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

335conclusion, we have shown that (3.53) may be written under the form:⎤∑ ⎢(3.63) ⎣−∑δ k σ(1),k σ(2) ,k σ(3) ⎥⎦y k1 y k2 y k3 ,k 4,k 1 ,k 2 ,k 3 ,k 4⎡σ∈F (2,1),(1,2)4i 1 , i 2 , i 3X k σ(4)y 2This rule is confirmed by inspecting (3.54) (as well as all the otherterms of Y i1 ,i 2 ,i 3and of Y i1 ,i 2 ,i 3 ,i 4). Indeed, the permutations σ of the set{k 1 , k 2 , k 3 } which stabilize the monomial y k1 y k2 ,k 3consist just of the identitypermutation and the transposition of k 2 and k 3 . The coset S 3 /H (1,1),(1,2)3has the three representatives( )1 2 3(3.64),1 2 3(1 2 32 1 3),(1 2 32 3 1which appear in the upper index position of each of the ten lines of (3.54).It follows that (3.54) may be written under the form(3.65)⎡∑⎢∑⎣k 1 ,k 2 ,k 3τ∈S 3 4− ∑ σ∈S 2 4∑σ∈F (1,1),(1,2)3∑τ∈F (1,1),(1,2)3δ k σ(1),k σ(2) ,k σ(3)i τ(1) ,i τ(2) ,i τ(3)δ k σ(1),k σ(2)i τ(1) ,i τ(2)X k σ(3)Y xi τ(4) y 2 −x i τ(3) x i τ(4) y⎤),⎥⎦y k1 y k2 ,k 3.3.66. General complete expression of Y i1 ,...,i κ. As in the incomplete expression(3.39) of Y i1 ,...,i κ, consider integers 1 λ 1 < · · · < λ d κ andµ 1 1, . . ., µ d 1 satisfying µ 1 λ 1 +· · ·+µ d λ d κ+1. By H µ1 λ 1 +···+H µd λ d,we denote the subgroup of permutations τ ∈ S µ1 λ 1 +···+H µd λ dthat leave unchangedthe general monomial (3.38), namely that satisfy∏y kσ(1:ν1 :1),...,k · · ·∏σ(1:ν1 :λ 1y) kσ(d:νd :1),...,k =σ(d:νd :λ d )1ν 1 µ 1 1ν d µ d(3.67)= ∏∏y k1:ν1 :1,...,k 1:ν1 :λ 1· · · y kd:νd :1,...,k d:νd :λ d.1ν 1 µ 1 1ν d µ dThe structure of this group may be described as follows. For every e =1, . . ., d, an arbitrary permutation σ of the set(3.68){e: 1:1, . . ., e: 1:λ } {{ } e , e: 2:1, . . ., e: 2:λ e , · · · , e: µ } {{ } e :1, . . .,e: µ e :λ } {{ } eλ eλ eλ} {{e}µ e}