“Cayley's Ω process”的意思、由来-开放百科全书

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Cayley's Ω process

释义

Applications
References

{{About|the mathematical process|the industrial OMEGA process|OMEGA process}}

In mathematics, Cayley's Ω process, introduced by {{harvs|txt|authorlink=Arthur Cayley|first=Arthur |last=Cayley|year=1846}}, 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 n² variables x_ij, the omega operator is given by the determinant

For binary forms f in x₁, y₁ and g in x₂, y₂ 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 x₁, y₁ and g to a form in x₂, y₂
Apply the Ω operator r times to the function fg, that is, f times g in these four variables
Substitute x for x₁ and x₂, y for y₁ and y₂ 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

{{harvtxt|Weyl|1946}} used to find generators for the invariants of various classical groups acting on natural polynomial algebras.{{harvtxt|Hilbert|1890}} 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.

References

{{citation|first=Arthur|last=Cayley|title=On linear transformations|journal=Cambridge and Dublin mathematical journal|volume=1|year=1846|pages=104–122|url=https://books.google.com/books?id=PBcAAAAAMAAJ&dq=Cambridge%20and%20Dublin%20mathematical%20journal%201846&pg=PR3#v=onepage&q=Cambridge%20and%20Dublin%20mathematical%20journal%201846&f=false}} Reprinted in {{citation|last=Cayley|title=The collected mathematical papers|volume=1|year=1889|publisher=Cambridge University press|place=Cambridge|pages=95–112}}
{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Theorie der algebraischen Formen | doi=10.1007/BF01208503 | year=1890 | journal=Mathematische Annalen | issn=0025-5831 | volume=36 | issue=4 | pages=473–534}}
{{Citation | doi=10.1090/S0002-9947-1989-0986027-X | last1=Howe | first1=Roger | author1-link=Roger Evans Howe | title=Remarks on classical invariant theory. | jstor=2001418 | mr=0986027 | year=1989 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=313 | issue=2 | pages=539–570 | publisher=American Mathematical Society}}
{{Citation | last1=Olver | first1=Peter J. |author1-link=Peter J. Olver | title=Classical invariant theory | publisher=Cambridge University Press | isbn=978-0-521-55821-1 | year=1999}}
{{Citation | last1=Sturmfels | first1=Bernd | author1-link=Bernd Sturmfels | title=Algorithms in invariant theory | publisher=Springer-Verlag | location=Berlin, New York | series=Texts and Monographs in Symbolic Computation | isbn=978-3-211-82445-0 | mr=1255980 | year=1993}}
{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The Classical Groups: Their Invariants and Representations | url=https://books.google.com/books?id=zmzKSP2xTtYC | accessdate=03/2007/26 | publisher=Princeton University Press | isbn=978-0-691-05756-9 | mr=0000255 | year=1946}}

1 : Invariant theory

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