Universal fermionization of bosons on permutative representations of the Cuntz algebra ${\mathcal{O}}_2$

Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. We show a fermionization of bosons which universally holds on any permutative representation of the Cuntz algebra ${\mathcal{O}}_2$. As examples, we show fermionizations on the Fock space and the infinite wedge.