  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.
