In mathematical invariant theory, a transvectant is an
invariant formed from ''n'' invariants in ''n'' variables using
Cayley's Ω process
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 ...
.
Definition
If ''Q''
1,...,''Q''
''n'' are functions of ''n'' variables x = (''x''
1,...,''x''
''n'') and ''r'' ≥ 0 is an integer then the ''r''
th transvectant of these functions is a function of ''n'' variables given by
:
where Ω is
Cayley's Ω process
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 ...
, the tensor product means take a product of functions with different variables x
1,..., x
''n'', and tr means set all the vectors x
''k'' equal.
Examples
The zeroth transvectant is the product of the ''n'' functions.
The first transvectant is the
Jacobian determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
of the ''n'' functions.
The second transvectant is a constant times the completely polarized form of the
Hessian of the ''n'' functions.
Footnotes
References
*
* {{Citation , last1=Olver , first1=Peter J. , author1-link=Peter J. Olver , last2=Sanders , first2=Jan A. , title=Transvectants, modular forms, and the Heisenberg algebra , doi=10.1006/aama.2000.0700 , mr=1783553 , year=2000 , journal=Advances in Applied Mathematics , issn=0196-8858 , volume=25 , issue=3 , pages=252–283, citeseerx=10.1.1.46.803
Invariant theory