If π is a representation of a Lie algebra, then π π ¯ ¯ is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication. In general, every irreducible representation π of s l ( n , C ) can be written uniquely as π = π + π, where π π ± ± ( X ) = 1 2 ( π π ( X ) ± ± i π π ( i − − 1 X ) ) , with π π + holomorphic (complex linear) and π π − − anti-holomorphic (conjugate linear). For s l ( 2 , C ) , since π π μ μ is holomorphic, π π μ μ ¯ ¯ is anti-holomorphic. Direct examination of the explicit expressions for π π μ μ , 0 and π π 0 , ν ν in equation (S8) below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression (S8) also allows for identification of π π + and π π − − for π π μ μ , ν ν as π π μ μ , ν ν + = π π μ μ ⊕ ⊕ ν ν + 1 , π π μ μ , ν ν − − = π π ν ν ⊕ ⊕ μ μ + 1 ¯ ¯ .