Transvectant Explained

In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's Ω process.

Definition

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$$\backslash operatorname\; \backslash Omega^r(Q\_1\backslash otimes\backslash cdots\; \backslash otimes\; Q\_n)$$whereis 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.

Examples

The zeroth transvectant is the product of the n functions.$$\backslash operatorname\; \backslash Omega^0(Q\_1\backslash otimes\backslash cdots\; \backslash otimes\; Q\_n)\; =\; \backslash prod\_k\; Q\_k$$The first transvectant is the Jacobian determinant of the n functions.$$\backslash operatorname\; \backslash Omega^1(Q\_1\backslash otimes\backslash cdots\; \backslash otimes\; Q\_n)\; =\; \backslash det\; \backslash begin\; \backslash partial\_k\; Q\_l\; \backslash end$$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:$$\backslash operatorname\; \backslash Omega^k(f\; \backslash otimes\; g)\; =\; \backslash sum\_^k\; (-1)^l\; \backslash binom\; kl\; \backslash partial\_x^\; \backslash partial\_y^l\; f\; \backslash partial\_y^\; \backslash partial\_l^l\; g$$which can be more succinctly written as$$f\; \backslash left(\backslash overleftarrow\; \backslash cdot\; \backslash overrightarrow-\backslash overleftarrow\; \backslash cdot\; \backslash overrightarrow\backslash right)^k\; g$$where the arrows denote the function to be taken the derivative of. This notation is used in Moyal product