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ON THE LINEAR COMBINANTS OF A BINARY PENCIL

Published online by Cambridge University Press:  01 September 2009

ABDELMALEK ABDESSELAM and
JAYDEEP CHIPALKATTI
ABDELMALEK ABDESSELAM
Affiliation:
Kerchof Hall Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville, VA 22904-4137, USA e-mail: malek@virginia.edu
JAYDEEP CHIPALKATTI
Affiliation:
Machray Hall, Department of Mathematics, University of Manitoba, Winnipeg MB R3T 2N2, Canada e-mail: chipalka@cc.umanitoba.ca
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Abstract

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Let A, B denote binary forms of order d, and let 2r−1 = (A, B)2r−1 be the sequence of their linear combinants for $1 \le r \le \lfloor\frac{d+1}{2}\rfloor$. It is known that 1, 3 together determine the pencil {A + λ B}λ∈P1 and hence indirectly the higher combinants 2r−1. In this paper we exhibit explicit formulae for all r ≥ 3, which allow us to recover 2r−1 from the knowledge of 1 and 3. The calculations make use of the symbolic method in classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the plethysm representation ∧2Sd for the group SL2. We give an example for the group SL3 to show that such a result may hold for other categories of representations.

Keywords

13A5022E70

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009, pp. 481 - 498
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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