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

eing related to the number 2 in the splitting 3 = 1 + 2 of the lower indicesof the monomial y l 11 y l 22 , it follows that the index l α attached to the second Xterm must be the index l 2 of the monomial y l 22 .This rule is still ambiguous. Indeed, let us examine the third line of (4.15).We have the splitting 10 = 6 + 4, homologous to the splitting of relativeweights 5 = (1+1+1)+2 in the monomial y l 11 y l 21 y l 31 y l 42 . Of course, it is clearthat we must choose the index l 4 for the Kronecker symbol associated to thesecond X term −4 X y 3, thus obtaining −δ j l 44 X y l 1y l 2y l 3 . However, sincethe monomial y l 11 y l 21 y l 31 has three indices l 1 , l 2 and l 3 , there arises a question:what index l α must we choose for the Kronecker symbol δ j l αattached to thefirst X term 6 X y 3: the index l 1 , the index l 2 or the index l 3 ?The answer is simple: any of the three indices l 1 , l 2 or l 3 works. Indeed,since the monomial y l 11 y l 21 y l 31 is symmetric with respect to all permutationsof the set of three indices {l 1 , l 2 , l 3 }, we have(4.17)∑ m ]m∑][−δ j l 16X y l 2y l 3y l 4 y l 11 y l 21 y l 31 y l 42 =[−δ j l 26X y l 1y l 3y l 4 y l 11 y l 21 y l 31 y l 42 =l 1 ,l 2 ,l 3 ,l 4 =1=l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1345[−δ j l 36X y l 1y l 2y l 4]y l 11 y l 21 y l 31 y l 42 .In fact, we have systematically used such symmetries during the intermediatecomputations (not exposed here) which we achieved manually to obtainthe final expressions of Y j 1, of Y j 2, of Y j 3 and of Y j 4. To fix ideas, we havealways choosen the first index. Here, the first index is l 1 ; in the first sum ofline 9 of (4.8), the first index l α for the second weight 12 is l 2 .This rule may be confirmed by inspecting all the monomials of Y j 2, ofY j 3, of Y j 4 (and also of Y j 5, which we have computed in a manuscript, butnot copied in this Latex file).From these considerations, we deduce that for the general formula, theweight decomposition is simply µ 1 λ 1 , . . .,µ d λ d and that the Kronecker symbolδ j α is intrinsically associated to the weights. In conclusion, building oninductive reasonings, we have formulated the following statement.Theorem 4.18. For one independent variable x, for several dependent variables(y 1 , . . ., y m ) and for κ 1, the general expression of the coefficient