45 research outputs found
Quintics with Finite Simple Symmetries
We construct all quintic invariants in five variables with simple Non-Abelian
finite symmetry groups. These define Calabi-Yau three-folds which are left
invariant by the action of A_5, A_6 or PSL_2(11).Comment: 18 pages, typos corrected, matches published versio
Adventures in Invariant Theory
We provide an introduction to enumerating and constructing invariants of
group representations via character methods. The problem is contextualised via
two case studies arising from our recent work: entanglement measures, for
characterising the structure of state spaces for composite quantum systems; and
Markov invariants, a robust alternative to parameter-estimation intensive
methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example
Ueber Systeme höherer complexer Zahlen
http://tartu.ester.ee/record=b1868469~S23*es
Standard model plethystics
We study the vacuum geometry prescribed by the gauge invariant operators of the minimal supersymmetric standard model via the plethystic program. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the minimal supersymmetric standard model gauge invariants. All data in Mathematica format are also presented
Development of a unified tensor calculus for the exceptional Lie algebras
The uniformity of the decomposition law, for a family F of Lie algebras which
includes the exceptional Lie algebras, of the tensor powers ad^n of their
adjoint representations ad is now well-known. This paper uses it to embark on
the development of a unified tensor calculus for the exceptional Lie algebras.
It deals explicitly with all the tensors that arise at the n=2 stage, obtaining
a large body of systematic information about their properties and identities
satisfied by them. Some results at the n=3 level are obtained, including a
simple derivation of the the dimension and Casimir eigenvalue data for all the
constituents of ad^3. This is vital input data for treating the set of all
tensors that enter the picture at the n=3 level, following a path already known
to be viable for a_1. The special way in which the Lie algebra d_4 conforms to
its place in the family F alongside the exceptional Lie algebras is described.Comment: 27 pages, LaTeX 2
Ueber die lineare Transformation der elliptischen Functionen.
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