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Maximal proper extremal subset in the crystal basis of a finite-dimensional irreducible module

  • You Zhou
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Abstract

The notion of extremal subsets was introduced by Assaf, Dranowski, and González as a natural generalization of Demazure crystals. In this paper, we show that the crystal basis of a finite-dimensional irreducible module over a quantum group of finite type has a unique maximal proper extremal subset, whose complement is the set of elements that generate the whole crystal basis as an extremal subset. We conjecture that this complement consists of all nonzero elements obtained by successively applying the lowering Kashiwara operators to a distinguished element, which is constructed from the lowest element of the crystal basis. We prove this conjecture in type A.

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Cite

BibTeXRIS
BibTeX
@misc{zhou2026maximal,
  title = {{Maximal proper extremal subset in the crystal basis of a finite-dimensional irreducible module}},
  author = {You Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23078},
  eprint = {2609.23078},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23078},
}
RIS
TY  - UNPB
TI  - Maximal proper extremal subset in the crystal basis of a finite-dimensional irreducible module
AU  - You Zhou
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23078
UR  - https://arxiv.org/abs/2609.23078
AB  - The notion of extremal subsets was introduced by Assaf, Dranowski, and González as a natural generalization of Demazure crystals. In this paper, we show that the crystal basis of a finite-dimensional irreducible module over a quantum group of finite type has a unique maximal proper extremal subset, whose complement is the set of elements that generate the whole crystal basis as an extremal subset. We conjecture that this complement consists of all nonzero elements obtained by successively applying the lowering Kashiwara operators to a distinguished element, which is constructed from the lowest element of the crystal basis. We prove this conjecture in type A.
ER  -