■単純リー環を使った面数数え上げ(その39)
ワイソフ算術を使って,Cnより深切稜の場合の多胞体のf0,f1,fn-1も求めておきたい.
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【1】正単体版
f0=(n−1)n(n+1)/2,n>3
m=n,f1=m/2・f0=n/2・f0
{3,3}(011)→f0=12(OK)
{3,3,3}(0110)→f0=30(OK)
{3,3,3,3}(01100)→f0=60(OK)
{3,3,3,3,3}(011000)→f0=105(OK)
f1=n/2・f0
{3,3}(011)→f1=18(OK)
{3,3,3}(0110)→f1=60(OK)
{3,3,3,3}(01100)→f1=150(OK)
{3,3,3,3,3}(011000)→f1=315(OK)
fn-1=2(n+1)
{3,3}(011)→f2=8(OK)
{3,3,3}(0110)→f3=10(OK)
{3,3,3,3}(01100)→f4=12(OK)
{3,3,3,3,3}(011000)→f5=14(OK)
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【2】正軸体版
f0=4n(n−1)(n−2)
m=2n−4,f1=m/2・f0=(n−2)・f0,n>3
{3,4}(011)→f0=24(OK)
{3,3,4}(0110)→f0=96(OK)
{3,3,3,4}(01100)→f0=240(OK)
{3,3,3,3,4}(011000)→f0=480(OK)
f1=(n−2)・f0,n>3
{3,4}(011)→f1=36(OK)
{3,3,4}(0110)→f1=192(OK)
{3,3,3,4}(01100)→f1=720(OK)
{3,3,3,3,4}(011000)→f1=1920(OK)
fn-1=2^n+2n
{3,4}(011)→f2=14(OK)
{3,3,4}(0110)→f3=24(OK)
{3,3,3,4}(01100)→f4=42(OK)
{3,3,3,3,4}(01100)→f5=76(OK)
【雑感】しかしながら,重複が生ずるため,意味論的解釈は難しいようである.
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