Irreducible representations of $\mathbb{Z}_2^2$-graded supersymmetry algebra and their applications

We give a brief review on recent developments of \mathbb{Z}_2^nℤ2n-graded symmetry in physics in which hidden \mathbb{Z}_2^nℤ2n-graded symmetries and \mathbb{Z}_2^nℤ2n-graded extensions of known systems are discussed. This elucidates physical relevance of the \mathbb{Z}_2^nℤ2n-graded algebras. As an example of physically interesting algebra, we take \mathbb{Z}_2^2ℤ22-graded supersymmetry (SUSY) algebras and consider their irreducible representations (irreps). A list of irreps for N = 1, 2N=1,2 algebras is presented and as an application of the irreps, \mathbb{Z}_2^2ℤ22-graded SUSY classical actions are constructed.