Multilinear Cayley factorization. (English) Zbl 0747.15009
The author develops an algorithm which solves the Cayley factorization of a homogeneous bracket polynomial \(P\) which is multilinear. The algorithm has seven steps.
In step 1 we first find the atomic extensors for the bracket polynomial \(P(a,b,\ldots,z)\) of rank \(d\) which is multilinear in the \(N\) points \(a,b,\ldots,z\). In step 2 we rewrite \(P\) as a bracket polynomial which is dotted in each atomic extensor. In step 3 we apply straightening with the \(d\) elements of \(E\) first in the linear order, if there exists an atomic extensor \(E\) of step 1. The result must have \(E\) as the first row of every resulting tableau.
In step 4 we find a primitive factor if any, for pairs of extensors \(E=\{e_ 1,e_ 2,\ldots,e_ k\}\), \(F=\{f_ 1,f_ 2,\ldots,f_ \ell\}\) such that \(k+\ell\geq d\). Step 5 checks if such \(E\) and \(F\) do not exist. Then no factorization is possible. In step 6 we recompute the atomic extensors by trying to extend the current ones. Finally in step 7 we go to step 2 and repeat. An example is worked out.
In step 1 we first find the atomic extensors for the bracket polynomial \(P(a,b,\ldots,z)\) of rank \(d\) which is multilinear in the \(N\) points \(a,b,\ldots,z\). In step 2 we rewrite \(P\) as a bracket polynomial which is dotted in each atomic extensor. In step 3 we apply straightening with the \(d\) elements of \(E\) first in the linear order, if there exists an atomic extensor \(E\) of step 1. The result must have \(E\) as the first row of every resulting tableau.
In step 4 we find a primitive factor if any, for pairs of extensors \(E=\{e_ 1,e_ 2,\ldots,e_ k\}\), \(F=\{f_ 1,f_ 2,\ldots,f_ \ell\}\) such that \(k+\ell\geq d\). Step 5 checks if such \(E\) and \(F\) do not exist. Then no factorization is possible. In step 6 we recompute the atomic extensors by trying to extend the current ones. Finally in step 7 we go to step 2 and repeat. An example is worked out.
Reviewer: J.K.Sengupta (Santa Barbara)
MSC:
| 15A75 | Exterior algebra, Grassmann algebras |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
multilinear Cayley factorization; algorithm; homogeneous bracket polynomial; atomic extensors; primitive factorReferences:
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