SEMINAL CONTRIBUTION TO THE SYMBOLIC METHOD IN INVARIANT THEORY

CLEBSCH, A.

Ueber symbolische Darstellung algebraischer Formen.

Berlin, G. Reimer, 1861. 4to. As extracted from "Journal für die reine und angewandte Mathematik, 59. Band, 1861". Without backstrip. Fine and clean. [Clebsch:] Pp. 1-62.

First printing of Clebsch's early and founding paper on the symbolic method in invariant theory. The Symbolic method is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a tensor product of copies of it.

"Clebsch completed the symbolic calculus for forms and invariants created by Aronhold, and henceforth one spoke of the Clebsch-Aronhold symbolic notation. Clebsch’s own contributions in this field of algebraic geometry include the following. With the help of suitable eliminations he determined a surface of order 11n - 24 intersecting a given surface of order n in points where there is a tangent that touches the surface at more than three coinciding points. For a given cubic surface he calculated the tenth-degree equation on the resolution of which the determination of the Sylvester pentahedron of that surface depends. For a plane quartic curve Clebsch found a remarkable invariant that, when it vanishes, makes it possible to write the curve equation as a sum of five fourth-degree powers. At the end of his life Clebsch inaugurated the notion of a "connex," a geometrical object in the plane obtained by setting a form containing both point and line coordinates equal to zero." (DSB)

Order-nr.: 48050