有限维不可约模的晶体基中的极大真极值子集
Maximal proper extremal subset in the crystal basis of a finite-dimensional irreducible module
AI总结：
本文证明了有限型量子群有限维不可约模的晶体基存在唯一极大真极值子集，并猜想其补集由下降Kashiwara算子作用于特定元素所得，且在A型中验证了该猜想。
AI中文摘要：
极值子集的概念由Assaf、Dranowski和González引入，作为Demazure晶体的自然推广。本文证明了有限型量子群上有限维不可约模的晶体基具有唯一的极大真极值子集，其补集是由生成整个晶体基作为极值子集的元素组成的集合。我们猜想该补集包含所有通过对一个特定元素连续应用下降Kashiwara算子得到的非零元素，该特定元素由晶体基的最低元素构造。我们在A型情形证明了这一猜想。
英文摘要：
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.
