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

must be equal to the number of indices k α minus γ. This rule is confirmedby examining the term (second and third line of (3.24))∑ [δ k 1,k 2i 1 , i 2Y x i 3y 2 + δ k 1,k 2i 3 , i 1Y x i 2y 2 + δ k 1,k 2i 2 , i 3Y x i 1y 2−(3.45)k 1 ,k 2]−δ k 1i 1X k 2x i 2x i 3y − δk 1i 2X k 2x i 1x i 3y − δk 1i 3X k 2yx i 1x i 2y k1 y k2 ,developing [3 Y xy 2 − 3 X x 2 y] (y 1 ) 2 .Also, we may examine the following term331(3.46)n∑k 1 ,k 2 =1[δ k 1,k 2i 1 , i 2Y x i 3x i 4y 2 + δ k 1,k 2i 1 , i 3Y x i 2x i 4y 2 + δ k 1,k 2i 1 , i 4Y x i 2x i 3y 2++δ k 1,k 2i 2 , i 3Y x i 1x i 4y 2 + δ k 1,k 2i 2 , i 4Y x i 1x i 3y 2 + δ k 1,k 2i 3 , i 4Y x i 1x i 2y 2−−δ k 1i 1X k 1x i 2x i 3x i 4y − δk 1i 2X k 1x i 1x i 2x i 3y − δk 1i 3X k 1x i 1x i 2x i 4y −−δ k 1i 4X k 1x i 1x i 2x i 3y]y k1 y k2 ,extracted from Y i1 ,i 2 ,i 3 ,i 4and developing [6 Y x 2 y 2 − 4 X x 3 y] (y 1 ) 2 . Wewould like to mention that we have not written the complete expression ofY i1 ,i 2 ,i 3 ,i 4, because it would cover two and a half printed pages.By inspecting the way how the indices are permuted in the multiple Kroneckersymbols of the first two lines of this expression (3.46), we observethat the six terms correspond exactly to the six possible choices of two complementaryordered couples of integers in the set {1, 2, 3, 4}, namely(3.47){1, 2} ∪ {3, 4}, {1, 3} ∪ {2, 4}, {1, 4} ∪ {2, 3},{2, 3} ∪ {1, 4}, {2, 4} ∪ {1, 3}, {3, 4} ∪ {1, 2}.At this point, we start to devise the general combinatorics. Before proceedingfurther, we need some notation.3.48. Permutation groups. For every p ∈ N with p 1, we denote by S pthe full permutation group of the set {1, 2, . . ., p − 1, p}. Its cardinal equalsp!. The letters σ and τ will be used to denote an element of S p . If p 2, andif q ∈ N satisfies 1 q p−1, we denote by S q p the subset of permutationsσ ∈ S p satisfying the two collections of inequalities(3.49)σ(1) < σ(2) < · · · < σ(q) and σ(q +1) < σ(q +2) < · · · < σ(p).The cardinal of S q p equals C q p = p!q! (p−q)! .