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

344Then we examine four instances extracted from the complete expression ofY4:j ⎧m∑ []4Y jxy l 1y l 2y − l 3 δj l 16X x 2 y l 2y l 3 y l 11 y l 21 y l 31 ,(4.15)⎪⎨⎪⎩l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 ,l 2 ,l 3 =1[]12Y jxy l 1y − l 2 δj l 16X x 2 y l 2 − δ j l 212X x 2 y l 1 y l 11 y l 22 ,[−δ j l 16X y l 2y l 3y l 4 − δ j l 44X y l 1y l 2y l 3]y l 11 y l 21 y l 31 y l 42 ,[−δ j l 14X y l 2y l 3 − δ j l 36X y l 1y l 2]y l 11 y l 21 y l 33 ,and we compare them to the corresponding terms of Y 4 :⎧ [ ]4Yxy 3 − 6X x 2 y 2 (y1 ) 3 ,(4.16)⎪⎨⎪⎩[ ]12Yxy 2 − 18X x 2 y y1 y 2 ,[ ]−10Xy 3 (y1 ) 3 y 2 ,[ ]−10Xy 2 (y1 ) 2 y 3 .In the development from (4.16) to (4.15), we see that the four integers justbefore X , namely 6 = 6, 18 = 6 + 12, 10 = 6 + 4 and 10 = 4 + 6, aresplit in a certain manner. Also, a single Kronecker symbol δ j l αis added as afactor. What are the rules?In the second splitting 18 = 6 + 12, we see that the relative weight of6 and of 12 is the same as the relative weight of 1 and 2 in the splitting3 = 1 + 2 issued from the lower indices of the corresponding monomialy l 11 y l 22 . Similarly, in the third splitting 10 = 6 + 4, the relative weight of6 and of 4 is the same as the relative weight of 1 + 1 + 1 and of 2 issuedfrom the lower indices of the corresponding monomial y l 11 y l 21 y l 31 y l 42 . Thisrule may be confirmed by inspecting all the other monomials of Y 2 , Y2,jof Y 3 , Y j 3 and of Y 4 , Y4. j For a general κ 1, the splitting of integersjust amounts to decompose the sum appearing inside the brackets of (4.14)as µ 1 λ 1 , µ 2 λ 2 , . . .,µ d λ d . In fact, when we wrote (4.14), we emphasized inadvance the decomposable factor (µ 1 λ 1 + · · · + µ d λ d ).Next, we have to determine what is the subscript α in the Kronecker symbolδ j l α. We claim that in the four instances (4.15), the subscript α is intrinsicallyrelated to weight splitting. Indeed, recall that in the second lineof (4.15), the number 6 of the splitting 18 = 6 + 12 is related to the number1 in the splitting 3 = 1 + 2 of the lower indices of the monomial y l 11 y l 22 . Itfollows that the index l α must be the index l 1 of the monomial y l 11 . Similarly,also in the second line of (4.15), the number 12 of the splitting 18 = 6 + 12