0P7M Lemma 6.2.5. Consider σ∈Hom𝒮n(I,J)\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J) and σ′∈Hom𝒮n(J,K)\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,K). We have dm(σ′∘σ)=dm(σ′)⋅dm(σ)\operatorname{dm}\nolimits(\sigma^{\prime}\circ\sigma)=\operatorname{dm}\nolimits(\sigma^{\prime})\cdot\operatorname{dm}\nolimits(\sigma).
0P7N Proof. We have m(σ′∘σ)=⟦σ′⟧⋅εI+⟦σ⟧εI=m(σ′)+m(σ)+⟦σ′⟧⋅(εI−εJ),m(\sigma^{\prime}\circ\sigma)=\llbracket\sigma^{\prime}\rrbracket\cdot\varepsilon_{I}+\llbracket\sigma\rrbracket\varepsilon_{I}=m(\sigma^{\prime})+m(\sigma)+\llbracket\sigma^{\prime}\rrbracket\cdot(\varepsilon_{I}-\varepsilon_{J}), hence m(σ′)+m(σ)−m(σ′∘σ)=⟦σ′⟧⋅ρ(⟦σ⟧).m(\sigma^{\prime})+m(\sigma)-m(\sigma^{\prime}\circ\sigma)=\llbracket\sigma^{\prime}\rrbracket\cdot\rho(\llbracket\sigma\rrbracket). The lemma follows. ∎