Kronecker products of projective representations of translation groups

Irreducible projective representations of the translation group of a finite N×N two-dimensional lattice can be labeled by symbols 〈n, l;q〉, where N = νn, gcd(l, n) = 1 and q denotes an irreducible representation of Z ν . Obtained matrices are n-dimensional and the factor system of this representation does not depend on q and equals m (l) n ([n1, n2], [n ′ 1, n ′ 2]) = exp(2πi ln2n ′ 1/N). For a given n the N × N lattice can be viewed as a ν × ν lattice consisting of n × n magnetic cells. The Kronecker product of such representations is another projective representation which can be decomposed into irreducible ones. It is interesting that such product can lead to the magnetic periodicity different from n, n′ and even nn′ or lcm(n, n′). For example, a product 〈n, l;q〉⊗ 〈n, l;q′〉 for even N decomposes into representations 〈 2 , l;k〉: there are four representations with ki − qi − q ′ i = 0, ν and each of them appears n/2 times. Similarly, the coupling of d representations 〈n, 1;q〉, j = 1, 2, . . . , d with n = dM changes the magnetic period from n to M . It is caused by multiplication of the charge by d what corresponds to a system of d electrons.