[MathJax off]

5 Module Polynomials

In this chapter we consider finitely generated modules over the monoid rings considered previously. We call an element of this module a module polynomial, and we describe functions to construct module polynomials and the standard algebraic operations for such polynomials.

A module polynomial modpoly is recorded as a list of pairs, [ gen, monpoly ], where gen is a module generator (basis element), and monpoly is a monoid polynomial. The module polynomial is printed as the formal sum of monoid polynomial multiples of the generators. Note that the monoid polynomials are the coefficients of the module polynomials and appear to the right of the generator, as we choose to work with right modules.

The examples we are aiming for are the identities among the relators of a finitely presented group (see section 5.4).

5.1 Construction of module polynomials

5.1-1 ModulePoly

The function ModulePoly returns a module polynomial. The terms of the polynomial may be input as a list of generators followed by a list of monoid polynomials or as one list of [generator, monoid polynomial] pairs.

Assuming that Fgens is the free group on the module generators and Fmon is the free group on the monoid generators, the function ZeroModulePoly returns the zero module polynomial, which has no terms, and is an element of the module.

gap> q8R := FreeRelatorGroup( q8 );; 
gap> genq8R := GeneratorsOfGroup( q8R ); 
[ q8_R1, q8_R2, q8_R3, q8_R4 ]
gap> q8Rlabs := [ "q", "r", "s", "t" ];; 
gap> Print( rmp1, "\n" ); 
 - 7*q8_M4 + 5*q8_M1 + 9*<identity ...>
gap> M := GeneratorsOfGroup( fmq8 ); 
[ q8_M1, q8_M2, q8_M3, q8_M4 ]
gap> mp2 := MonoidPolyFromCoeffsWords( [4,-5], [ M[4], M[1] ] );;
gap> Print( mp2, "\n" ); 
4*q8_M4 - 5*q8_M1
gap> zeromp := ZeroModulePoly( q8R, freeq8 );
zero modpoly 
gap> s1 := ModulePoly( [ genq8R[4], genq8R[1] ], [ rmp1, mp2 ] );
q8_R1*(4*q8_M4 - 5*q8_M1) + q8_R4*( - 7*q8_M4 + 5*q8_M1 + 9*<identity ...>)

5.1-2 PrintLnModulePoly

The function PrintModulePoly prints a module polynomial, using the function PrintUsingLabels. Two lists of labels are involved: those for the fp-group being investigated, and those for the free relator group of this group. The function PrintLnModulePoly does exactly the same, and then appends a newline.

gap> q8Rlabs := [ "q", "r", "s", "t" ];; 
gap> PrintLnModulePoly( s1, genfgmon, q8labs, genq8R, q8Rlabs );
q*(4*B + -5*a) + t*(-7*B + 5*a + 9*id)
gap> s2 := ModulePoly( [ genq8R[3], genq8R[2], genq8R[1] ], 
>       [ -1*rmp1, 3*mp2, (rmp1+mp2) ] );;
gap> PrintLnModulePoly( s2, genfgmon, q8labs, genq8R, q8Rlabs );
q*(-3*B + 9*id) + r*(12*B + -15*a) + s*(7*B + -5*a + -9*id)

5.2 Components of a module polynomial

5.2-1 Terms

The function Terms returns the terms of a module polynomial as a list of pairs. In LeadTerm, the generators are ordered, and the term of modpoly with the highest value generator is defined to be the leading term. The monoid polynomial (coefficient) part of the leading term is returned by the function LeadMonoidPoly.

The function Length counts the number of module generators which occur in modpoly (a generator occurs in a polynomial if it has nonzero coefficient). The function One returns the identity in the free group on the generators.

gap> [ Length(s1), Length(s2) ];
[ 2, 3 ]
gap> One( s1 );
<identity ...>
gap> terms := Terms( s1 );;
gap> for t in terms do 
>        PrintModulePolyTerm( t, genfmq8, q8labs, genq8R, q8Rlabs ); 
>        Print( "\n" );
>    od; 
q*(4*B + -5*a)
t*(-7*B + 5*a + 9*id)
gap> t1 := LeadTerm( s1 );;
gap> PrintModulePolyTerm( t1, genfmq8, q8labs, genq8R, q8Rlabs );
t*(-7*B + 5*a + 9*id)
gap> t2 := LeadTerm( s2 );;
gap> PrintModulePolyTerm( t2, genfmq8, q8labs, genq8R, q8Rlabs );
s*(7*B + -5*a + -9*id) 
gap> p1 := LeadMonoidPoly( s1 ); 
 - 7*q8_M4 + 5*q8_M1 + 9*<identity ...>
gap> p2 := LeadMonoidPoly( s2 );
7*q8_M4 - 5*q8_M1 - 9*<identity ...>

5.3 Module Polynomial Operations

5.3-1 AddTermModulePoly

The function AddTermModulePoly adds a term [gen, monpoly] to a module polynomial modpoly.

Tests for equality and arithmetic operations are performed in the usual way. Module polynomials may be added or subtracted. A module polynomial can also be multiplied on the right by a word or by a scalar. The effect of this is to multiply the monoid polynomial parts of each term by the word or scalar. This is made clearer in the example.

gap> mp0 := MonoidPolyFromCoeffsWords( [6], [ M[2] ] );;
gap> s0 := AddTermModulePoly( s1, genq8R[3], mp0 ); 
q8_R1*(4*q8_M4 - 5*q8_M1) + q8_R3*(6*q8_M2) + q8_R4*( - 7*q8_M4 + 5*q8_M1 + 
9*<identity ...>)
gap> Print( s1 + s2, "\n" );
q8_R1*( q8_M4 - 5*q8_M1 + 9*<identity ...>) + q8_R2*(12*q8_M4 - 
15*q8_M1) + q8_R3*(7*q8_M4 - 5*q8_M1 - 9*<identity ...>) + q8_R4*( - 
7*q8_M4 + 5*q8_M1 + 9*<identity ...>)
gap> Print( s1 - s0, "\n" );
q8_R3*( - 6*q8_M2)
gap> Print( s1 * 1/2, "\n" );
q8_R1*(2*q8_M4 - 5/2*q8_M1) + q8_R4*( - 7/2*q8_M4 + 5/2*q8_M1 + 9/
2*<identity ...>)
gap> Print( s1 * M[1], "\n" );
q8_R1*(4*q8_M4*q8_M1 - 5*q8_M1^2) + q8_R4*( - 7*q8_M4*q8_M1 + 5*q8_M1^2 + 
9*q8_M1)

generated by GAPDoc2HTML