//////////////////////////////////////////////////////////////// version="version fpalgebras.lib 4.1.1.0 Mar_2018 "; // $Id: b10e9fb4df79a9a03382db96573bf1583ae973a5 $ category="Noncommutative"; info=" LIBRARY: fpalgebras.lib AUTHORS: Karim Abou Zeid, karim.abou.zeid at rwth-aachen.de Grischa Studzinski, grischa.studzinski at rwth-aachen.de Support: Project II.6 in the transregional collaborative research centre SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG OVERVIEW: Generation of various algebras, including group algebras of finitely presented groups in the Letterplace ring PROCEDURES: operatorAlgebra(string a, int d); baumslagSolitar(int n, int m, int d, list #); baumslagGroup(int m, int n, int d); crystallographicGroupP1(int d); crystallographicGroupPM(int d); crystallographicGroupPG(int d); crystallographicGroupP2MM(int d); crystallographicGroupP2(int d); crystallographicGroupP2GG(int d); crystallographicGroupCM(int d); crystallographicGroupC2MM(int d); crystallographicGroupP4(int d); crystallographicGroupP4MM(int d); crystallographicGroupP4GM(int d); crystallographicGroupP3(int d); crystallographicGroupP31M(int d); crystallographicGroupP3M1(int d); crystallographicGroupP6(int d); crystallographicGroupP6MM(int d); dyckGroup1(int n, int d, intvec P); dyckGroup2(int n, int d, intvec P); dyckGroup3(int n, int d, intvec P); fibonacciGroup(int m, int d); tetrahedronGroup(int g, int d); triangularGroup(int g, int d); "; LIB "freegb.lib"; LIB "general.lib"; //////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////// // Operator Algebras /////////////////////////////////////////////// //////////////////////////////////////////////////////////////////// proc operatorAlgebra(string a, int d) "USAGE: operatorAlgebra(a,d); a a string, d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - a gives the name of the algebra - d gives the degreebound for the Letterplace ring a must be one of the following: integrodiff3 toeplitz weyl1 usl2 usl2h shift1inverse exterior2 quadrowmm shift1 weyl1inverse This is a collection of common algebras " { if (d < 2) { ERROR("Degbound d is too small. Must be at least 2."); } int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } if (a == "integrodiff3") { ring r = 0,(d,I,x),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = d(1)*x(2)-x(1)*d(2)-1, I(1)*x(2)-x(1)*I(2)+I(1)*I(2), d(1)*I(2)-1; } if (a == "toeplitz") { ring r = 0,(y,x),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = y(1)*x(2)-1; } if (a == "weyl1") { ring r = 0,(d,x),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = d(1)*x(2)-x(1)*d(2)-1; } if (a == "usl2") { ring r = 0,(h,f,e),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = f(1)*e(2)-e(1)*f(2)+h(1), h(1)*e(2)-e(1)*h(2)-2*e(1), h(1)*f(2)-f(1)*h(2)+2*f(1); } if (a == "usl2h") { ring r = 0,(H,h,f,e),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = f(1)*e(2)-e(1)*f(2)+h(1)*H(2), h(1)*e(2)-e(1)*h(2)-2*e(1)*H(2), h(1)*f(2)-f(1)*h(2)+2*f(1)*H(2), f(1)*H(2)-H(1)*f(2), e(1)*H(2)-H(1)*e(2), h(1)*H(2)-H(1)*h(2); } if (a == "shift1inverse") { ring r = 0,(d,x,t),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = d(1)*x(2)-x(1)*d(2)-d(1), t(1)*x(2)-1, x(1)*t(2)-1; } if (a == "exterior2") { ring r = 0,(y,x),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = y(1)*x(2)+x(1)*y(2), x(1)*x(2), y(1)*y(2); } if (a == "quadrowmm") { ring r = 0,(y,x),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = y(1)*x(2)-x(1)*y(2), x(1)*x(2), y(1)*y(2); } if (a == "shift1") { ring r = 0,(s,x),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = s(1)*x(2)-x(1)*s(2)-s(1); } if (a == "weyl1inverse") { ring r = 0,(d,x,t),dp; def R = makeLetterplaceRing(d); setring(R); ideal I = d(1)*x(2)-x(1)*d(2)-1, t(1)*x(2)-1, x(1)*t(2)-1; } if (!defined(I)) { ERROR("Illegal argument for algebra"); } export(I); if (baseringdef == 1) {setring save;} return(R); } //////////////////////////////////////////////////////////////////// // Baumslag //////////////////////////////////////////////////////// // from Grischa Studzinski ///////////////////////////////////////// //////////////////////////////////////////////////////////////////// proc baumslagSolitar(int n, int m, int d, list #) "USAGE: baumslagSolitar(m,n,d[,IsGroup]); n an integer, m an integer, d an integer, IsGroup an optional integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - in the group case: A = a^(-1), B = b^(-1) - negativ input is only allowed in the group case! - d gives a degreebound and must be >m,n This is a family " { int isGroup = 0; if (size(#) > 0) {isGroup = #[1];} if (isGroup != 0) { int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } if (m < 0 || n < 0) {ERROR("Exponent can't be negativ in monoid rings!");} if (d < 1 || d < m || d < n) {ERROR("Degree bound must be positiv and greater then m,n!");} int i; ring mr = 0,(a,b),Dp; def Mr = makeLetterplaceRing(d); setring Mr; poly p,q; if (n==0) {p = b(1);} else { p = a(1)*b(2); for (i = 1; i < n; i++) {p = lpMult(a(1),p);} } if (m==0) {q = b(1);} else { q = b(1)*a(2); for (i = 1; i < m; i++) {q = lpMult(q,a(1));} } ideal I = p - q; export(I); if (baseringdef == 1) {setring save;} return(Mr); } else { int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } int i; if (d < 1 || d < absValue(m) || d < absValue(n)) {ERROR("Degree bound must be positiv and greater then |m|,|n|!");} ring gr = 0,(a,b,A,B),Dp; def Gr = makeLetterplaceRing(d); setring Gr; poly p,q; if (n==0) {p = b(1);} else {if (n > 0) { p = a(1)*b(2); for (i = 1; i < n; i++) {p = lpMult(a(1),p);} } else { p = A(1)*b(2); for (i = 1; i < -n; i++) {p = lpMult(A(1),p);} } } if (m==0) {q = b(1);} else {if (m > 0) { q = b(1)*a(2); for (i = 1; i < m; i++) {q = lpMult(q,a(1));} } else { q = A(1)*b(2); for (i = 1; i < -m; i++) {q = lpMult(q,A(1));} } } ideal I = p - q, a(1)*A(2) - 1, b(1)*B(2) - 1, a(1)*A(2) - A(1)*a(2), b(1)*B(2) - B(1)*b(2); export(I); if (baseringdef == 1) {setring save;} return(Gr); } } example { "EXAMPLE:"; echo = 2; def R = baumslagSolitar(2,3,4); setring R; I; } proc baumslagGroup(int m, int n, int d) "USAGE: baumslagGroup(m,n,d); m an integer, n an integer, d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - Baumslag group with the following presentation < a, b | a^m = b^n = 1 > -d gives the degreebound for the Letterplace ring This is a family " { if (m < 0 || n < 0 ) {ERROR("m,n must be non-negativ integers!");} if (d < 1 || d < m || d < n) {ERROR("degreebound must be positiv and larger than n and m!");} int i; ring r = 0,(a,b),dp; def R = makeLetterplaceRing(d); setring R; poly p,q; p = 1; q = 1; for (i = 1; i <= m; i++){p = lpMult(p,a(1));} for (i = 1; i <= n; i++){q = lpMult(q,b(1));} ideal I = p-1,q-1; export(I); return(R); } example { "EXAMPLE:"; echo = 2; def R = baumslagGroup(2,3,4); setring R; I; } //////////////////////////////////////////////////////////////////// // Crystallographic Groups ////////////////////////////////////////// // from Grischa Studzinski ///////////////////////////////////////// //////////////////////////////////////////////////////////////////// proc crystallographicGroupP1(int d) "USAGE: crystallographicGroupP1(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p1 group with the following presentation < x, y | [x, y] = 1 > -d gives the degreebound for the Letterplace ring " { if (d < 2){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, X(1)*x(2)-1, x(1)*X(2)-1, y(1)*Y(2)-1, Y(1)*y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP1(5); setring R; I; } // old? there is already another crystallographicGroupP2 proc /* proc crystallographicGroupP2(int d) */ /* " */ /* p2 group with the following presentation */ /* < x, y, r | [x, y] = r^2 = 1, r^-1*x*r = x^-1, r^-1*y*r = y^-1 > */ /* Note: r = r^-1 */ /* " */ /* { */ /* if (d < 3){ERROR("Degreebound is to small for choosen example!");} */ /* int baseringdef; */ /* if (defined(basering)) // if a basering is defined, it should be saved for later use */ /* { */ /* def save = basering; */ /* baseringdef = 1; */ /* } */ /* ring r = 2,(x,y,r,X,Y),dp; */ /* def R = makeLetterplaceRing(d); */ /* setring R; */ /* ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2), r(1)*r(2)-1, r(1)*x(2)*r(3)-X(1), r(1)*y(2)*r(3)-Y(1),x(1)*X(2)-1, */ /* X(1)*x(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; */ /* I = simplify(I,2); */ /* export(I); */ /* if (baseringdef == 1) {setring save;} */ /* return(R); */ /* } */ proc crystallographicGroupPM(int d) "USAGE: crystallographicGroupPM(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - pm group with the following presentation < x, y, m | [x, y] = m^2 = 1, m^-1*x*m = x, m^-1*y*m = y^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 3){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,m,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-m(1)*m(2), m(1)*m(2)-1, m(1)*x(2)*m(3)-x(1), m(1)*y(2)*m(3)-Y(1),x(1)*X(2)-1, X(1)*x(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupPM(5); setring R; I; } proc crystallographicGroupPG(int d) "USAGE: crystallographicGroupPG(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - pg group with the following presentation < x, y, t | [x, y] = 1, t^2 = x, t^-1*y*t = y^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 3){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,t,X,Y,T),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, t(1)*t(2) - x(1), T(1)*y(2)*t(3)-Y(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1, t(1)*T(2)-1, T(1)*t(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupPG(5); setring R; I; } proc crystallographicGroupP2MM(int d) "USAGE: crystallographicGroupP2MM(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p2mm group with the following presentation < x, y, p, q | [x, y] = [p, q] = p^2 = q^2 = 1, p^-1*x*p = x, q^-1*x*q = x^-1, p^-1*y*p = y^-1, q^-1*y*q = y > - d gives the degreebound for the Letterplace ring " { if (d < 3){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,p,q,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, p(1)*q(2)-q(1)*p(2)-1, p(1)*p(2) - 1, q(1)*q(2) - 1, p(1)*y(2)*p(3)-Y(1), p(1)*x(2)*p(3)-x(1), q(1)*y(2)*q(3)-y(1), q(1)*x(2)*q(3)-X(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1, x(1)*y(2)-y(1)*x(2)- p(1)*p(2), x(1)*y(2)-y(1)*x(2)- q(1)*q(2), p(1)*p(2)-q(1)*q(2); I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP2MM(5); setring R; I; } proc crystallographicGroupP2(int d) "USAGE: crystallographicGroupP2(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p2 group with the following presentation < x, y, m, t | [x, y] = t^2 = 1, m^2 = y, t^-1*x*t = x, m^-1*x*m = x^-1, t^-1*y*t = y^-1, t^-1*m*t = m^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 3){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,m,t,X,Y,M),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), m(1)*m(2)-y(1), t(1)*t(2) - 1, t(1)*x(2)*t(3)-x(1), M(1)*x(2)*m(3)-X(1), t(1)*y(2)*t(3)-Y(1), t(1)*m(2)*t(3)-M(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1, m(1)*M(2)-1, M(1)*m(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP2(5); setring R; I; } proc crystallographicGroupP2GG(int d) "USAGE: crystallographicGroupP2GG(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p2gg group with the following presentation < x, y, u, v | [x, y] = (u*v)^2 = 1, u^2 = x, v^2 = y, v^-1*x*v = x^-1, u^-1*y*u = y^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 4){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,u,v,X,Y,U,V),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-u(1)*v(2)*u(3)*v(4), u(1)*v(2)*u(3)*v(4)-1, u(1)*u(2)-x(1), v(1)*v(2) - y(1), V(1)*x(2)*v(3)-X(1), U(1)*y(2)*u(3)-Y(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1, u(1)*U(2)-1, U(1)*u(2)-1, v(1)*V(2)-1, V(1)*v(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP2GG(5); setring R; I; } proc crystallographicGroupCM(int d) "USAGE: crystallographicGroupCM(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - cm group with the following presentation < x, y, t | [x, y] = t^2 = 1, t^-1*x*t = x*y, t^-1*y*t = y^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 3){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,t,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), t(1)*t(2)-1, t(1)*x(2)*t(3)-x(1)*y(2), t(1)*y(2)*t(3)-Y(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupCM(5); setring R; I; } proc crystallographicGroupC2MM(int d) "USAGE: crystallographicGroupC2MM(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - c2mm group with the following presentation < x, y, m, r | [x, y] = m^2 = r^2 = 1, m^-1*y*m = y^-1, m^-1*x*m = x*y, r^-1*y*r = y^-1, r^-1*x*r = x^-1, m^-1*r*m = r^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 3){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,m,r,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-m(1)*m(2), x(1)*y(2)-y(1)*x(2)-r(1)*r(2), m(1)*m(2)-1, r(1)*r(2)-1, m(1)*m(2)-r(1)*r(2), m(1)*y(2)*m(3)-Y(1), m(1)*x(2)*m(3)-x(1)*y(2), (1)*y(2)*r(3)-Y(1), r(1)*x(2)*r(3)-X(1), m(1)*r(2)*m(3)-r(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupC2MM(5); setring R; I; } proc crystallographicGroupP4(int d) "USAGE: crystallographicGroupP4(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p4 group with the following presentation < x, y, r | [x, y] = r^4 = 1, r^-1*x*r = x^-1, r^-1*x*r = y > - d gives the degreebound for the Letterplace ring " { if (d < 5){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4), r(1)*r(2)*r(3)*r(4)-1, r(1)*r(2)*r(3)*x(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP4(5); setring R; I; } proc crystallographicGroupP4MM(int d) "USAGE: crystallographicGroupP4MM(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p4mm group with the following presentation < x, y, r, m | [x, y] = r^4 = m^2 = 1, r^-1*y*r = x^-1, r^-1*x*r = y, m^-1*x*m = y, m^-1*r*m = r^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 5){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,m,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4), r(1)*r(2)*r(3)*r(4)-1, r(1)*r(2)*r(3)*x(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP4MM(5); setring R; I; } proc crystallographicGroupP4GM(int d) "USAGE: crystallographicGroupP4GM(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p4gm group with the following presentation < x, y, r, t | [x, y] = r^4 = t^2 = 1, r^-1*y*r = x^-1, r^-1*x*r = y, t^-1*x*t = y, t^-1*r*t = x^-1*r^-1> - d gives the degreebound for the Letterplace ring " { if (d < 5){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,t,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4), r(1)*r(2)*r(3)*r(4)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), t(1)*t(2)-1, r(1)*r(2)*r(3)*r(4)-t(1)*t(2), r(1)*r(2)*r(3)*y(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), t(1)*r(2)*t(3)-X(1)*r(2)*r(3)*r(4), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP4GM(5); setring R; I; } proc crystallographicGroupP3(int d) "USAGE: crystallographicGroupP3(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p3 group with the following presentation < x, y, r | [x, y] = r^3 = 1, r^-1*x*r = x^-1*y, r^-1*y*r = x^-1> - d gives the degreebound for the Letterplace ring " { if (d < 4){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3), r(1)*r(2)*r(3)-1, r(1)*r(2)*x(3)*r(4)-X(1)*y(2), r(1)*r(2)*y(3)*r(4)-X(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP3(5); setring R; I; } proc crystallographicGroupP31M(int d) "USAGE: crystallographicGroupP31M(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p31m group with the following presentation < x, y, r, t | [x, y] = r^2 = t^2 = (t*r)^3 = 1, r^-1*x*r = x, t^-1*y*t = y, t^-1*x*t = x^-1*y, r^-1*y*r = x*y^-1 > - d gives the degreebound for the Letterplace ring " { if (d < 6){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,t,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2), x(1)*y(2)-y(1)*x(2)-t(1)*t(2), r(1)*r(2)-1, t(1)*t(2)-1, t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-1, r(1)*r(2)-t(1)*t(2), x(1)*y(2)-y(1)*x(2)-t(1)*r(2)*t(3)*r(4)*t(5)*r(6), t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-r(1)*r(2), t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-t(1)*t(2), r(1)*x(2)*r(3)-x(1), t(1)*y(2)*t(3)-y(1), t(1)*x(2)*t(3)-X(1)*y(2), r(1)*y(2)*r(3)-x(1)*Y(2), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP31M(6); setring R; I; } proc crystallographicGroupP3M1(int d) "USAGE: crystallographicGroupP3M1(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p3m1 group with the following presentation < x, y, r, m | [x, y] = r^3 = m^2 = 1, m^-1*r*m = r^2, r^-1*x*r = x^-1*y, r^-1*y*r = x^-1, m^-1*x*m = x^-1, m^-1*y*m = x^-1*y > - d gives the degreebound for the Letterplace ring " { if (d < 4){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,m,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3), x(1)*y(2)-y(1)*x(2)-m(1)*m(2), r(1)*r(2)*r(3)-1, m(1)*m(2)-1, r(1)*r(2)*r(3)-m(1)*m(2), m(1)*r(2)*m(3)-r(1)*r(2), r(1)*r(2)*x(3)*r(4)-X(1)*y(2), r(1)*r(2)*y(3)*r(4)-X(1),m(1)*x(2)*m(3)-X(1), m(1)*y(2)*m(3)-X(1)*y(2), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP3M1(5); setring R; I; } proc crystallographicGroupP6(int d) "USAGE: crystallographicGroupP6(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p6 group with the following presentation < x, y, r | [x, y] = r^6 = 1, r^-1*x*r = y, r^-1*y*r = x^-1*y> - d gives the degreebound for the Letterplace ring " { if (d < 7){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4)*r(5)*r(6), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-1, r(1)*r(2)*r(3)*r(4)*r(5)*x(6)*r(7)-y(1), r(1)*r(2)*r(3)*r(4)*r(5)*y(6)*r(7)-X(1)*y(2), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP6(7); setring R; I; } proc crystallographicGroupP6MM(int d) "USAGE: crystallographicGroupP6MM(d); d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - p6mm group with the following presentation < x, y, r, m | [x, y] = r^6 = m^2 = 1, r^-1*y*r = x^-1*y, r^-1*x*r = y, m^-1*x*m = x^-1, m^-1*y*m = x^-1*y, m^-1*r*m = r^-1*y> - d gives the degreebound for the Letterplace ring " { if (d < 7){ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,r,m,X,Y),dp; def R = makeLetterplaceRing(d); setring R; ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4)*r(5)*r(6), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-1, x(1)*y(2)-y(1)*x(2)-m(1)*m(2), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-m(1)*m(2), m(1)*m(2)-1, m(1)*x(2)*m(3)-X(1), m(1)*y(2)*m(3)-X(1)*y(2), r(1)*r(2)*r(3)*r(4)*r(5)*x(6)*r(7)-y(1), r(1)*r(2)*r(3)*r(4)*r(5)*y(6)*r(7)-X(1)*y(2), m(1)*r(2)*m(3)- r(1)*r(2)*r(3)*r(4)*r(5)*y(6), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = crystallographicGroupP6MM(7); setring R; I; } //////////////////////////////////////////////////////////////////// // Dyck Group ////////////////////////////////////////////////////// // from Grischa Studzinski ///////////////////////////////////////// //////////////////////////////////////////////////////////////////// proc dyckGroup1(int n, int d, intvec P) "USAGE: dyckGroup1(n,d,P); n an integer, d an integer, P an intvec RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - The Dyck group with the following presentation < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 > - negative exponents are allowed - representation in the form x_i^p_i - x_(i+1)^p_(i+1) - d gives the degreebound for the Letterplace ring This is a family " { int baseringdef,i,j; if (n < 1) {ERROR("There must be at least one variable!");} if (d < n) {ERROR("Degreebound is to small!");} for (i = 1; i <= size(P); i++) {if (d < absValue(P[i])){ERROR("Degreebound is to small!");}} if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x(1..n),Y(1..n)),dp; def R = makeLetterplaceRing(d); setring R; ideal I; poly p,q; p = 1; q = 1; for (i = 1; i<= n; i++) {p = lpMult(p,var(i));} I = p-1; for (i = n; i > 0; i--) { if (P[i] >= 0) {for (j = 1; j <= P[i]; j++){q = lpMult(q,var(i));}} else {for (j = 1; j <= -P[i]; j++){q = lpMult(q,var(i+n));}} I = p - q,I; p = q; q = 1; } I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; intvec P = 1,2,3; def R = dyckGroup1(3,5,P); setring R; I; } proc dyckGroup2(int n, int d, intvec P) "USAGE: dyckGroup2(n,d,P); n an integer, d an integer, P an intvec RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - The Dyck group with the following presentation < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 > - negative exponents are allowed - representation in the form x_i^p_i - 1 - d gives the degreebound for the Letterplace ring This is a family " { int baseringdef,i,j; if (n < 1) {ERROR("There must be at least one variable!");} if (d < n) {ERROR("Degreebound is to small!");} for (i = 1; i <= size(P); i++) {if (d < absValue(P[i])){ERROR("Degreebound is to small!");}} if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x(1..n),Y(1..n)),dp; def R = makeLetterplaceRing(d); setring R; ideal I; poly p; p = 1; for (i = 1; i<= n; i++) {p = lpMult(p,var(i));} I = p-1; for (i = n; i > 0; i--) { p = 1; if (P[i] >= 0) {for (j = 1; j <= P[i]; j++){p = lpMult(p,var(i));}} else {for (j = 1; j <= -P[i]; j++){p = lpMult(p,var(i+n));}} I = p - 1,I; } I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; intvec P = 1,2,3; def R = dyckGroup2(3,5,P); setring R; I; } proc dyckGroup3(int n, int d, intvec P) "USAGE: dyckGroup2(n,d,P); n an integer, d an integer, P an intvec RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - The Dyck group with the following presentation < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 > - only positive exponents are allowed - no inverse generators needed - d gives the degreebound for the Letterplace ring This is a family " { int baseringdef,i,j; if (n < 1) {ERROR("There must be at least one variable!");} if (d < n) {ERROR("Degreebound is to small!");} for (i = 1; i <= size(P); i++) {if (P[i] < 0){ERROR("Exponents must be positive!");}} for (i = 1; i <= size(P); i++) {if (d < P[i]){ERROR("Degreebound is to small!");}} if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,x(1..n),dp; def R = makeLetterplaceRing(d); setring R; ideal I; poly p; p = 1; for (i = 1; i<= n; i++) {p = lpMult(p,var(i));} I = p-1; for (i = n; i > 0; i--) { p = 1; for (j = 1; j <= P[i]; j++){p = lpMult(p,var(i));} I = p - 1,I; } I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; intvec P = 1,2,3; def R = dyckGroup3(3,5,P); setring R; I; } //////////////////////////////////////////////////////////////////// // Fibonacci Group ///////////////////////////////////////////////// // from Grischa Studzinski ///////////////////////////////////////// //////////////////////////////////////////////////////////////////// proc fibonacciGroup(int m, int d) "USAGE: fibonacciGroup(m,d); m an integer, d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - The Fibonacci group F(2, m) with the following presentation < x_1, x_2, ... , x_m | x_i * x_(i + 1) = x_(i + 2) > - d gives the degreebound for the Letterplace ring This is a family " { // TODO: basefield Q oder F2? // TODO: inverse Elemente! if (m < 3) {ERROR("At least three generators are required!");} if (d < 2) {ERROR("Degree bound must be at least 2!");} int baseringdef,i; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x(1..m),Y(1..m)),dp; def R = makeLetterplaceRing(d); setring R; ideal I; poly p; for (i = 1; i < m-1; i++) { p = lpMult(var(i),var(i+1))-var(i+2); I = I,p; } for (i = 1; i <= m; i++) { p = lpMult(var(i),var(i+m))-1; I = I,p; p = lpMult(var(i+m),var(i))-1; I = I,p; } I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = fibonacciGroup(3,5); setring R; I; } //////////////////////////////////////////////////////////////////// // Tetrahedron Groups /////////////////////////////////////////////// // from Grischa Studzinski ///////////////////////////////////////// //////////////////////////////////////////////////////////////////// proc tetrahedronGroup(int g, int d) "USAGE: tetrahedronGroup(g,d); g an integer, d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - g gives the number of the example (1 - 5) - d gives the degreebound for the Letterplace ring This is a family The examples are found in Classification of the finite generalized tetrahedron groups by Gerhard Rosenberger and Martin Scheer. The 5 examples are denoted in Proposition 1.9 and concern finite generalized tetrahedron group in the Tsarnarov-case, which are not equivalent to a presentation for an ordinary tetrahedron group. " { if (g < 1 || g > 5) {ERROR("There are only 5 examples!");} if ((g == 1 && d < 6)||(g == 2 && d < 6)||(g == 3 && d < 5)||(g == 4 && d < 4)||(g == 5 && d < 5)) {ERROR("Degreebound is to small for choosen example!");} int baseringdef,i,j; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(x,y,z),dp; def R = makeLetterplaceRing(d); setring R; ideal I; if (g == 1) {I = x(1)*x(2)*x(3)*x(4)*x(5)-1, y(1)*y(2)-1, z(1)*z(2)*z(3)-1, x(1)*y(2)*x(3)*y(4)*x(5)*y(6)-1, x(1)*x(2)*z(3)*x(4)*x(5)*z(6)-1, y(1)*z(2)*y(3)*z(4)-1; } if (g == 2) {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)*z(5)-1,x(1)*y(2)*x(3)*y(4)-1,x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*z(3)*y(4)*z(5)*z(6)-1; } if (g == 3) {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)-1, x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1; } if (g == 4) {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)-1,x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1; } if (g ==5) {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)*z(5)-1,x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1; } I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = tetrahedronGroup(3,5); setring R; I; } //////////////////////////////////////////////////////////////////// // Triangular Groups /////////////////////////////////////////////// // from Grischa Studzinski ///////////////////////////////////////// //////////////////////////////////////////////////////////////////// proc triangularGroup(int g, int d) "USAGE: triangularGroup(g,d); g an integer, d an integer RETURN: ring NOTE: - the ring contains the ideal I, which contains the required relations - g gives the number of the example (1 - 14) - d gives the degreebound for the Letterplace ring This is a family The examples are found in Classification of the finite generalized tetrahedron groups by Gerhard Rosenberger and Martin Scheer. The 14 examples are denoted in theorem 2.12 " { if (g < 1 || g > 14) {ERROR("There are only 14 examples!");} if ((g == 1 && d < 20)||(g == 2 && d < 21)||(g == 3 && d < 10)||(g == 4 && d < 12)||(g == 5 && d < 10)||(g == 6 && d < 18)||(g == 7 && d < 20)||(g == 8 && d < 16)||(g == 9 && d < 10)||(g == 10 && d < 14)||(g == 11 && d < 16)||(g == 12 && d < 24)||(g == 13 && d < 28)||(g == 14 && d < 37)) {ERROR("Degreebound is to small for choosen example!");} int baseringdef; if (defined(basering)) // if a basering is defined, it should be saved for later use { def save = basering; baseringdef = 1; } ring r = 2,(a,b),dp; def R = makeLetterplaceRing(d); setring R; ideal I; if (g == 1) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*b(10)*a(11)*b(12)*a(13)*b(14)*a(15)*b(16)*b(17)*a(18)*b(19)*b(20)-1; } if (g == 2) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*a(12)*b(13)*b(14)*a(15)*b(16)*a(17)*b(18)*a(19)*b(20)*b(21)-1; } if (g == 3) {I = a(1)*a(2)*a(3)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1; } if (g == 4) {I = a(1)*a(2)*a(3)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*a(4)*b(5)*b(6)*a(7)*b(8)*a(9)*a(10)*b(11)*b(12)-1; } if (g == 5) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1, a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1; } if (g == 6) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*b(8)*b(9)*a(10)*b(11)*a(12)*b(13)*a(14)*b(15)*b(16)*b(17)*b(18)-1; } if (g == 7) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1, a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*b(8)*b(9)*b(10)*a(11)*b(12)*a(13)*b(14)*b(15)*a(16)*b(17)*b(18)*b(19)*b(20)-1; } if (g == 8) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*b(8)*a(9)*b(10)*a(11)*b(12)*a(13)*b(14)*b(15)*b(16)-1; } if (g == 9) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1; } if (g == 10) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*a(12)*b(13)*b(14)-1; } if (g == 11) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*b(9)*a(10)*b(11)*a(12)*b(13)*a(14)*b(15)*b(16)-1; } if (g == 12) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*b(12)*a(13)*b(14)*a(15)*b(16)*a(17)*b(18)*b(19)*a(20)*b(21)*a(22)*b(23)*b(24)-1; } if (g == 13) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*a(9)*b(10)*b(11)*a(12)*b(13)*b(14)*a(15)*b(16)*a(17)*b(18)*a(19)*b(20)*a(21)*b(22)*a(23)*b(24)*b(25)*a(26)*b(27)*b(28)-1; } if (g == 14) {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*b(9)*a(10)*b(11)*b(12)*a(13)*b(14)*a(15)*b(16)*b(17)*a(18)*b(19)*b(20)*a(21)*b(22)*a(23)*b(24)*a(25)*b(26)*a(27)*b(28)*b(29)*a(30)*b(31)*a(32)*b(33)*b(34)*a(35)*b(36)*b(37)-1; } I = simplify(I,2); export(I); if (baseringdef == 1) {setring save;} return(R); } example { "EXAMPLE:"; echo = 2; def R = triangularGroup(3,10); setring R; I; }