) for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when 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 𝔤 is a classical Lie algebra, a rank 2 Kac-Moody algebra or a nonexceptional affine Lie algebra.

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BibTeX RIS

BibTeX

@article{kanakubo2026conjecture,
  title = {{A conjecture on monomial realizations and polyhedral realizations for crystal bases}},
  author = {Yuki Kanakubo},
  journal = {Algebraic Combinatorics},
  year = {2026},
  month = oct,
  doi = {10.5802/alco.518},
  url = {https://alco.centre-mersenne.org/articles/10.5802/alco.518/},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - A conjecture on monomial realizations and polyhedral realizations for crystal bases
AU  - Yuki Kanakubo
PY  - 2026
DA  - 2026/10/27
JO  - Algebraic Combinatorics
DO  - 10.5802/alco.518
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.518/
AB  - Crystal bases are powerful combinatorial tools in the representation theory of quantum groups U q ( 𝔤 ) for a symmetrizable Kac-Moody algebras 𝔤 . The polyhedral realizations are combinatorial descriptions of the crystal base B ( ∞ ) for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when 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 𝔤 is a classical Lie algebra, a rank 2 Kac-Moody algebra or a nonexceptional affine Lie algebra.
SN  - 2589-5486
ER  -

Abstract

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