I have this $2^n*2^n$ matrix that represent the evolution of a system of $n$ spin. I know that I can have only one excited spin in my configuration a time. (eg: 0110 nor 0101 ar not permitted, but 0100 it is)
$s_+$ is defined with $s_x+is_y$ and $s_-$ is defined with $s_x-is_y$ (creation and annichilation operators)
I've created the hamiltonian composing pauli matrices with kronecker product. I now that I have only few operation in my algebra: $I, s_z, s_+[k].s_-[k-1]$ where k is the spin number where this operation is made. This algebra sends valid state to valid states, preserving the constant of my motion.
How can i reduce the dimension of my matrix deleting row and columns to remain in a proper subspace? How can I have a $n*n$ matrix??