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

336which leaves unchanged the monomial(3.69)∏y kσ(e:νe:1) ,...,k σ(e:νe:λe)=1ν eµ e∏1ν eµ ey ke:νe:1 ,...,k e:νe:λe.uniquely decomposes as the composition of• µ e arbitrary permutations of the µ e groups of λ e integers {e : ν e :1, . . ., e:ν e :λ e }, of total cardinal (λ e !) µe ;• an arbitrary permutation between these µ e groups, of total cardinal µ e !.Consequently( )(3.70) Card H (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d= µ 1 !(λ 1 !) µ1 · · ·µ d !(λ d !) µ d.Finally, define the coset(3.71) F (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d:= S µ1 λ 1 +···+µ d λ d/H (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ dwith(3.72)( )Card F (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d= Card (S µ 1 λ 1 +···+µ d λ d)(Card)H (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d= (µ 1λ 1 + · · · + µ d λ d )!.µ 1 !(λ 1 !) µ 1 · · ·µd !(λ d !) µ dIn conclusion, by means of this formalism, we may write down the completegeneralization of Theorem 2.24 to several independent variables.