Transvectant

If Q₁,...,Q_n are functions of n variables x = (x₁,...,x_n) and r ≥ 0 is an integer then the r^th transvectant of these functions is a function of n variables given by$$\backslash operatorname\{Tr\}\; \backslash Omega^r(Q\_1\backslash otimes\backslash cdots\; \backslash otimes\; Q\_n)$$where$$$$

\Omega = \begin{vmatrix} \frac{\partial}{\partial x_{11}} & \cdots &\frac{\partial}{\partial x_{1n}} \\ \vdots& \ddots & \vdots\\ \frac{\partial}{\partial x_{n1}} & \cdots &\frac{\partial}{\partial x_{nn}} \end{vmatrix}

is Cayley's Ω process, and the tensor product means take a product of functions with different variables x¹,..., xⁿ, and the trace operator Tr means setting all the vectors x^k equal.

The zeroth transvectant is the product of the n functions.$$\backslash operatorname\{Tr\}\; \backslash Omega^0(Q\_1\backslash otimes\backslash cdots\; \backslash otimes\; Q\_n)\; =\; \backslash prod\_k\; Q\_k$$The first transvectant is the Jacobian determinant of the n functions.$$\backslash operatorname\{Tr\}\; \backslash Omega^1(Q\_1\backslash otimes\backslash cdots\; \backslash otimes\; Q\_n)\; =\; \backslash det\; \backslash begin\{bmatrix\}\; \backslash partial\_k\; Q\_l\; \backslash end\{bmatrix\}$$The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.

When $n\; =\; 2$, the binary transvectants have an explicit formula:{{sfn|Olver|1999|p=88}}$$\backslash operatorname\{Tr\}\; \backslash Omega^k(\; f\; \backslash otimes\; g\; )\; =\; \backslash sum\_\{l=0\}^k\; (-1)^l\; \backslash binom\; kl\; \backslash partial\_x^\{k-l\}\; \backslash partial\_y^l\; f\; \backslash partial\_y^\{k-l\}\; \backslash partial\_l^l\; g$$which can be more succinctly written as$$f\; \backslash left(\backslash overleftarrow\{\backslash partial\_\{x\}\}\; \backslash cdot\; \backslash overrightarrow\{\backslash partial\_\{y\}\}-\backslash overleftarrow\{\backslash partial\_\{y\}\}\; \backslash cdot\; \backslash overrightarrow\{\backslash partial\_\{x\}\}\backslash right)^k\; g$$where the arrows denote the function to be taken the derivative of. This notation is used in Moyal product.

{{Math theorem

| math_statement = All polynomial covariants and invariants of any system of binary forms can be expressed as linear combinations of iterated transvectants.

| name = First Fundamental Theorem of Invariant Theory

| note = {{sfn|Olver|1999|p=90}}

}}

{{Reflist}}

• {{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=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 }}

Category:Invariant theory