Molien series and low-degree invariants for a natural
action of SO(3)wrZ2
Molien series and low-degree invariants for a natural
action of SO(3)wrZ2
We investigate the invariants of the 25-dimensional real representation of the group SO(3)wr2 given by the left and right actions of SO(3) on 5 × 5 matrices together with matrix transposition; the action on column vectors is the irreducible five-dimensional representation of SO(3). The 25-dimensional representation arises naturally in the study of nematic liquid crystals, where the second-rank orientational order parameters of a molecule are represented by a symmetric 3 × 3 traceless symmetric matrix, and where a rigid rotation in R3 induces a linear transformation of this space of matrices. The entropy contribution to a free energy density function in this context turns out to have SO(3)wrZ2 symmetry. Although it is unrealistic to expect to describe the complete algebraic structure of the ring of invariants, we are able to calculate as a rational function the Molien series that gives the number of linearly independent invariants at each homogeneous degree. The form of the function indicates a basis of 19 primary invariants and suggests there are N = 1453926048 linearly independent secondary invariants; we prove that their number is an integer multiple of N/4. The algebraic structure of invariants up to degree 4 is investigated in detail.
1-24
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231
Lauterbach, Reiner
f541c8b3-23d9-4282-b431-7e01e3060937
Turzi, Stefano
e33458ff-7b11-4c77-b40f-7c8f3146f5a0
9 January 2015
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231
Lauterbach, Reiner
f541c8b3-23d9-4282-b431-7e01e3060937
Turzi, Stefano
e33458ff-7b11-4c77-b40f-7c8f3146f5a0
Chillingworth, David, Lauterbach, Reiner and Turzi, Stefano
(2015)
Molien series and low-degree invariants for a natural
action of SO(3)wrZ2.
Journal of Physics A: Mathematical and Theoretical, 48 (1), .
(doi:10.1088/1751-8113/48/1/015203).
Abstract
We investigate the invariants of the 25-dimensional real representation of the group SO(3)wr2 given by the left and right actions of SO(3) on 5 × 5 matrices together with matrix transposition; the action on column vectors is the irreducible five-dimensional representation of SO(3). The 25-dimensional representation arises naturally in the study of nematic liquid crystals, where the second-rank orientational order parameters of a molecule are represented by a symmetric 3 × 3 traceless symmetric matrix, and where a rigid rotation in R3 induces a linear transformation of this space of matrices. The entropy contribution to a free energy density function in this context turns out to have SO(3)wrZ2 symmetry. Although it is unrealistic to expect to describe the complete algebraic structure of the ring of invariants, we are able to calculate as a rational function the Molien series that gives the number of linearly independent invariants at each homogeneous degree. The form of the function indicates a basis of 19 primary invariants and suggests there are N = 1453926048 linearly independent secondary invariants; we prove that their number is an integer multiple of N/4. The algebraic structure of invariants up to degree 4 is investigated in detail.
Text
jpaper_submitted.pdf
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Accepted/In Press date: 17 October 2014
e-pub ahead of print date: 4 December 2014
Published date: 9 January 2015
Organisations:
Pure Mathematics
Identifiers
Local EPrints ID: 366919
URI: http://eprints.soton.ac.uk/id/eprint/366919
ISSN: 1751-8113
PURE UUID: 670c6fcb-6589-488a-8788-76606c63c143
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Date deposited: 15 Jul 2014 09:39
Last modified: 14 Mar 2024 17:19
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Author:
Reiner Lauterbach
Author:
Stefano Turzi
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