Molien Function Research Articles

The action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of Molien functions

The present article completes the mathematical description initiated in the paper by Dhont and Zhilinskií (2013 The action of the orthogonal group on planar vectors: invariants, covariants and syzygies J. Phys. A: Math. Theor. 46 455202) of the algebraic structures that emerge from the symmetry-adapted polynomials in the coordinates of n planar vectors under the action of the SO(2) group. The set of -covariant polynomials contains all the polynomials that transform according to the weight of SO(2) and is a free module for but a non-free module for . The sum of the rational functions of the Molien function for -covariants describes the decomposition of the ring of invariants or the module of -covariants as a direct sum of submodules. A method for extracting the generating function for -covariants from the comprehensive generating function for all polynomials is introduced. The approach allows the direct construction of the integrity basis for the module of -covariants decomposed as a direct sum of submodules and gives insight into the expressions for the Molien functions found in our earlier paper. In particular, a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed, where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed.

The action of the orthogonal group on planar vectors: invariants, covariants and syzygies

The construction of invariant and covariant polynomials from the x, y components of n planar vectors under the SO(2) and O(2) orthogonal groups is addressed. Molien functions determined under the SO(2) symmetry group are used as a guide to propose integrity bases for the algebra of invariants and the modules of covariants. The Molien functions that describe the structure of the algebra of invariants and the free modules of (m)-covariants, m ⩽ n − 1, are written as a ratio of a numerator in λ with positive coefficients over a (1 − λ2)2n − 1 denominator. This form of single rational function is standard in invariant theory and has a clear symbolic interpretation. However, its usefulness is lost for the non-free modules of (m)-covariants, m ⩾ n, due to negative coefficients in the numerator. We propose for these non-free modules a new representation of the Molien function as a sum of n rational functions with positive coefficients in the numerators and different numbers of terms in the denominators. This non-standard form is symbolically interpreted in terms of a generalized integrity basis. Integrity bases are explicitly given for n = 2, 3, 4 planar vectors and m ranging from 0 to 5. The integrity bases obtained under the SO(2) symmetry group are subsequently extended to the O(2) group.

Explicit and spontaneous breaking of SU(3) into its finite subgroups

We investigate the breaking of SU(3) into its subgroups from the viewpoints of explicit and spontaneous breaking. A one-to-one link between these two approaches is given by the complex spherical harmonics, which form a complete set of SU(3)-representation functions. An invariant of degrees p and q in complex conjugate variables corresponds to a singlet, or vacuum expectation value, in a (p,q)-representation of SU(3). We review the formalism of the Molien function, which contains information on primary and secondary invariants. Generalizations of the Molien function to the tensor generating functions are discussed. The latter allows all branching rules to be deduced. We have computed all primary and secondary invariants for all proper finite subgroups of order smaller than 512, for the entire series of groups \Delta(3n^2), \Delta(6n^2), and for all crystallographic groups. Examples of sufficient conditions for breaking into a subgroup are worked out for the entire T_{n[a]}-, \Delta(3n^2)-, \Delta(6n^2)-series and for all crystallographic groups \Sigma(X). The corresponding invariants provide an alternative definition of these groups. A Mathematica package, SUtree, is provided which allows the extraction of the invariants, Molien and generating functions, syzygies, VEVs, branching rules, character tables, matrix (p,q)_{SU(3)}-representations, Kronecker products, etc. for the groups discussed above.

SU(6) Casimir invariants and SU(2) ⊗ SU(3) scalars for a mixed qubit-qutrit state

In the present paper, a few steps are undertaken towards the description of the “qubit–qutrit” pair – a quantum bipartite system composed of two- and three-level subsystems. Calculations of the Molien functions and Poincare series for the qubit-qubit and qubit-qutrit “local unitary invariants” are outlined and compared with the known results. The requirement of the positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the enveloping algebra $ \mathfrak{su}$ (6). Bibliography: 26 titles.

On the Classification of Automorphic Lie Algebras

It is shown that the problem of reduction can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of these algebras, beyond the context of integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group Dn are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of sl2-Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.

Invariants, equivariants and characters in symmetric bifurcation theory

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions’. We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group O(3).

Qubits and invariant theory

The invariants of a mixed two-qubit system are discussed. These are polynomials in the elements of the corresponding density matrix. They are counted by means of grouptheoretic branching rules and the Molien function is determined. The fundamental invariants are then explicitly constructed and the relations between them are found in the form of syzygies. In this way, complete sets of primary and secondary invariants are identified: there are 10 of the former and 15 of the latter.

Molien function for duality

The Molien function counts the number of independent group invariants of a representation. For chiral superfields, it is invariant under duality by construction. We illustrate how it calculates the spectrum of supersymmetric gauge theories.

Molien Functions and Commensurability

Molien Functions and Commensurability

Invariant polynomials and Molien functions

The Molien function associated with a finite-dimensional representation of a compact Lie group is a useful tool in representation theory because it acts as a generating function for counting the number of invariant polynomials on the representation space. The main purpose of this paper is to introduce a more general (and apparently new) generating function which, in the same sense, counts not only the number of invariant real polynomials in a real representation or the number of invariant complex polynomials in a complex representation, as a function of their degree, but encodes the number of invariant real polynomials in a complex representation, as a function of their bidegree (the first and second component of this bidegree being the number of variables in which the polynomial is holomorphic and antiholomorphic, respectively). This is obviously an additional and non-trivial piece of information for representations which are truly complex (i.e., not self-conjugate) or are pseudo-real, but it provides additional insight even for real representations. In addition, we collect a number of general formulas for these functions and for their coefficients and calculate them in various irreducible representations of various classical groups, using the software package MAPLE.

Symmetry Analysis of the Vibrational Dynamics of the H3D2+and H2D3+Complexes

A symmetry analysis of the H3D2+and H2D3+complexes in a model with one large amplitude motion, the propeller-like internal rotation, is presented. Symmetry coordinates and symmetry adapted polynomial expansions of the potential, dipole moment, and polarizability functions are derived within the framework of the extended molecular symmetry groupG3(2, 2) using the projection operator and Molien function techniques.

Molien functions and commensurability of the helicoidal ordering

The Molien functions of the line groups for the representations with different k in the Brillouin zone (including rational and irrational ones) are calculated. It is illustrated how the resulting invariants enable a complete symmetry treatment of the commensurate and the incommensurate phase transitions.

SYMMETRY THEORY OF THE MAGNETIC PHASE TRANSITION OF FeRh ALLOY

The formula of the Molien function in the case of corepresentation is derived by application of the symmetry theory for second order phase transition to magnetic systems, it is different from that of unitary groups. The analysis of the magnetic phase transition of FeRh alloy shows that eqivalent representation in the sense of corepresentation may lead to different symmetry breaking for the reason of none-quivalent characters. The result that a continuous magnetic phase transition Oh1⊕θOh1→C4v1⊕θ[D4h1-C4v1] exists theoretically is supported by experiments.

A generalized Molien function for field theoretical Hamiltonians

A generating function, or Molien function, the coefficients of which give the number of independent polynomial invariants in G, has been useful in the Landau and renormalization group theories of phase transitions. Here a generalized Molien function for a field theoretical Hamiltonian (with short-range interactions) of the most general form invariant in a group G is derived. This form is useful for more general renormalization group calculations. Its Taylor series is calculated to low order for the FΓ−2 representation of the space group R3̄c and also for the l=1 (faithful) representation of SO(3).

Calculation of the Molien generating function for invariants of space groups

The new algorithms for computing the Molien generating function for a representation of a finite group obtained in the preceding paper are applied to obtain an expression which can be used for irreducible representation (*kn) of any crystallographic space group G. It proves convenient to express M (Γ,G;z) as a sum: M (Γ,G;z) =1/‖P‖Σkckm̄ (Γ,gk;z), where the partial Molien function m̄ (Γ;gk;z) is labelled by a coset representative gk carrying the class index k of the point group P=G/T, T being the translation group, the sum is over classes k, and ck is the order of class Ck(P). The resulting form was used to compute M (Γ,G;z) for irreducible representations *Γn, *Xn, *Rn of nonsymmorphic space group A-15 or O3h-Pm3n in which many high temperature superconducting crystals occur. Certain of these representations (matrix groups) are identified as generalized Coxeter groups, i.e., unitary groups generated by reflections. The Molien function for these groups as the required form given by Shephard and Todd: M (Γ,G;z) =[Πi(1−zdi)]−1. The di satisfy dimensionality theorems.

Erratum: Calculation of the Molien generating function for invariants of space groups

The new algorithms for computing the Molien generating function for a representation of a finite group obtained in the preceding paper are applied to obtain an expression which can be used for irreducible representation (*kn) of any crystallographic space group G. It proves convenient to express M (Γ,G;z) as a sum: M (Γ,G;z) =1/‖P‖Σkckm (Γ,gk;z), where the partial Molien function m (Γ;gk;z) is labelled by a coset representative gk carrying the class index k of the point group P=G/T, T being the translation group, the sum is over classes k, and ck is the order of class Ck(P). The resulting form was used to compute M (Γ,G;z) for irreducible representations *Γn, *Xn, *Rn of nonsymmorphic space group A‐15 or O3h‐Pm3n in which many high temperature superconducting crystals occur. Certain of these representations (matrix groups) are identified as generalized Coxeter groups, i.e., unitary groups generated by reflections. The Molien function for these groups as the required form given by Shephard and Todd: M (Γ...

New algorithms for the Molien function

Two new forms are given for the Molien (generating) function for multiplicity cn1 of the identity representation in the symmetrized nth power of representation Γ of finite group G. These are M (Γ,G;z) =Σncn1zn=1/‖G‖Σg exp [Σlzlχ (gl)/l] =1/‖G‖ΣgΠj[1−zγj(g)]−δ j, where g is an element in G γj(g) an irreducible character in the Abelian subgroup A generated by g, δj the subduction coefficient of Γ of G↓γj of A; we usually take Γ irreducible. These forms have the merit of only requiring characters. In the following paper these algorithms are used to compute the Molien function for space group irreducible representation.