"Root systems and Dynkin diagrams(mathematica)"의 두 판 사이의 차이

2010년 3월 14일 (일) 13:00 판

A_n root systems

(* A_n type Cartan matrix *)
r := 2
rt[i_] := UnitVector[r + 1, i] - UnitVector[r + 1, i + 1]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
A // MatrixForm

B_n root systems

Clear[rt]
(*B_r type Cartan matrix*)
r := 4
rt[i_] :=
If[i < r, UnitVector[r, i] - UnitVector[r, 1 + i], UnitVector[r, r]]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
Print["root vectors"]
Table[rt[i], {i, 1, r}] // TableForm
Print["Cartan matrix"]
A // MatrixForm

C_n root systems

Clear[rt]
(*C_r type Cartan matrix*)
r := 4
rt[i_] :=
If[i < r, UnitVector[r, i] - UnitVector[r, 1 + i], 2*UnitVector[r, r]]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
Print["root vectors"]
Table[rt[i], {i, 1, r}] // TableForm
Print["Cartan matrix"]
A // MatrixForm

D_n root systems

Clear[rt]
(*D_r type Cartan matrix*)
r := 6
rt[i_] :=
If[i < r, UnitVector[r, i] - UnitVector[r, 1 + i],
UnitVector[r, r - 1] + UnitVector[r, r]]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
Print["root vectors"]
Table[rt[i], {i, 1, r}] // TableForm
Print["Cartan matrix"]
A // MatrixForm

E_7 root system

[5]http://en.wikipedia.org/wiki/E7_%28mathematics%29

Clear[rt]
(*E_ 7 type Cartan matrix*)
r := 7
alp := Sum[UnitVector[r + 1, i]/2, {i, 1, 4}] -
Sum[UnitVector[r + 1, i]/2, {i, 5, 8}]
rt[i_] :=
Piecewise[{{UnitVector[r + 1, i + 2] - UnitVector[r + 1, 1 + i],
i < 7}, {alp, i == 7}}, {i, 1, r}]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
Print["root vectors"]
Table[rt[i], {i, 1, r}] // TableForm
Print["Cartan matrix"]
A // MatrixForm

E_8 root system

Clear[rt]
(*E_ 8 type Cartan matrix*)
alp := -Sum[UnitVector[r, i]/2, {i, 1, r}]
r := 8
rt[i_] :=
Piecewise[{{UnitVector[r, i] - UnitVector[r, 1 + i],
i < 7}, {UnitVector[r, i] + UnitVector[r, i - 1], i == 7}, {alp,
i == 8}}, {i, Range[r]}]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
Print["root vectors"]
Table[rt[i], {i, 1, r}] // TableForm
Print["Cartan matrix"]
A // MatrixForm

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dilogarithm and Nahm's conjecture (mathematica)

@@ 39번째 줄: / 39번째 줄: @@
-<h5>E_8 root systems</h5>
+<h5>E_7 root system</h5>
+* [http://en.wikipedia.org/wiki/E7_%28mathematics%29 ]http://en.wikipedia.org/wiki/E7_%28mathematics%29
+Clear[rt]<br> (*E_ 7 type Cartan matrix*)<br> r := 7<br> alp := Sum[UnitVector[r + 1, i]/2, {i, 1, 4}] -<br>   Sum[UnitVector[r + 1, i]/2, {i, 5, 8}]<br> rt[i_] :=<br>  Piecewise[{{UnitVector[r + 1, i + 2] - UnitVector[r + 1, 1 + i],<br>     i < 7}, {alp, i == 7}}, {i, 1, r}]<br> a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[a[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm
+<h5>E_8 root system</h5>
 Clear[rt]<br> (*E_ 8 type Cartan matrix*)<br> alp := -Sum[UnitVector[r, i]/2, {i, 1, r}]<br> r := 8<br> rt[i_] :=<br>  Piecewise[{{UnitVector[r, i] - UnitVector[r, 1 + i],<br>     i < 7}, {UnitVector[r, i] + UnitVector[r, i - 1], i == 7}, {alp,<br>     i == 8}}, {i, Range[r]}]<br> a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[a[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm