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).

Construction of module polynomials
The function ModulePoly returns a module polynomial. The terms of the polynomial maybe 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> frq8 := FreeRelatorGroup( q8 );; gap> genfrq8 := GeneratorsOfGroup( frq8 ); [ q8_R1, q8_R2, q8_R3, q8_R4 ] gap> Display( rmp1 ); - 7*q8_M4 + 5*q8_M1 + 9*<identity ...> gap> mp2 := MonoidPolyFromCoeffsWords( [4,-5], [ M[4], M[1] ] );; gap> Display( mp2 ); 4*q8_M4 - 5*q8_M1 gap> s1 := ModulePoly( [ genfrq8[4], genfrq8[1] ], [ rmp1, mp2 ] ); q8_R1*(4*q8_M4 - 5*q8_M1) + q8_R4*( - 7*q8_M4 + 5*q8_M1 + 9*<identity ...>) gap> s2 := ModulePoly( [ genfrq8[3], genfrq8[2], genfrq8[1] ], gap> [ -1*rmp1, 3*mp2, (rmp1+mp2) ] ); q8_R1*( - 3*q8_M4 + 9*<identity ...>) + q8_R2*(12*q8_M4 - 15*q8_M1) + q8_R3*( 7*q8_M4 - 5*q8_M1 - 9*<identity ...>) gap> zeromp := ZeroModulePoly( frq8, freeq8 ); zero modpoly
Components of a module polynomial
The first function 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.

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.

gap> [ Length(s1), Length(s2) ]; [ 2, 3 ] gap> One( s1 ); <identity ...> gap> Terms( s1 ); [ [ q8_R1, <monpoly> ], [ q8_R4, <monpoly> ] ] gap> Display( LeadTerm( s1 ) ); [ q8_R4, - 7*q8_M4 + 5*q8_M1 + 9*<identity ...> ] gap> Display( LeadTerm( s2 ) ); [ q8_R3, 7*q8_M4 - 5*q8_M1 - 9*<identity ...> ] gap> Display( LeadMonoidPoly( s1 ) ); - 7*q8_M4 + 5*q8_M1 + 9*<identity ...> gap> Display( LeadMonoidPoly( s2 ) ); 7*q8_M4 - 5*q8_M1 - 9*<identity ...>
Module Polynomial Operations
=,+,* for module polynomials
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> Display(mp0); 6*q8_M2 gap> s0 := AddTermModulePoly( s1, genfrq8[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> Display( s1 + s2 ); 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> Display( s1 - s0 ); q8_R3*( - 6*q8_M2) gap> Display( s1 * 1/2 ); q8_R1*(2*q8_M4 - 5/2*q8_M1) + q8_R4*( - 7/2*q8_M4 + 5/2*q8_M1 + 9/ 2*<identity ...>) gap> Display( s1 * M[1] ); 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)
Identities among relators
The identities among the relators for a finitely presented group are constructed as logged module polynomials. The procedure, described in and based on work in , is to construct a full set of Y-sequences for the group; convert these into module polynomials (eliminating empty sequences); and then apply simplification rules (including the primary identity property) to eliminate obvious duplicates and conjugates.

It is not guaranteed that a minimal set of identities is obtained. For q8 a set of seven identities is obtained, whereas a minimal set contains only six. See Example 5.1 of for further details.

gap> yseqs := IdentityYSequences( q8 );; gap> Length( yseqs ); 32 gap> polys := IdentityModulePolys( q8 );; gap> Length( polys ); 22 gap> idsq8 := IdentitiesAmongRelators( q8 );; gap> Length( idsq8 ); 2 gap> Length( idsq8[1] ); 7 gap> Display( idsq8[1] ); [ ( q8_Y3*( q8_M1*q8_M4), q8_R1*( q8_M1 - <identity ...>) ), ( q8_Y10*( -q8_M1*q8_M4), q8_R2*( q8_M2 - <identity ...>) ), ( q8_Y17*( <identity ...>), q8_R1*( -q8_M3 - q8_M2) + q8_R3*( q8_M1^ 2 + q8_M3 + q8_M1 + <identity ...>) ), ( q8_Y31*( q8_M1*q8_M4), q8_R3*( q8_M3 - q8_M2) + q8_R4*( q8_M1 - <identity \ ...>) ), ( q8_Y32*( -q8_M1*q8_M4), q8_R2*( -q8_M1^ 2) + q8_R3*( -q8_M3 - <identity ...>) + q8_R4*( q8_M2 + <identity ...>) ), ( q8_Y12*( q8_M1*q8_M4), q8_R1*( -q8_M2) + q8_R3*( q8_M1*q8_M2 + q8_M4) + q8\ _R4*( q8_M2 - <identity ...>) ), ( q8_Y16*( -<identity ...>), q8_R1*( -<identity ...>) + q8_R2*( -q8_M1) + q8\ _R4*( q8_M3 + q8_M1) ) ]
The root identities are identities of the form r^wr^{-1} where r = w^n is a relator and n>1.

For q8 only two of the four relators are proper powers, q=a^4 and r=b^4, so the root identities are q^aq^{-1} and r^br^{-1}.
gap> RootIdentities( q8 ); [ ( q8_Y3*( q8_M1*q8_M4), q8_R1*( q8_M1 - <identity ...>) ), ( q8_Y10*( -q8_M1*q8_M4), q8_R2*( q8_M2 - <identity ...>) ) ] gap> RootIdentities(s3); [ ( s3_Y4*( s3_M2*s3_M1), s3_R1*( s3_M1 - <identity ...>) ), ( s3_Y8*( s3_M2*s3_M1), s3_R2*( s3_M2 - <identity ...>) ), ( s3_Y7*( s3_M2*s3_M1), s3_R3*( s3_M2 - s3_M1) ) ]