Difference between revisions of "ApCoCoA-1:Extended Hecke groups"

From ApCoCoAWiki
Line 41: Line 41:
  
 
====Example in Symbolic Data Format====
 
====Example in Symbolic Data Format====
  <FREEALGEBRA createdAt="2014-07-03" createdBy="strohmeier">
+
<FREEALGEBRA createdAt="2014-07-03" createdBy="strohmeier">
  <vars>r,x,y</vars>
+
<vars>r,x,y</vars>
  <uptoDeg>53</uptoDeg>
+
<uptoDeg>53</uptoDeg>
  <basis>
+
<basis>
  <ncpoly>r*r-1</ncpoly>
+
<ncpoly>r*r-1</ncpoly>
  <ncpoly>x*x-1</ncpoly>
+
<ncpoly>x*x-1</ncpoly>
  <ncpoly>y^3-1</ncpoly>
+
<ncpoly>y^3-1</ncpoly>
  <ncpoly>r*x*r*x-1</ncpoly>
+
<ncpoly>r*x*r*x-1</ncpoly>
  <ncpoly>y*r*y*r-1</ncpoly>
+
<ncpoly>y*r*y*r-1</ncpoly>
  </basis>
+
</basis>
  <Comment>The partial LLex Gb has 69 elements</Comment>
+
<Comment>The partial LLex Gb has 69 elements</Comment>
  <Comment>Extended_Hecke_Group_p3</Comment>
+
<Comment>Extended_Hecke_Group_p3</Comment>
  </FREEALGEBRA>
+
</FREEALGEBRA>

Revision as of 09:12, 20 July 2014

Description

Reference

Ma. Louise Antonette N. De Las Peñas, Ma. Carlota B. Decena and Glenn R. Laigo, ON THE SUBGROUPS OF THE EXTENDED HECKE, HECKE AND PICARD GROUPS, Proceedings of the 5th Asian Mathematical Conference, Malaysia 2009.

Computation

/*Use the ApCoCoA package ncpoly.*/
 
// Define the variable p of the extended Hecke group
MEMORY.P := 3;

Use ZZ/(2)[r,x,y];
NC.SetOrdering("LLEX");

Define CreateRelationsExtendedHeckegroup()
  Relations:=[];
  
  //add the relation r^2 = 1 
  Append(Relations,[[r,r],[1]]);
  
  //add the relation x^2 = 1
  Append(Relations,[[x,x],[1]]);
  
  //add the relation y^p = 1
  Append(Relations,[[y^MEMORY.P],[1]]);
  
  //add the relation (rx)^2 = 1
  Append(Relations,[[r,x,r,x],[1]]);
  
  // add the relation (yr)^2 = 1 
  Append(Relations,[[y,r,y,r],[1]]);
   Return Relations;
EndDefine;

Relations:=CreateRelationsExtendedHeckegroup();
Relations;

Gb:=NC.GB(Relations,31,1,100,1000);
Gb;

Example in Symbolic Data Format

<FREEALGEBRA createdAt="2014-07-03" createdBy="strohmeier">
	<vars>r,x,y</vars>
	<uptoDeg>53</uptoDeg>
	<basis>
	<ncpoly>r*r-1</ncpoly>
	<ncpoly>x*x-1</ncpoly>
	<ncpoly>y^3-1</ncpoly>
	<ncpoly>r*x*r*x-1</ncpoly>
	<ncpoly>y*r*y*r-1</ncpoly>
	</basis>
	<Comment>The partial LLex Gb has 69 elements</Comment>
	<Comment>Extended_Hecke_Group_p3</Comment>
</FREEALGEBRA>