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

Correspondingly, we identify the set(3.35){1,... ,λ 1 ,... ,µ 1 λ 1 ,...... ,µ 1 λ 1 + µ 2 λ 2 ,... ...,µ 1 λ 1 + µ 2 λ 2 + · · · + µ d λ d }of all integers α from 1 to µ 1 λ 1 + µ 2 λ 2 + · · · + µ d λ d with the followingspecific set(3.36) {1:1:1,... ,1:1:λ} {{ 1 ,...,1:µ} 1 :λ 1 ,... ,2:µ 2 :λ 2 ,... ,d:µ d :λ d },λ}1{{ }µ 1 λ 1} {{ }µ 1 λ 1 +µ 2 λ 2} {{ }µ 1 λ 1 +µ 2 λ 2 +···+µ d λ dwritten in a lexicographic way which emphasizes clearly the subdivision ind collections of µ d groups of λ d integers.With this notation at hand, we see that the development, in several independentvariables, of the general monomial (y λ1 ) µ1 · · ·(y λd ) µ d, may bewritten as follows:(3.37)y k1:1:1 ,...,k 1:1:λ1 · · · y k1:µ1 :1,...,k 1:µ1 :λ 1· · · y kd:1:1 ,...,k d:1:λd · · · · · · y kd:µd :1,...,k d:µd :λ d.Formally speaking, this expression is better than (3.32). Using product symbols,we may even write it under the slightly more compact form∏∏(3.38)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 dNow that we have translated the monomial, we may add all the summationsymbols: the general expression of Y κ (which generalizes our three previousexamples (3.26)) will be of the form:(3.39)Y κ = Y x i 1···x iκ +n∑k 1:1:1 ,...,k 1:1:λ1 =1[?]∏∑κ+1d=1· · ·∑1λ 1