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

230− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 a µ | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1) a − · · · −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 a µ |a ν |a ν+1 | · · · |a n+1) b −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |a ν a µ |a ν+1 | · · · |a n+1) OK −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 a µ | · · · |a n+1) c − · · · −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1 a µ) d .The ante-penultimate underlined term “OK” will be kept untouched. To thepairs of (subtracted) □-binomials that are underlined with a, b, c, d appended(including of course all terms present in the four “· · · ”), we need anelementary instance of the Plücker identities.To state it generally, let m 2, let C 1 , C 2 , . . ., C m , D, E be (m + 2)column vectors in K m and introduce the following notation for the m ×(m + 2) matrix consisting of these vectors:[C 1 |C 2 | · · · |C m |D|E].Extracting columns from this matrix, we shall construct m×m determinantsthat are modification of the following “ground” determinant:|C 1 | · · · |C m || ≡ ∣ ∣ ∣ ∣C 1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C m∣ ∣∣ ∣ .We use a double vertical line in the beginning and in the end to denote adeterminant. Also, we emphasize two distinct columns, the j 1 -th and thej 2 -th, where j 2 > j 1 , since we will modify them. For instance in this matrix,let us replace these two columns by the column D and by the column E,which yields the determinant:∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2E| · · · |C m∣ ∣∣ ∣.In this notation, one should understand that only the j 1 -th and the j 2 -thcolumns are distinct from the columns of the fundamental m × m “ground”determinant.Lemma. ([17], p. 155) The following quadratic identity between determinantsholds true:∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2E| · · · |C n∣ ∣∣ ∣ ·∣ ∣∣ ∣C1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C n∣ ∣∣ ∣ == ∣ ∣ ∣ ∣C 1 | · · · | j 1D| · · · | j 2C j2 | · · · |C n∣ ∣∣ ∣ · ∣∣ ∣ ∣C 1 | · · · | j 1C j1 | · · · | j 2E| · · · |C n∣ ∣ ∣ ∣ −− ∣ ∣ ∣ ∣ C1 | · · · | j 1E| · · · | j 2C j 2| · · · |C n∣ ∣∣ ∣ ·∣ ∣ ∣ ∣C1 | · · · | j 1C j1 | · · · | j 2D| · · · |C n∣ ∣∣ ∣ .Admitting this elementary statement without redoing its proof and applyingit to all the above underlined pairs of (subtracted) monomials, afterchecking that all final signs are “−”, we obtain the following neat expression