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

B ∈ Λ(d)
ε ∗ i (B) := dim C Ker
(
)
⊕B
V (d)
τ
i −→ ⊕ V (d) in(τ)
τ∈H;out(τ)=i
P (Γ)-module i-th socle
Λ ∈ IrrΛ(d) Λ generic B ∈ Λ
ε ∗ i (Λ) := ε ∗ i (B)
ϕ ∗ i (Λ) := ε ∗ i (Λ) + (s(i), wt(Λ))
4.3 ε i = i-th top IrrΛ(d) ε ∗ i = i-th socle
IrrΛ(d) = ⊔ ( ) p
IrrΛ(d) , where ( IrrΛ(d) ) p
:= { Λ ∈ IrrΛ(d) | ε ∗ i i i (Λ) = p } .
p≥0
4.3
(ẽ ∗ i ) max : ( IrrΛ(d) ) l ∼
→ ( IrrΛ(d ′ ) ) 0
i
i
d ′ := d − ls(i) 21
top socle (4.3.1)
map
0 → S(i) ⊕l φ ′′
−→ V (d) φ′
−→ V (d ′ ) → 0. (4.5.1)
(4.3.2) (ẽ ∗ i ) max
(ẽ ∗ i ) max i-th cosocle
(ẽ ∗ i ) max
ẽ ∗ i : B → B ⊔ {0}, ˜f
∗
i : B → B
Theorem 19 ([10]). (1) ( B; wt, ε ∗ i , ϕ ∗ i , ẽ ∗ i , ˜f
) (
i
∗ root datum Q I , {s(i)}, (·, ·) )
crystal
(2) crystal ( B; wt, ε ∗ i , ϕ ∗ i , ẽ ∗ i , ˜f
)
i
∗ B(∞)
(2) Theorem 18 Theorem 19
(
B; wt, εi , ϕ i , ẽ i , ˜f
) (
i
∼= B(∞) ∼ = B; wt, ε
∗
i , ϕ ∗ i , ẽ ∗ i , ˜f
)
i
∗
crystal
“ε i = ε ∗ i ”
top socle
21 d ′ = d ′′
–182–