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In mathematics, Cayley's Ω process, introduced by , is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action. As a partial differential operator acting on functions of ''n''2 variables ''x''''ij'', the omega operator is given by the determinant : For binary forms ''f'' in ''x''1, ''y''1 and ''g'' in ''x''2, ''y''2 the Ω operator is . The ''r''-fold Ω process Ω''r''(''f'', ''g'') on two forms ''f'' and ''g'' in the variables ''x'' and ''y'' is then # Convert ''f'' to a form in ''x''1, ''y''1 and ''g'' to a form in ''x''2, ''y''2 # Apply the Ω operator ''r'' times to the function ''fg'', that is, ''f'' times ''g'' in these four variables # Substitute ''x'' for ''x''1 and ''x''2, ''y'' for ''y''1 and ''y''2 in the result The result of the ''r''-fold Ω process Ω''r''(''f'', ''g'') on the two forms ''f'' and ''g'' is also called the ''r''-th transvectant and is commonly written (''f'', ''g'')''r''. ==Applications== Cayley's Ω process appears in Capelli's identity, which used to find generators for the invariants of various classical groups acting on natural polynomial algebras. used Cayley's Ω process in his proof of finite generation of rings of invariants of the general linear group. His use of the Ω process gives an explicit formula for the Reynolds operator of the special linear group. Cayley's Ω process is used to define transvectants. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cayley's Ω process」の詳細全文を読む スポンサード リンク
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