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

328Consider now the product (y λ1 ) µ 1(y λ2 ) µ 2. How should it develope in thecase of several independent variables? For instance, in the expression ofY i1 ,i 2 ,i 3, we have developed the product (y 1 ) 2 y 2 as y k1 y k2 y k3 ,k 4. Thus, areasonable proposal of formalism would be that the product (y λ1 ) µ 1(y λ2 ) µ 2should be developed as a product of the form(3.30)wherey k1 ,...,k λ1y kλ1 +1,...,k 2λ1 · · · y k(µ1 −1)λ 1 +1,...,k µ1 λ 1y kµ1 λ 1 +1,...,k µ1 λ 1 +λ 2· · · y kµ1 λ 1 +(µ 2 −1)λ 2 +1,...,k µ1 λ 1 +µ 2 λ 2,(3.31) k 1 , . . .,k λ1 , . . .,k µ1 λ 1, . . ., k µ1 λ 1 +µ 2 λ 2= 1, . . ., n.However, when trying to write down the development of the general monomial(y λ1 ) µ 1(y λ2 ) µ2 · · ·(y λd ) µ d, we would obtain the complicated product(3.32)y k1 ,...,k λ1y kλ1 +1,...,k 2λ1 · · · y k(µ1 −1)λ 1 +1,...,k µ1 λ 1y kµ1 λ 1 +1,...,k µ1 λ 1 +λ 2...y kµ1 λ 1 +(µ 2 −1)λ 2 +1,...,k µ1 λ 1 +µ 2 λ 2......... ............... ............... ...............y kµ1 λ 1 +···+µ d−1 λ d−1 +1,...,k µ1 λ 1 +···+µ d−1 λ d−1 +λ d· · ·· · · y kµ1 λ 1 +···+µ d−1 λ d−1 +(µ d −1)λ d +1,...,k µ1 λ 1 +···+µ d λ d.Essentially, this product is still readable. However, in it, some of the integersk α have a too long index α, often involving a sum. Such a lengthof α would be very inconvenient in writing down and in reading the generalKronecker symbols δ kα 1 ,...,kα λi 1 ,......,i λwhich should appear in the final expressionof Y i1 ,...,i κ. One should read in advance Theorem 3.73 below to observethe presence of such multiple Kronecker symbols. Consequently, forα = 1, . . ., µ 1 λ 1 , . . .,µ 1 λ 1 + · · · + µ d λ d , we have to denote the indicesk α differently.Notational convention 3.33. We denote d collection of µ d groups of λ d (apriori distinct) integers k α = 1, . . ., n by(3.34)k 1:1:1 , . . .,k} {{ 1:1:λ1 , . . .,k} 1:µ1 :1, . . .,k 1:µ1 :λ} {{ }1λ 1 λ} {{1}µ 1,k 2:1:1 , . . .,k} {{ 2:1:λ2 , . . .,k} 2:µ2 :1, . . .,k 2:µ2 :λ} {{ }2λ 2 λ} {{2}µ 2,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .k d:1:1 , . . .,k} {{ d:1:λd , . . ., k} d:µd :1, . . .,k d:µd :λ} {{ } dλ d.λ dµ d} {{ }