In mathematical invariant theory, a transvectant is an

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

formed from ''n'' invariants in ''n'' variables using Cayley's Ω process.

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 :

tr \Omega^r(Q_1\otimes\cdots \otimes Q_n)

where Ω is Cayley's Ω process, the tensor product means take a product of functions with different variables x¹,..., 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 A Hessian is an inhabitant of the German state of Hesse. Hessian may also refer to: Named from the toponym *Hessian (soldier), eighteenth-century German regiments in service with the British Empire **Hessian (boot), a style of boot **Hessian f ...

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