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

332Lemma 3.50. For κ 1, the development of (2.20) to several independentvariables (x 1 , . . .,x n ) is:(3.51)⎡⎤n∑Y i1 ,i 2 ,...,i κ= Y x i 1x i 2···x iκ + ⎣ ∑δ k 1i τ(1)Y i x τ(2)···x i τ(κ) y − X k 1 ⎦ yx i 1x i 2 ···x iκ k1 +++n∑k 1 ,k 2 =1n∑⎡k 1 ,k 2 ,k 3 =1+ · · · · · · +++n∑k 1 ,...,k κ=1n∑⎣ ∑τ∈S 2 κ⎡⎣ ∑⎡τ∈S 3 κk 1 ,...,k κ,k κ+1 =1k 1 =1τ∈S 1 κδ k 1, k 2i τ(1) ,i τ(2)Y xi τ(3)···x i τ(κ) y 2 − ∑τ∈S 1 κδ k 1, k 2 , k 3i τ(1) ,i τ(2) ,i τ(3)Y xi τ(4)···x i τ(κ) y 3 −− ∑τ∈S 2 κ⎣δ k 1,...,k κi 1 ,..., i κY y κ −δ k 1i τ(1)X k 2x i τ(2) ···x i τ(κ) yδ k 1, k 2i τ(1),iτ(2)X k 3x i τ(3)···x i τ(κ) y 2 ⎤⎦ y k1 y k2 y k3 +∑τ∈S κ−1κ⎤⎦ y k1 y k2 +δ k 1,......,k κ−1i τ(1) ,...,i τ(κ−1)X kκx i τ(κ) y κ−1 ⎤⎦ y k1 · · · y kκ +[−δ k 1,...,k κi 1 ,..., i κX k κ+1y κ ]y k1 · · · y kκ y kκ+1 + remainder.Here, the term remainder collects all remaining monomials in the pure jetvariables y k1 ,...,k λ.3.52. Continuation. Thus, we have devised how the part of Y i1 ,...,i κwhichinvolves only the jet variables y kα must be written. To proceed further, weshall examine the following term, extracted from Y i1 ,i 2 ,i 3(lines 12 and 13of (3.24))(3.53) ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y− δ k 2,k 3 ,k 12 i 1 , i 2 , i 3X k 4y− δ k 3,k 2 ,k 12 i 1 , i 2 , i 3X k 4y− 2k 1 ,k 2 ,k 3 ,k 4−δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y 2 − δ k 3,k 1 ,k 4i 1 , i 2 , i 3X k 2y 2 − δ k 1,k 3 ,k 4i 1 , i 2 , i 3X k 2y 2 ]y k1 y k2 y k3 ,k 4,which developes the term [−6 X y 2] (y 1 ) 2 y 2 of Y 3 (third line of (2.8)). Duringthe computation which led us to the final expression (3.24), we organizedthe formula in order that, in the six Kronecker symbols, the lower indicesi 1 , i 2 , i 3 are all written in the same order. But then, what is the rule for theappearance of the four upper indices k 1 , k 2 , k 3 , k 4 ?In April 2001, we discovered the rule by inspecting both (3.53) andthe following complicated term, extracted from the complete expression of