User:Drorbn/06-1350-HW4: Difference between revisions

Revision as of 18:38, 18 November 2006

Our generators are $T$ , $R$ , $\Phi$ and $B^{\pm }$ :

The picture is

In formulas, this is

(1230)^{\star }B^{+}(1213)^{\star }B^{+}(1023)^{\star }B^{+}=(1123)^{\star }B^{+}(1203)^{\star }B^{+}(1231)^{\star }B^{+}

.

Linearized and written in functional form, this becomes

$\rho _{3}(x_{1},x_{2},x_{3},x_{4})=$	$b^{+}(x_{1},x_{2},x_{3})+b^{+}(x_{1}+x_{3},x_{2},x_{4})+b^{+}(x_{1},x_{3},x_{4})$
	$-b^{+}(x_{1}+x_{2},x_{3},x_{4})-b^{+}(x_{1},x_{2},x_{4})-b^{+}(x_{1}+x_{4},x_{2},x_{3}).$

@@ Line 1: / Line 1: @@
 ===The Generators===
-Our generators are <math>T</math>, <math>R</math>, <math>Y=\Phi</math> and <math>B^{\pm}</math> (we might consider splitting <math>Y</math> into two, <math>Y^{up}</math> and <math>Y^{dn}</math>):
+Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
-{| align=center
+{| align=center cellpadding=10 style="border: solid orange 1px"
 |- align=center valign=middle
 |align=left|Picture
-|
 |
 |
@@ Line 15: / Line 14: @@
 |<math>T</math>
 |<math>R</math>
-|<math>Y^{up}</math>
+|<math>\Phi</math>
-|<math>Y^{dn}</math>
 |<math>B^+</math>
 |<math>B^-</math>
@@ Line 23: / Line 21: @@
 |<math>t</math>
 |<math>r</math>
-|<math>y^u</math>
+|<math>\varphi</math>
-|<math>y^d</math>
 |<math>b^+</math>
 |<math>b^-</math>
@@ Line 36: / Line 33: @@
 In formulas, this is
 <center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
+Linearized and written in functional form, this becomes
-Thus the R3 component of <math>d</math> is
+{| align=center
-<center><math>(1230)^\star b^+ +(1213)^\star b^+ +(1023)^\star b^+ -(1123)^\star b^+ -(1203)^\star b^+ -(1231)^\star b^+</math>.</center>
+|-
+|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
+|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
+|-
+|
+|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math>
+|}
 ===The Syzygies===