Annexe: Preuve de l'irreductibilit\303\251 de V(w_1) \134otimes V(w_2) \134otimes V(w_3) with(LinearAlgebra):TMat:=proc(A,B) local C,m,n,i,j; n,n:=Dimension(A): m,m:=Dimension(B): C:=Matrix(n*m); for i from 1 to n do for j from 1 to n do C[[(i-1)*m+1..i*m],[(j-1)*m+1..j*m]]:=A[i,j]*B od; od; return C; end;

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e1:=Matrix([[0,1],[0,0]]);f1:=Matrix([[0,0],[1,0]]);f0:=Matrix([[0,x],[0,0]]);e0:=Matrix([[0,0],[1/x,0]]);K1:=Matrix([[q,0],[0,1/q]]);K0:=Matrix([[1/q,0],[0,q]]);E:=IdentityMatrix(2);dE:=IdentityMatrix(4);

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de1:=TMat(e1,K1)+TMat(E,e1); de0:=TMat(e0,K0)+TMat(E,subs(x=y,e0)); df1:=TMat(f1,E)+TMat(K0,f1); df0:=TMat(f0,E)+TMat(K1,subs(x=y,f0)); dK0:=TMat(K0,K0);dK1:=TMat(K1,K1);

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dde1:=TMat(de1,K1)+TMat(dE,e1); ddf1:=TMat(df1,E)+TMat(dK0,f1); dde0:=TMat(de0,K0)+TMat(dE,subs(x=z,e0)); ddf0:=TMat(df0,E)+TMat(dK1,subs(x=z,f0)); ddK0:=TMat(dK0,K0);ddK1:=TMat(K1,dK1);

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v := Vector([1, 0, 0, 0, 0, 0, 0, 0]); v1 := simplify(MatrixVectorMultiply(ddf1, v)); v2 := simplify(MatrixVectorMultiply(dde0, v));

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w3 := MatrixVectorMultiply(dde0, v1); w4 := MatrixVectorMultiply(ddf1, v1); w5 := MatrixVectorMultiply(dde0, v2); w6 := MatrixVectorMultiply(ddf1, v2);

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w:=Vector([0, 0, 0, 0, 0, 0, 0, 1]); w1:= MatrixVectorMultiply(dde1, w); w2:= MatrixVectorMultiply(ddf0, w);

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v3:= MatrixVectorMultiply(dde1, w1); v4:= MatrixVectorMultiply(ddf0, w1); v5:= MatrixVectorMultiply(dde1, w2); v6:= MatrixVectorMultiply(dde1, w2);

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V1:=Basis([v1,v2,v3,v4,v5,v6]);V2:=Basis([w1,w2,w3,w4,w5,w6]);

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On obtient 4 sous espaces engendr\303\251s respectivement par {v}, V1, V2 et {w}, dont la r\303\251union donne l'espace tout entier. Z1 := convert(IntersectionBasis([NullSpace(dde1), NullSpace(ddf0)]), list); V:=Z1[2];

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MatrixVectorMultiply(dde0,MatrixVectorMultiply(ddf1,V));

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solve(q^2*(1+(q^3*(-y+q^2*z)/(q^4*z-x)-q)/q)/x+q*(q^3*(-y+q^2*z)/(q^4*z-x)-q-(-y+q^2*z)*q/(q^4*z-x))/y+(1-q^2*(-y+q^2*z)/(q^4*z-x))/z=0,{x,y,z});

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Z2 := convert(IntersectionBasis([NullSpace(dde0), NullSpace(ddf1)]), list); W:=Z2[1];

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MatrixVectorMultiply(dde1,MatrixVectorMultiply(ddf0,W));

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solve(q*y*(x*(q^2*y-x)/(q*y*(q^4*z-x))-x/(q*y))+x+q*(z*(x*(q^2*y-x)/(q*y*(q^4*z-x))-x/(q*y))-x*q*z*(q^2*y-x)/(y*(q^4*z-x)))+q^2*(z-z*(q^2*y-x)/(q^4*z-x)),{x,y,z});

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L'intersection des noyaux de dde1 et ddf0 nous donne un vecteur autre que v=(1,0,0,0,0,0,0,0). Avec les bonnes hypoth\303\251ses, par dde0 et ddf1 on peut rejoindre le sous espaces engendr\303\251 par w=(0,0,0,0,0,0,0,1), et ainsi remonter vers v. De m\303\252me pour l'intersection des noyeux de dde0 et ddf1. TTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NTcwMFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhRiciIiJGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NTc2NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhIiIiRidGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NTgyOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhRiclInhHRidGJg==TTdSMApJNlJUQUJMRV9TQVZFLzE3NDExNjU4OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhKiQlInhHISIiRidGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NTg5MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjJSJxRyIiIUYoKiRGJyEiIkYmTTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NTk1NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjKiQlInFHISIiIiIhRipGKEYmTTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NjAyMFgsJSlhbnl0aGluZ0c2IyUpaWRlbnRpdHlHNiJbZ2whIiIhISEjISIjIiNGJw==TTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NjE0OFgsJSlhbnl0aGluZ0c2IyUpaWRlbnRpdHlHNiJbZ2whIiIhISEjISIlIiVGJw==TTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1NzgxMlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhRidGJ0YnIiIiRidGJ0YnJSJxR0YnRidGJ0YnKiRGKSEiIkYoRidGJg==TTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1OTA5MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhKiQlInlHISIiKiYlInhHRiolInFHRipGJ0YnRidGJyomRixGKkYtIiIiRidGJ0YnRihGJ0YnRidGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3MzU2MDMwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhKiQlInFHISIiIiIiRidGJ0YnRidGK0YnRidGJ0YpRidGJ0YnRidGJg==TTdSMApJNlJUQUJMRV9TQVZFLzE3MzU1MzAwNFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIhRidGJ0YnKiYlInFHIiIiJSJ5R0YqRidGJ0YnJSJ4R0YnRidGJ0YnRiwqJkYpISIiRitGKkYnRiY=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TTdSMApJNlJUQUJMRV9TQVZFLzE3NTkzMjk3MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhI1tvIikiKSIiIUYnRidGJ0YnRidGJ0YnKiYlInFHIiIjJSJ6RyIiIkYnRidGJ0YnRidGJ0YnKiZGKUYsJSJ5R0YsRidGJ0YnRidGJ0YnRidGJ0YtRitGJ0YnRidGJ0YnJSJ4R0YnRidGJ0YnRidGJ0YnRidGL0YnRidGK0YnRidGJ0YnRidGL0YnKiZGKSEiIkYuRixGJ0YnRidGJ0YnRidGL0YnRjAqJkYpISIjRitGLEYnRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE3NTkzMzAzNlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhI1tvIikiKSokJSJxRyEiJCIiIUYqRipGKkYqRipGKkYqKiRGKCEiIkYqRipGKkYqRipGKkYqRipGK0YqRipGKkYqRipGKkYqRipGKEYqRipGKkYqRipGKkYqRipGK0YqRipGKkYqRipGKkYqRipGKEYqRipGKkYqRipGKkYqRipGKEYqRipGKkYqRipGKkYqRioqJEYoIiIkRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE3NTkzNTAyMFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhI1tvIikiKSokJSJxRyIiJCIiIUYqRipGKkYqRipGKkYqRihGKkYqRipGKkYqRipGKkYqRihGKkYqRipGKkYqRipGKkYqKiRGKCEiIkYqRipGKkYqRipGKkYqRipGKEYqRipGKkYqRipGKkYqRipGK0YqRipGKkYqRipGKkYqRipGK0YqRipGKkYqRipGKkYqRioqJEYoISIkRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE3NzYzNzM4NFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIiIiIUYoRihGKEYoRihGKEYmTTdSMApJNlJUQUJMRV9TQVZFLzIwMzkxNTA1NlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiISokJSJxRyEiIyokRikhIiJGJyIiIkYnRidGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzIwNTY4MDA4OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiISokJSJ6RyEiIiomJSJ5R0YqJSJxR0YqRicqJiUieEdGKkYtISIjRidGJ0YnRiY=TTdSMApJNlJUQUJMRV9TQVZFLzIwNTY3OTk0NFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicsJiomJSJ5RyEiIiUicUdGKyIiIiomRixGKyUiekdGK0YtRicsJiomJSJ4R0YrRiwhIiNGLSokRi9GK0YtLCYqJkYyRitGLEYrRi1GKUYtRidGJg==TTdSMApJNlJUQUJMRV9TQVZFLzIwNTcxNzUwNFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicsJiokJSJxRyEiJCIiIiokRiohIiJGLEYnLCZGLEYsKiRGKiEiI0YsLCZGKkYsRi1GLEYnRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE3NTk4MDU1MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicsJiooJSJ5RyEiIiUicUciIiIlInpHRitGLSooRi5GK0YqRitGLEYrRi1GJywmKiYlInhHRitGLkYrRi0qKEYsISIjRi5GK0YyRitGLSwmKihGMkYrRipGK0YsRitGLSooRipGK0YsISIkRjJGK0YtRidGJg==TTdSMApJNlJUQUJMRV9TQVZFLzE3NTk4MTI1MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicsJiomJSJ5RyEiIiUicUdGKyIiIiomRixGKyUiekdGK0YtRicsJiomJSJ4R0YrRiwhIiNGLSokRi9GK0YtLCYqJkYyRitGLEYrRi1GKUYtRidGJg==TTdSMApJNlJUQUJMRV9TQVZFLzE3NjAwMDc4MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRidGJ0YnRidGJyIiIkYmTTdSMApJNlJUQUJMRV9TQVZFLzE3NjAwMDM0OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicqJCUicUchIiNGJyokRikhIiIiIiJGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3NzgyMDkxNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRiclInhHRicqJiUicUchIiIlInlHIiIiKiZGKiEiIyUiekdGLUYnRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE3Nzg0MDMxMlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiISwmKiQlInFHISIkIiIiKiRGKiEiIkYsLCZGLEYsKiRGKiEiI0YsRicsJkYqRixGLUYsRidGJ0YnRiY=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TTdSMApJNlJUQUJMRV9TQVZFLzE3NzAyMzgzNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiISwmKiYlInFHISIiJSJ5RyIiIkYtKiYlInhHRi1GKkYrRi0sJiomRiohIiMlInpHRi1GLUYvRi1GJywmKiZGM0YtRipGK0YtRilGLUYnRidGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3NjAwMDM0OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicqJCUicUchIiNGJyokRikhIiIiIiJGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE3NzgyMDkxNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRiclInhHRicqJiUicUchIiIlInlHIiIiKiZGKiEiIyUiekdGLUYnRiY=TTdSMApJNlJUQUJMRV9TQVZFLzIwNTY3OTk0NFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicsJiomJSJ5RyEiIiUicUdGKyIiIiomRixGKyUiekdGK0YtRicsJiomJSJ4R0YrRiwhIiNGLSokRi9GK0YtLCYqJkYyRitGLEYrRi1GKUYtRidGJg==TTdSMApJNlJUQUJMRV9TQVZFLzE3MTc3NDMxNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIiIiIUYoRihGKEYoRihGKEYmTTdSMApJNlJUQUJMRV9TQVZFLzE3MTc3NDM4MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiISwmKiglInFHIiIkLCYlInlHISIiKiZGKiIiIyUiekciIiJGMkYyLCYqJkYqIiIlRjFGMkYyJSJ4R0YuRi5GMkYqRi5GMkYnLCQqKEYsRjJGKkYyRjNGLkYuRidGJ0YnRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE3MTc3NDM4MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiISwmKiglInFHIiIkLCYlInlHISIiKiZGKiIiIyUiekciIiJGMkYyLCYqJkYqIiIlRjFGMkYyJSJ4R0YuRi5GMkYqRi5GMkYnLCQqKEYsRjJGKkYyRjNGLkYuRidGJ0YnRiY=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TTdSMApJNlJUQUJMRV9TQVZFLzE5OTM4NTEzNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRidGJ0YnRidGJyIiIkYmTTdSMApJNlJUQUJMRV9TQVZFLzE5OTM4NTA3MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSIiIUYnRicsJiosJSJ4RyIiIiUicUchIiIsJiomRiwiIiMlInlHRitGK0YqRi1GK0YxRi0sJiomRiwiIiUlInpHRitGK0YqRi1GLUYrKihGLEYtRipGK0YxRi1GLUYnRissJCosRixGK0Y1RitGLkYrRjFGLUYyRi1GLUYnRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE5OTc2MzQ3NlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIikiKSwqKiglInFHIiIiJSJ5R0YqLCYqLCUieEdGKkYpISIiLCYqJkYpIiIjRitGKkYqRi5GL0YqRitGLywmKiZGKSIiJSUiekdGKkYqRi5GL0YvRioqKEYpRi9GLkYqRitGL0YvRipGKkYuRioqJkYpRiosJiomRjZGKkYsRipGKiouRi5GKkYpRipGNkYqRjBGKkYrRi9GM0YvRi9GKkYqKiZGKUYyLCZGNkYqKihGNkYqRjBGKkYzRi9GL0YqRioiIiFGP0Y/Rj9GP0Y/Rj9GJg==