## Affine Geometries of Paths Possessing an Invariant Integral by Eisenhart L. P. By Eisenhart L. P.

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Example text

Let A be an affine C-algebra and M ∈ repn A. M is Zariski open in its closure O(M ). M . then, by the above remark Consider the ’orbit-map’ GLn O(M ) = φ(GLn ) contains a Zariski open subset U of O(M ) contained in the image of φ which is O(M ). U is also open in O(M ). Next, we claim that O(M ) contains a closed orbit. Indeed, assume O(M ) is not closed, then the complement CM = O(M ) − O(M ) is a proper Zariski closed subset whence dim C < dim O(M ). But, C is the union of GLn -orbits O(Mi ) with dim O(Mi ) < dim O(M ).

In particular, we apply this to the underlying vectorspace of Mψ which is V = Cn (column vectors) on which GLn acts by left multiplication. We define Mj = ⊕i>j Vλ,i and claim that this defines a finite filtration on Mψ with associated graded A-module Mρ . For any a ∈ A (it suffices to vary a over the generators of A) we can consider the linear maps φij (a) : Vλ,i ⊂ a. ✲ V = Mψ ✲ Mψ = V ✲ ✲ Vλ,j (that is, we express the action of a in a blockmatrix Φa with respect to the decomposition of V ). λ(t)−1 with corresponding blocks Vλ,i φij (a) ✲ Vλ,j ✻ λ(t)−1 Vλ,i that is φij (a) = t j−i λ(t) ❄ ✲ Vλ,j φij (a) φij (a).

D} and let σ = (i1 i1 . . iα )(j1 j2 . . jβ ) . . (z1 z2 . . zζ ) be a decomposition of σ ∈ Sd into cycles including those of length one. The map T assigns to σ a formal necklace Tσ (x1 , . . , xd ) defined by Tσ (x1 , . . , xd ) = t(xi1 xi2 . . xiα )t(xj1 xj2 . . xjβ ) . . t(xz1 xz2 . . (v1 ⊗ . . ⊗ vd ) = vσ(1) ⊗ . . ⊗ vσ(d) hence determines a linear map λσ ∈ End(V ⊗d ). Recall from section 3 that under the natural identifications (Mn⊗d )∗ (V ∗⊗d ⊗ V ⊗d )∗ End(V ⊗d ) the map λσ defines the multilinear map µσ : Mn ⊗ .