28.1 Functions and Variables for Groups

Function: todd_coxeter
    todd_coxeter (relations, subgroup)
    todd_coxeter (relations) ¶

Find the order of G/H where G is the Free Group modulo relations, and H is the subgroup of G generated by subgroup. subgroup is an optional argument, defaulting to []. In doing this it produces a multiplication table for the right action of G on G/H, where the cosets are enumerated [H,Hg2,Hg3,...]. This can be seen internally in the variable todd_coxeter_state.

Example:

(%i1) symet(n):=create_list(
        if (j - i) = 1 then (p(i,j))^^3 else
            if (not i = j) then (p(i,j))^^2 else
                p(i,i) , j, 1, n-1, i, 1, j);
                                                       <3>
(%o1) symet(n) := create_list(if j - i = 1 then p(i, j)
                                <2>
 else (if not i = j then p(i, j)    else p(i, i)), j, 1, n - 1, 
i, 1, j)

(%i2) p(i,j) := concat(x,i).concat(x,j);
(%o2)        p(i, j) := concat(x, i) . concat(x, j)

(%i3) symet(5);
         <2>           <3>    <2>           <2>           <3>
(%o3) [x1   , (x1 . x2)   , x2   , (x1 . x3)   , (x2 . x3)   , 
            <2>           <2>           <2>           <3>    <2>
          x3   , (x1 . x4)   , (x2 . x4)   , (x3 . x4)   , x4   ]

(%i4) todd_coxeter(%o3);
Rows tried 426
(%o4)                          120

(%i5) todd_coxeter(%o3,[x1]);
Rows tried 213
(%o5)                          60

(%i6) todd_coxeter(%o3,[x1,x2]);
Rows tried 71
(%o6)                          20

Categories: Group theory ·