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

164and the third term III involves at least once the monomial U i 1l 1 ,l 2 ,l 3(note thatthere is no term involving simultaneously U i 1l 1 ,l 2and U i 1l 1 ,l 2 ,l 3):(7.28) ⎧III = ∑ ∑ [(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3R j − u i 1 δj i 1δ l 1,l 2k 2 ,k 3Q l 3xk1+ δ l 1,l 2k 3 ,k 1Q l 3xk2+i 1 l 1 ,l 2 ,l 3⎪⎨)]+δ l 1,l 2k 1 ,k 2Q l 3xk3U i 1l 1 ,l 2 ,l 3+ ∑ ∑ [−δ j i 1δ l 2,l 3 ,l 4k 1 ,k 2 ,k 3Q l 1− u i 2i 1 ,i 2 l 1 ,l 2 ,l 3 ,l 4⎪⎩−δ j i 2(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4u i 1 + δl 4,l 1 ,l 2k 1 ,k 2 ,k 3Q l 3u i 1 + δl 3,l 4 ,l 1k 1 ,k 2 ,k 3Q l 2u i 1)]U i 1l 1U i 2l 2 ,l 3 ,l 4.Before giving the partial expression of R κ we introduce some notations. Forp ∈ N with p ≥ 1, let S p be the group of permutations of {1, 2, . . ., p}. Forq ∈ N with 1 ≤ q ≤ p − 1, let S q p be the set of permutations σ ∈ S p suchthat σ(1) < σ(2) < · · · < σ(q) and σ(q + 1) < σ(q + 2) < · · · < σ(p). Itscardinal is C q p . Let C p be the group of cyclic permutations of {1, 2, . . ., p}.Reasoning recursively from the formula of R j k 1 ,k 2 ,k 3given by (7.28), we maygeneralize Lemma 8.1:Lemma 8.1. For every κ ≥ 4 and for every j = 1, . . ., m, k 1 , . . .,k κ =1, . . ., n, we have:(7.28) R j k 1 ,k 2 ,...,k κ= I 1 + · · · + I 9 + Remainderwhere I 1 = Rx j k1 x k2 ...x kκ,⎡I 2 = ∑ ∑⎣ ∑i 1 l 1 σ∈S 1 κ⎡∑⎣ ∑l 1 ,l 2I 3 = ∑ i 1I 4 = ∑ i 1σ∈S 2 κ−δ j i 1⎛⎝ ∑δ l 1kσ(1)R j x kσ(2)···x kσ(κ) u i 1 − δj i 1Q l 1xk1 ...x kκ⎤⎦U i 1l 1,δ l 1,l 2k σ(1) ,k σ(2)R j x kσ(3)···x kσ(κ) u i 1 −σ∈S 1 κ⎡∑⎣ ∑l 1 ,...,l κ−2σ∈S κ−2κ⎛−δ j ⎝ ∑i 1σ∈S κ−3κδ l 1kσ(1)Q l 2⎞xkσ(2)···x⎠kσ(κ)⎤⎦U i 1l 1 ,l 2,δ l 1,......,l κ−2k σ(1) ,...,k σ(κ−2)R j x kσ(κ−1) x kσ(κ) u i 1 −δ l 1,......,l κ−3k σ(1) ,...,k σ(κ−3)Q l κ−2⎞⎠x kσ(κ−2) x kσ(κ−1) x kσ(κ)⎤⎦U i 1l 1 ,...,l κ−2,