Proceedings of the 44th Symposium on Ring Theory and ...

Definition 12. B crystal B
B crystal graph
b ′ = ˜f i b b b ′ i ∈ I
i
b ✲ b ′
Example 13. Γ A n quiver orientation ω 1 ∈ P
n + 1
(ω 1 , s(i)) = δ 1,i
B(ω 1 ) := {b 0 , b 1 , · · · , b n }
wt, ε i , ϕ i , ẽ i , ˜f i
k∑
wt(b k ) := ω 1 − s(i) (0 ≤ k ≤ n),
{
1 (i = k),
ε i (b k ) :=
0 (o<strong>the</strong>rwise),
{
b k−1 (i = k),
ẽ i b k :=
0 (o<strong>the</strong>rwise),
i=1
{
1 (i = k + 1),
ϕ i (b k ) :=
0 (o<strong>the</strong>rwise),
{
b k+1 (i = k + 1),
˜f i b k :=
0 (o<strong>the</strong>rwise).
B(ω 1 ) crystal crystal graph
b 0
1 ✲ b 2 1
✲ b 3 2
✲ · · · n ✲ b n
B(ω 1 ) A n vector crystal basiscrystal
Example 14. Γ loop quiver j ∈ I Z
B j := {b j (k) | k ∈ Z}
wt, ε i , ϕ i , ẽ i , ˜f i
wt(b j (k)) := ks(j) (k ∈ Z),
{
{
−k (i = j),
k (i = j),
ε i (b j (k)) :=
ϕ i (b j (k)) :=
−∞ (i ≠ j),
−∞ (i ≠ j),
{
{
b j (k + 1) (i = j),
b j (k − 1) (i = j),
ẽ i b j (k) :=
˜f i b j (k) :=
0 (i ≠ j),
0 (i ≠ j).
B j crystal crystal graph
j j
· · · ✲ ❡
b j (k − 1)
✲ ❡
b j (k)
j j j
✲ ❡ ✲ ❡
b j (k + 1) b j (k + 2)
–177–
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