Transvectant

In mathematical invariant theory, a transvectant is an invariant 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$\operatorname \Omega^r(Q_1\otimes\cdots \otimes Q_n)$where$\Omega = \begin \frac & \cdots &\frac \\ \vdots& \ddots & \vdots\\ \frac & \cdots &\frac \end$is Cayley's Ω process, and the tensor product means take a product of functions with different variables x¹,..., x^''n'', and the trace operator Tr means setting all the vectors x^''k'' equal.

Examples

The zeroth transvectant is the product of the ''n'' functions.$\operatorname \Omega^0(Q_1\otimes\cdots \otimes Q_n) = \prod_k Q_k$The first transvectant is the Jacobian determinant of the ''n'' functions.$\operatorname \Omega^1(Q_1\otimes\cdots \otimes Q_n) = \det \begin \partial_k Q_l \end$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:$\operatorname \Omega^k( f \otimes g ) = \sum_^k (-1)^l \binom kl \partial_x^ \partial_y^l f \partial_y^ \partial_l^l g$which can be more succinctly written as$f \left(\overleftarrow \cdot \overrightarrow-\overleftarrow \cdot \overrightarrow\right)^k g$where the arrows denote the function to be taken the derivative of. This notation is used in Moyal product.

Applications

References

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* {{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