Definition 3.2

For two crystals B 1 subscript 𝐵 1 B_{1} italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and B 2 subscript 𝐵 2 B_{2} italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , a morphism of crystals from B 1 subscript 𝐵 1 B_{1} italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to B 2 subscript 𝐵 2 B_{2} italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is a map ψ : B 1 ⊔ { 0 } → B 2 ⊔ { 0 } : 𝜓 → square-union subscript 𝐵 1 0 square-union subscript 𝐵 2 0 \psi:B_{1}\sqcup\{0\}\to B_{2}\sqcup\{0\} italic_ψ : italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊔ { 0 } → italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊔ { 0 } such that

(3.2) ψ ⁢ ( 0 ) = 0 , 𝜓 0 0 \displaystyle\psi(0)=0, italic_ψ ( 0 ) = 0 ,
ψ ⁢ ( e ~ i ⁢ b ) = e ~ i ⁢ ψ ⁢ ( b ) for ⁢ b ∈ B 1 with e ~ i ⁢ b ∈ B 1 , ψ ⁢ ( b ) ∈ B 2 , ψ ⁢ ( e ~ i ⁢ b ) ∈ B 2 , formulae-sequence 𝜓 subscript ~ 𝑒 𝑖 𝑏 subscript ~ 𝑒 𝑖 𝜓 𝑏 formulae-sequence for 𝑏 subscript 𝐵 1 with formulae-sequence subscript ~ 𝑒 𝑖 𝑏 subscript 𝐵 1 formulae-sequence 𝜓 𝑏 subscript 𝐵 2 𝜓 subscript ~ 𝑒 𝑖 𝑏 subscript 𝐵 2 \displaystyle\psi({\tilde{e}_{i}}b)=\tilde{e}_{i}\psi(b)\ \ \text{for}\ b\in B% _{1}\ \ \text{with}\ \ {\tilde{e}_{i}}b\in B_{1},\ \psi(b)\in B_{2},\ \psi(% \tilde{e}_{i}b)\in B_{2}, italic_ψ ( ~ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b ) = ~ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ψ ( italic_b ) for italic_b ∈ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with ~ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b ∈ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ψ ( italic_b ) ∈ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_ψ ( ~ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b ) ∈ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,
ψ ⁢ ( f ~ i ⁢ b ) = f ~ i ⁢ ψ ⁢ ( b ) for ⁢ b ∈ B 1 with f ~ i ⁢ b ∈ B 1 , ψ ⁢ ( b ) ∈ B 2 , ψ ⁢ ( f ~ i ⁢ b ) ∈ B 2 , formulae-sequence 𝜓 subscript ~ 𝑓 𝑖 𝑏 subscript ~ 𝑓 𝑖 𝜓 𝑏 formulae-sequence for 𝑏 subscript 𝐵 1 with formulae-sequence subscript ~ 𝑓 𝑖 𝑏 subscript 𝐵 1 formulae-sequence 𝜓 𝑏 subscript 𝐵 2 𝜓 subscript ~ 𝑓 𝑖 𝑏 subscript 𝐵 2 \displaystyle\psi({\tilde{f}_{i}}b)=\tilde{f}_{i}\psi(b)\ \ \text{for}\ b\in B% _{1}\ \ \text{with}\ \ {\tilde{f}_{i}}b\in B_{1},\ \psi(b)\in B_{2},\ \psi(% \tilde{f}_{i}b)\in B_{2}, italic_ψ ( ~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b ) = ~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ψ ( italic_b ) for italic_b ∈ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with ~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b ∈ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ψ ( italic_b ) ∈ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_ψ ( ~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_b ) ∈ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,
wt ⁡ ( ψ ⁢ ( b ) ) = wt ⁡ ( b ) for ⁢ b ∈ B 1 with ψ ⁢ ( b ) ∈ B 2 , formulae-sequence wt 𝜓 𝑏 wt 𝑏 formulae-sequence for 𝑏 subscript 𝐵 1 with 𝜓 𝑏 subscript 𝐵 2 \displaystyle{\operatorname{wt}}(\psi(b))={\operatorname{wt}}(b)\ \ \text{for}% \ b\in B_{1}\ \ \text{with}\ \ \psi(b)\in B_{2}, roman_wt ( italic_ψ ( italic_b ) ) = roman_wt ( italic_b ) for italic_b ∈ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with italic_ψ ( italic_b ) ∈ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,
ε i ⁢ ( ψ ⁢ ( b ) ) = ε i ⁢ ( b ) , φ i ⁢ ( ψ ⁢ ( b ) ) = φ i ⁢ ( b ) for ⁢ b ∈ B 1 with ψ ⁢ ( b ) ∈ B 2 . formulae-sequence subscript 𝜀 𝑖 𝜓 𝑏 subscript 𝜀 𝑖 𝑏 formulae-sequence subscript 𝜑 𝑖 𝜓 𝑏 subscript 𝜑 𝑖 𝑏 formulae-sequence for 𝑏 subscript 𝐵 1 with 𝜓 𝑏 subscript 𝐵 2 \displaystyle\varepsilon_{i}(\psi(b))=\varepsilon_{i}(b),\ \ \varphi_{i}(\psi(% b))=\varphi_{i}(b)\ \ \text{for}\ b\in B_{1}\ \ \text{with}\ \ \psi(b)\in B_{2}. italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ψ ( italic_b ) ) = italic_ε start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_b ) , italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ψ ( italic_b ) ) = italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_b ) for italic_b ∈ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with italic_ψ ( italic_b ) ∈ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .