CKP层次的两个玻色化：概述和特征恒等式
The two bosonizations of the CKP hierarchy: overview and character identities
摘要：
本文讨论了由日本物理学会的Date，Jimbo，Kashiwara和Miwa提出的CKP族的Hirota双线性方程。50（11），3813-3818（1981），及其代数性质。我们平行地回顾了CKP层级的两个玻色化：一个来自扭曲的海森堡代数（van de Leur，Orlov和Shiota，SIGMA 8，28（2012）），第二个来自非扭曲的海森堡代数（Anguelova，J. Math. Phys. 58（7），071707（2017））。特别是，我们叙述的分解成不可约海森堡模块和（扭曲）的费米子结构的最高权向量下的两个海森堡行动所跨越的空间。我们表明，这两个玻色化产生四个不同的对角化分次算子的CKP Fock空间，而不是所有的人之间相互交换。我们计算与这四个分级算子相关的各种分级维度。我们证明了一个和与产品的身份有关的玻色子与费米子的描述下untwisted海森堡行动，利用收费和度分级运营商。作为一个推论，由此产生的身份关系的CKP层次，戴森曲柄的分区和Hammond-Lewis birank的一个不同的整数bipartition。
英文摘要：
We discuss the Hirota bilinear equation for the CKP hierarchy introduced by Date, Jimbo, Kashiwara and Miwa in J. Phys. Soc. Japan. 50(11), 3813-3818 (1981), and its algebraic properties. We review in parallel the two bosonizations of the CKP hierarchy: one arising from a twisted Heisenberg algebra (van de Leur, Orlov and Shiota, SIGMA 8, 28 (2012)), and the second from an untwisted Heisenberg algebra (Anguelova, J. Math. Phys. 58(7), 071707 (2017)). In particular, we recount the decompositions into irreducible Heisenberg modules and the (twisted) fermionic structures of the spaces spanned by the highest weight vectors under the two Heisenberg actions. We show that the two bosonizations give rise to four different diagonalizable grading operators on the CKP Fock space, not all of them commuting among each other. We compute the various graded dimensions related to these four grading operators. We prove a sum-vs-product identity relating the bosonic vs fermionic descriptions under the untwisted Heisenberg action, utilizing the charge and the degree grading operators. As a corollary, the resulting identities relate the CKP hierarchy, the Dyson crank of a partition and the Hammond-Lewis birank of a distinct integer bipartition.
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