Coxeter-Dinkinov diagram

Coxeter-Dinkinov diagram (tudi Coxeterjev diagram ali Coxeterjev graf) je graf, ki ima s številkami označene stranice (imenujejo se veje) s katerimi se prikaže prostorske odnose med zbirko zrcal oziroma odbojnih hiperravnin. Opisujejo kalejdoskopsko konstrukcijo: vsak vozel grafa predstavlja ogledalo (v domeni facete). Oznaka pri vsaki veji določa stopnjo diedrskega kota dveh ogledal (v domeni grebena). Neoznačene veje pomenijo red 3.

Coxeter-Dinkinovi diagrami za osnovne končne Coxeterjeve grupe.

Coxeter-Dinkinovi diagrami za osnovne afine Coxeterjeve grupe.

Vsak diagram predstavlja Coxeterjevo grupo in tudi Coxeterjeve grupe so razvrščene po pripadajočih diagramih.

Podobni so Dinkinovi diagrami. Ti se od Coxeterjevih diagramov razlikujejo samo v tem, da so v Dinkinovih diagramih veje, ki imajo oznako 4 ali več, usmerjene. Coxeterjevi diagrami so neusmerjeni. Razen tega morajo Dinkinovi diagrami zadoščati še dodatni kristalografski omejitvi, ki zahteva, da so dovoljene veje samo 2, 3, 4 in 6.

Opis diagramov

Veje Coxeter-Dinkinovih diagramov so označene z racionalnimi števili ${\displaystyle p\,}$, kar predstavlja diedrski kot v velikosti 180°/p. Če je p enako 2, je kot 90° in se lahko v diagramu veja izpusti. Kadar je veja neoznačena, to pomeni, da zanjo velja ${\displaystyle p=3\,}$, kar pomeni kot 60º. Vzporedni zrcali imata oznako "∞"

Geometrijska ponazoritev

Coxeter-Dinkinov diagram se lahko prikaže kot domena ogledal. Ogledalo v tem primeru predstavlja hiperravnino s pomočjo sfernega ali evklidskega ali hiperboličnega prostora z dano razsežnostjo.

Takšna ponazoritev kaže osnovne domene za dvo in trirazsežne evklidske grupe in dvorazsežne sferne grupe.

Coxeterjeve grupe v ravnini s pripadajočimi diagrami. Ogledala domen so označena kot veje m1, m2 itd. Oglišča so obarvana v skladu z zaporedjem odboja. Prizmatske grupe ${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$ so prikazane kot podvojitev ${\displaystyle {\tilde {C}}_{2}}$, toda nastale bi lahko tudi kot pravokotne domene iz podvojitev ${\displaystyle {\tilde {G}}_{2}}$ trikotnikov. ${\displaystyle {\tilde {A}}_{2}}$ je podvojitev ${\displaystyle {\tilde {G}}_{2}}$ trikotnika.

Coxeterjeve grupe v trirazsežnem prostoru z diagrami. Zrcala (stranice trikotnika) so označena z nasprotnim ogliščem 0..3. Veje grafa so obarvane z zaporedjem odboja. ${\displaystyle {\tilde {C}}_{3}}$ izpolni 1/48 kocke. ${\displaystyle {\tilde {B}}_{3}}$ izpolni 1/24 kocke. ${\displaystyle {\tilde {A}}_{3}}$ izpolni1/12 kocke.

Coxeterjeve grupe na sferi s pripadajočimi diagrami. Osnovna domena je prikazana v rumeni barvi. Oglišča domen (in veje grafa) so obarvane v zaporedju zrcaljenja.

Uporaba v uniformnih politopih

Coxeter-Dinkinovi diagrami lahko opišejo skoraj vse vrste uniformnih politopov in uniformnih teselacij

Cartanove matrike

Vsakemu Coxeterjevemu diagramu pripada odgovarjajoča Cartanova matrika. Vse Cartanove matrike Coxeterjevih grup so simetrične. Elementi Cartanove matrike so a_i,j = a_j,i = -2*cos(π/p), kjer je:

• ${\displaystyle p\,}$ red veje med pari zrcal.

Determinanta Cartanove matrike določa ali je grupa končna (pozitivna), afina (nič) ali hiperbolična (negativna). Hiperbolična grupa ja kompaktna, če so vse njene podgrupe končne.

rang 2 Coxeterjeve grupe

red simetrije p	ime grupe	Coxeterjev diagram	Cartanova matrika
			${\displaystyle \left[{\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}}\right]}$	determinanta (4-a21*a12)
končne (determinanta>0)
2	I2(2) = A1xA1		${\displaystyle \left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]}$	4
3	I2(3) = A2		${\displaystyle \left[{\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}}\right]}$	3
4	I2(4) = BC2		${\displaystyle \left[{\begin{smallmatrix}2&-{\sqrt {2}}\\-{\sqrt {2}}&2\end{smallmatrix}}\right]}$	2
5	I2(5) = H2		${\displaystyle \left[{\begin{smallmatrix}2&-\phi \\-\phi &2\end{smallmatrix}}\right]}$	${\displaystyle 4\sin ^{2}(\pi /5)}$ =${\displaystyle (5-{\sqrt {5}})/2}$ ~1,38196601125
6	I2(6) = G2		${\displaystyle \left[{\begin{smallmatrix}2&-{\sqrt {3}}\\-{\sqrt {3}}&2\end{smallmatrix}}\right]}$	1
8	I2(8)		${\displaystyle \left[{\begin{smallmatrix}2&-2\cos(\pi /8)\\-2\cos(\pi /8)&2\end{smallmatrix}}\right]}$	${\displaystyle 2-{\sqrt {2}}}$ ~0,58578643763
10	I2(10)		${\displaystyle \left[{\begin{smallmatrix}2&-2\cos(\pi /10)\\-2\cos(\pi /10)&2\end{smallmatrix}}\right]}$	${\displaystyle 4\sin ^{2}(\pi /10)}$ =${\displaystyle (3-{\sqrt {5}})/2}$ ~0,38196601125
12	I2(12)		${\displaystyle \left[{\begin{smallmatrix}2&-2\cos(\pi /12)\\-2\cos(\pi /12)&2\end{smallmatrix}}\right]}$	${\displaystyle 2-{\sqrt {3}}}$ ~0,26794919243
p	I2(p)		${\displaystyle \left[{\begin{smallmatrix}2&-2\cos(\pi /p)\\-2\cos(\pi /p)&2\end{smallmatrix}}\right]}$	${\displaystyle 4\sin ^{2}(\pi /p)}$
afine (determinanta=0)
∞	I2(∞) = ${\displaystyle {\tilde {I}}_{1}}$ = ${\displaystyle {\tilde {A}}_{1}}$		${\displaystyle \left[{\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}}\right]}$	0

Končne Coxeterjeve grupe

Povezani končni Dinkinovi grafi za rang 1 do 9

rang	enostavne Lijeve grupe			posebne Liejeve grupe
	${\displaystyle {A}_{1+}}$	${\displaystyle {BC}_{2+}}$	${\displaystyle {D}_{2+}}$	${\displaystyle {E}_{3-8}}$	${\displaystyle {F}_{4}}$ / ${\displaystyle {G}_{2}}$	${\displaystyle {H}_{2-4}}$	${\displaystyle {I}_{2}(p)}$
1	A1=[]
2	A2=[3]	BC2=[4]	D2=A1xA1		G2=[6]	H2=[6]	I2[p]
3	A3=[32]	BC3=[3,4]	D3=A3	E3=A2xA1		H3
4	A4=[33]	BC4=[32,4]	D4=[31,1,1]	E4=A4	F4	H4
5	A5=[34]	BC5=[33,4]	D5=[32,1,1]	E5=D5
6	A6=[35]	BC6=[34,4]	D6=[33,1,1]	E6=[32,2,1]
7	A7=[36]	BC7=[35,4]	D7=[34,1,1]	E7=[33,2,1]
8	A8=[37]	BC8=[36,4]	D8=[35,1,1]	E8=[34,2,1]
9	A9=[38]	BC9=[37,4]	D9=[36,1,1]
10+	..	..	..	..

Afine Coxeterjeve grupe

Afini Dinkinovi grafi od 2 do10 vozlov

rang	${\displaystyle {\tilde {A}}_{1+}}$ (P2+)	${\displaystyle {\tilde {B}}_{3+}}$ (S4+)	${\displaystyle {\tilde {C}}_{1+}}$ (R2+)	${\displaystyle {\tilde {D}}_{4+}}$ (Q5+)	${\displaystyle {\tilde {E}}_{n}}$ (Tn+1) / ${\displaystyle {\tilde {F}}_{4}}$ (U5) / ${\displaystyle {\tilde {G}}_{2}}$ (V3)
2	${\displaystyle {\tilde {A}}_{1}}$=[∞]		${\displaystyle {\tilde {C}}_{1}}$=[∞]
3	${\displaystyle {\tilde {A}}_{2}}$=[3[3]]		${\displaystyle {\tilde {C}}_{2}}$=[4,4]		${\displaystyle {\tilde {G}}_{2}}$=[6,3]
4	${\displaystyle {\tilde {A}}_{3}}$=[3[4]]	${\displaystyle {\tilde {B}}_{3}}$=[4,31,1]	${\displaystyle {\tilde {C}}_{3}}$=[4,3,4]
5	${\displaystyle {\tilde {A}}_{4}}$=[3[5]]	${\displaystyle {\tilde {B}}_{4}}$=[4,3,31,1]	${\displaystyle {\tilde {C}}_{4}}$=[4,32,4]	${\displaystyle {\tilde {D}}_{4}}$=[31,1,1,1]	${\displaystyle {\tilde {F}}_{4}}$=[3,4,3,3]
6	${\displaystyle {\tilde {A}}_{5}}$=[3[6]]	${\displaystyle {\tilde {B}}_{5}}$=[4,32,31,1]	${\displaystyle {\tilde {C}}_{5}}$=[4,33,4]	${\displaystyle {\tilde {D}}_{5}}$=[31,1,3,31,1]
7	${\displaystyle {\tilde {A}}_{6}}$=[3[7]]	${\displaystyle {\tilde {B}}_{6}}$=[4,33,31,1]	${\displaystyle {\tilde {C}}_{6}}$=[4,34,4]	${\displaystyle {\tilde {D}}_{6}}$=[31,1,32,31,1]	${\displaystyle {\tilde {E}}_{6}}$=[32,2,2]
8	${\displaystyle {\tilde {A}}_{7}}$=[3[8]]	${\displaystyle {\tilde {B}}_{7}}$=[4,34,31,1]	${\displaystyle {\tilde {C}}_{7}}$=[4,35,4]	${\displaystyle {\tilde {D}}_{7}}$=[31,1,33,31,1]	${\displaystyle {\tilde {E}}_{7}}$=[33,3,1]
9	${\displaystyle {\tilde {A}}_{8}}$=[3[9]]	${\displaystyle {\tilde {B}}_{8}}$=[4,35,31,1]	${\displaystyle {\tilde {C}}_{8}}$=[4,36,4]	${\displaystyle {\tilde {D}}_{8}}$=[31,1,34,31,1]	${\displaystyle {\tilde {E}}_{8}}$=[35,2,1]
10	${\displaystyle {\tilde {A}}_{9}}$=[3[10]]	${\displaystyle {\tilde {B}}_{9}}$=[4,36,31,1]	${\displaystyle {\tilde {C}}_{9}}$=[4,37,4]	${\displaystyle {\tilde {D}}_{9}}$=[31,1,35,31,1]
11	...	...	...	...

Hiperbolične Coxeterjeve grupe

Kompaktne

Rang 3

Kompaktne hiperbolične Coxeterjeve grupe

linearne	ciklične
∞: [p,q], 2(p+q)<pq ... ... ...	∞ [(p,q,r)], p+q+r>9 ...

Rangi od 4 do 5

Kompaktne hiperbolične Coxeterjeve grupe

razsežnost Hd	rang	skupno število	linearne	razcepljene	ciklične
H3	4	9	3: ${\displaystyle {\bar {BH}}_{3}}$ = [4,3,5]: ${\displaystyle {\bar {K}}_{3}}$ = [5,3,5]: ${\displaystyle {\bar {J}}_{3}}$ = [3,5,3]:	${\displaystyle {\bar {DH}}_{3}}$ = [5,31,1]:	${\displaystyle {\widehat {AB}}_{3}}$ = [(3,3,3,4)]:   ${\displaystyle {\widehat {AH}}_{3}}$ = [(3,3,3,5)]:   ${\displaystyle {\widehat {BB}}_{3}}$ = [(3,4,3,4)]: ${\displaystyle {\widehat {BH}}_{3}}$ = [(3,4,3,5)]: ${\displaystyle {\widehat {HH}}_{3}}$ = [(3,5,3,5)]:
H4	5	5	3: ${\displaystyle {\bar {H}}_{4}}$ = [3,3,3,5]: ${\displaystyle {\bar {BH}}_{4}}$ = [4,3,3,5]: ${\displaystyle {\bar {K}}_{4}}$ = [5,3,3,5]:	${\displaystyle {\bar {DH}}_{4}}$ = [5,3,31,1]:	${\displaystyle {\widehat {AF}}_{4}}$ = [(3,3,3,3,4)]:

Nekompaktni

rang 3

linearni grafi	ciklični grafi
[p,∞] [∞,∞]	[(p,q,∞)] [(p,∞,∞)] [(∞,∞,∞)]

Rangi od 4 do 10

Znanih je skupno 48 nekompaktnih hiperboličnih Coxeterjevih grup z rangom od 4 do 10. V naslednji preglednici je vseh 58 razvrščenih v pet skupin.

Hiperbolične nekompaktne grupe

rang	skupno število	grupe
4	23	${\displaystyle {\widehat {BR}}_{3}}$ = [(3,3,4,4)]: ${\displaystyle {\widehat {CR}}_{3}}$ = [(3,4,4,4)]: ${\displaystyle {\widehat {RR}}_{3}}$ = [(4,4,4,4)]: ${\displaystyle {\widehat {AV}}_{3}}$ = [(3,3,3,6)]: ${\displaystyle {\widehat {BV}}_{3}}$ = [(3,4,3,6)]: ${\displaystyle {\widehat {HV}}_{3}}$ = [(3,5,3,6)]: ${\displaystyle {\widehat {VV}}_{3}}$ = [(3,6,3,6)]:	${\displaystyle {\bar {P}}_{3}}$ = [3,3[3]]: ${\displaystyle {\bar {BP}}_{3}}$ = [4,3[3]]: ${\displaystyle {\bar {HP}}_{3}}$ = [5,3[3]]: ${\displaystyle {\bar {VP}}_{3}}$ = [6,3[3]]: ${\displaystyle {\bar {DV}}_{3}}$ = [6,31,1]: ${\displaystyle {\bar {O}}_{3}}$ = [6,41,1]: ${\displaystyle {\bar {M}}_{3}}$ = [4,41,1]:	${\displaystyle {\bar {R}}_{3}}$ = [3,4,4]: ${\displaystyle {\bar {N}}_{3}}$ = [4,4,4]: ${\displaystyle {\bar {V}}_{3}}$ = [3,3,6]: ${\displaystyle {\bar {BV}}_{3}}$ = [4,3,6]: ${\displaystyle {\bar {HV}}_{3}}$ = [5,3,6]: ${\displaystyle {\bar {Y}}_{3}}$ = [3,6,3]: ${\displaystyle {\bar {Z}}_{3}}$ = [6,3,6]:	${\displaystyle {\bar {DP}}_{3}}$ = [3[ ]x[ ]]: ${\displaystyle {\bar {PP}}_{3}}$ = [3[3,3]]:
5	9	${\displaystyle {\bar {P}}_{4}}$ = [3,3[4]]: ${\displaystyle {\bar {BP}}_{4}}$ = [4,3[4]]: ${\displaystyle {\widehat {FR}}_{4}}$ = [(3,3,4,3,4)]: ${\displaystyle {\bar {DP}}_{4}}$ = [3[3]x[ ]]:	${\displaystyle {\bar {N}}_{4}}$ = [4,/3\,3,4]: ${\displaystyle {\bar {O}}_{4}}$ = [3,4,31,1]: ${\displaystyle {\bar {S}}_{4}}$ = [4,32,1]:	${\displaystyle {\bar {R}}_{4}}$ = [3,4,3,4]:	${\displaystyle {\bar {M}}_{4}}$ = [4,31,1,1]:
6	12	${\displaystyle {\bar {P}}_{5}}$ = [3,3[5]]: ${\displaystyle {\widehat {AU}}_{5}}$ = [(3,3,3,3,3,4)]: ${\displaystyle {\widehat {AR}}_{5}}$ = [(3,3,4,3,3,4)]:	${\displaystyle {\bar {S}}_{5}}$ = [4,3,32,1]: ${\displaystyle {\bar {O}}_{5}}$ = [3,4,31,1]: ${\displaystyle {\bar {N}}_{5}}$ = [4,3,/3\,3,4]:	${\displaystyle {\bar {U}}_{5}}$ = [3,3,3,4,3]: ${\displaystyle {\bar {X}}_{5}}$ = [3,3,4,3,3]: ${\displaystyle {\bar {R}}_{5}}$ = [3,4,3,3,4]:	${\displaystyle {\bar {Q}}_{5}}$ = [32,1,1,1]: ${\displaystyle {\bar {M}}_{5}}$ = [4,3,31,1,1]: ${\displaystyle {\bar {L}}_{5}}$ = [31,1,1,1,1]:
7	3	${\displaystyle {\bar {P}}_{6}}$ = [3,3[6]]:	${\displaystyle {\bar {Q}}_{6}}$ = [31,1,3,32,1]:	${\displaystyle {\bar {S}}_{6}}$ = [4,3,3,32,1]:
8	4	${\displaystyle {\bar {P}}_{7}}$ = [3,3[7]]:	${\displaystyle {\bar {Q}}_{7}}$ = [31,1,32,32,1]:	${\displaystyle {\bar {S}}_{7}}$ = [4,33,32,1]:	${\displaystyle {\bar {T}}_{7}}$ = [33,2,2]:
9	4	${\displaystyle {\bar {P}}_{8}}$ = [3,3[8]]:	${\displaystyle {\bar {Q}}_{8}}$ = [31,1,33,32,1]:	${\displaystyle {\bar {S}}_{8}}$ = [4,34,32,1]:	${\displaystyle {\bar {T}}_{8}}$ = [34,3,1]:
10	3		${\displaystyle {\bar {Q}}_{9}}$ = [31,1,34,32,1]:	${\displaystyle {\bar {S}}_{9}}$ = [4,35,32,1]:	${\displaystyle {\bar {T}}_{9}}$ = [36,2,1]:

Zunanje povezave

• Weisstein, Eric Wolfgang. »Coxeter-Dynkin Diagram«. MathWorld.