Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

Y i1 ,i 2 ,i 3 ,i 4written in one of our manuscripts:∑ [δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y x i 4y 2 + δ k 2,k 1 ,k 3i 1 , i 2 , i 3Y x i 4y 2 + δ k 2,k 3 ,k 1i 1 , i 2 , i 3Y x i 4y 2+k 1 ,k 2 ,k 3(3.54)+ δ k 1,k 2 ,k 3i 1 , i 2 , i 4Y x i 3y 2 + δ k 2,k 1 ,k 3i 1 , i 2 , i 4Y x i 3y 2 + δ k 2,k 3 ,k 1i 1 , i 2 , i 4Y x i 3y 2++ δ k 1,k 2 ,k 3i 1 , i 3 , i 4Y x i 2y 2 + δ k 2,k 1 ,k 3i 1 , i 3 , i 4Y x i 2y 2 + δ k 2,k 3 ,k 1i 1 , i 3 , i 4Y x i 2y 2++ δ k 1,k 2 ,k 3i 2 , i 3 , i 4Y x i 1y 2 + δ k 2,k 1 ,k 3i 2 , i 3 , i 4Y x i 1y 2 + δ k 2,k 3 ,k 1i 2 , i 3 , i 4Y x i 1y 2−− δ k 1,k 2i 1 , i 2X k 3x i 3x i 4y − δk 2,k 1i 1 , i 2X k 3x i 3x i 4y − δk 2,k 3i 1 , i 2X k 1x i 3x i 4y −− δ k 1,k 2i 1 , i 3X k 3x i 2x i 4y − δk 2,k 1i 1 , i 3X k 3x i 2x i 4y − δk 2,k 3i 1 , i 3X k 1x i 2x i 4y −− δ k 1,k 2i 1 , i 4X k 3x i 2x i 3y − δk 2,k 1i 1 , i 4X k 3x i 2x i 3y − δk 2,k 3i 1 , i 4X k 1x i 2x i 3y −− δ k 1,k 2i 2 , i 3X k 3x i 1x i 4y − δk 2,k 1i 2 , i 3X k 3x i 1x i 4y − δk 2,k 3i 2 , i 3X k 1x i 1x i 4y −− δ k 1,k 2i 2 , i 4X k 3x i 1x i 3y − δk 2,k 1i 2 , i 4X k 3x i 1x i 3y − δk 2,k 3i 2 , i 4X k 1x i 1x i 3y −−δ k 1,k 2i 3 , i 4X k 3x i 1x i 2y − δk 2,k 1i 3 , i 4X k 3x i 1x i 2y − δk 2,k 3i 3 , i 4X k 1x i 1x i 2y]y k1 y k2 ,k 3.This sum developes the term [12 Y xy 2 − 18 X x 2 y] y 1 y 2 of Y 3 (third lineof (2.9)). Let us explain what are the formal rules.In the bracketed terms of (3.53), there are no permutation of the indicesi 1 , i 2 , i 3 , but there is a certain unknown subset of all the permutations of thefour indices k 1 , k 2 , k 3 , k 4 . In the bracketed terms of (3.54), two combinatoricsare present:• there are some permutations of the indices i 1 , i 2 , i 3 , i 4 and• there are some permutations of the indices k 1 , k 2 , k 3 .Here, the permutations of the indices i 1 , i 2 , i 3 , i 4 are easily guessed, sincethey are the same as the permutations which were introduced in §3.48 above.Indeed, in the first four lines of (3.54), we see the four decompositions(3.55){i 1 , i 2 , i 3 }∪{i 4 }, {i 1 , i 2 , i 4 }∪{i 3 }, {i 1 , i 3 , i 4 }∪{i 2 }, {i 2 , i 3 , i 4 }∪{i 1 },of the set {i 1 , i 2 , i 3 , i 4 }, and in the last six lines of (3.54), we see the sixdecompositions(3.56){i 1 , i 2 } ∪ {i 3 , i 4 }, {i 1 , i 3 } ∪ {i 2 , i 4 }, {i 1 , i 4 } ∪ {i 2 , i 3 },{i 2 , i 3 } ∪ {i 1 , i 4 }, {i 2 , i 4 } ∪ {i 1 , i 3 }, {i 3 , i 4 } ∪ {i 1 , i 2 },so that (3.54) may be written under the form(3.57) ⎡∑⎣ ∑ ∑δ k τ(1),k τ(2) ,k τ(3)i τ(1) ,i τ(2) ,i τ(3)Y i − ∑x τ(4) y 2k 1 ,k 2 ,k 3 σ∈?τ∈S 3 4τ∈S 2 4∑σ∈?δ k τ(1),k τ(2)i τ(1) ,i τ(2)X k τ(3)333x i τ(3) x i τ(4) y⎤⎦ y k1 y k2 ,k 3,