Peter D. Jarvis

Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras

We present the commutation and anticommutation relations, satisfied by the generators of the graded general linear, special linear and orthosymplectic Lie algebras, in canonical two‐index matrix form. Tensor operators are constructed in the enveloping algebra, including powers of the matrix of generators. Traces of the latter are shown to yield a sequence of Casimir invariants. The transformation properties of vector operators under these algebras are also exhibited. The eigenvalues of the quadratic Casimir invariants are given for the irreducible representations of ggl(m ‖ n), gsl(m ‖ n), and osp(m ‖ n) in terms of the highest‐weight vector. In such representations, characteristic polynomial identities of order (m+n), satisfied by the matrix of generators, are obtained in factorized form. These are used in each case to determine the number of independent Casimir invariants of the trace form.

Local OSp(4/2) supersymmetry and extended BRS transformations for gravity

Abstract When the gravitational field and its fictitious partners are grouped into anOSp(4/2) supermultiplet, an extended BRS invariance emerges. The action and BRS transformations (ordinary and dual) are written in supersymmetric form, and the extended gauge identities are deduced, in parallel with recent work on Yang-Mills theory.

A supersymmetric Weinberg-Salam model

Abstract We formulate a unified Weinberg-Salam model in terms of the Yang-Mills theory of SU ( 2 1 ) over a Minkowski superspace. The gauge potentials comprise vector bosons and Higgs scalars. Fermions transform in a spinor representation of the space-time supersymmetry.

New branching rules induced by plethysm

We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T? of Young symmetry type ? where ? is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = ?fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s? ? {?} by the basic M series of complete symmetric functions and the L = M?1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains ?-generalized Newell?Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.

A Hopf laboratory for symmetric functions

An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bi-algebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focused on Laplace pairing, Sweedler cohomology for 1- and 2-cochains and twisted products (Rota cliffordizations) induced by branching operators in the symmetric function context. The latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type. A commentary on related issues in the combinatorial approach to quantum field theory is given.

Is the General Time-Reversible Model Bad for Molecular Phylogenetics?

The general time-reversible (GTR) model (Tavare 1986) has been the workhorse of molecular phylogenetics for the last decade. GTR sits at the top of the ModelTest hierarchy of models (Posada and Crandall 1998) and, usually with the addition of invariant sites and a gamma distribution of rates across sites, is currently by far the most commonly selected model for phylogenetic inference (see Table 1). However, a recent publication (Sumner et al. 2012) shows that GTR, along with several other commonly used models, has an undesirable mathematical property that may be a cause of concern for the thoughtful phylogeneticist. In mathematical terms, the problem is simple: matrix multiplication of two GTR substitution matrices does not return another GTR matrix. It is the purpose of this article to give examples that demonstrate why this lack of closure may pose a problem for phylogenetic analysis and thus add GTR to the growing list of factors that are known to cause model misspecification in phylogenetics.

The genetic code as a periodic table: algebraic aspects

The systematics of indices of physico-chemical properties of codons and amino acids across the genetic code are examined. Using a simple numerical labelling scheme for nucleic acid bases, A=(-1,0), C=(0,-1), G=(0,1), U=(1,0), data can be fitted as low order polynomials of the six coordinates in the 64-dimensional codon weight space. The work confirms and extends the recent studies by Siemion et al. (1995. BioSystems 36, 231-238) of the conformational parameters. Fundamental patterns in the data such as codon periodicities, and related harmonics and reflection symmetries, are here associated with the structure of the set of basis monomials chosen for fitting. Results are plotted using the Siemion one-step mutation ring scheme, and variants thereof. The connections between the present work, and recent studies of the genetic code structure using dynamical symmetry algebras, are pointed out.

Codon and nucleotide assignments in a supersymmetric model of the genetic code

Abstract The supersymmetric model we developed for the evolution of the genetic code is elaborated. Energy considerations in nucleic acid strand modelling, using sl(2) polarity spin and sl( 2 1 ) family box quartet symmetry, lead for the case of codons and anticodons to assignments of codons to 64-dimensional sl( 6 1 ) ⋍ A(5,0) multiplets.

Markov invariants, plethysms, and phylogenetics

We explore model-based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log-Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analyzing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.

Casimir invariants, characteristic identities, and Young diagrams for color algebras and superalgebras

The generalized commutation relations satisfied by generators of the general linear, special linear, and orthosymplectic color (super) algebras are presented in matrix form. Tensor operators, including Casimir invariants, are constructed in the enveloping algebra. For the general, special linear and orthosymplectic cases, eigenvalues of the quadratic and higher Casimir invariants are given in terms of the highest‐weight vector. Correspondingly, characteristic polynomial identities, satisfied by the matrix of generators, are obtained in factorized form. Classes of finite‐dimensional representations are identified using Young diagram techniques, and dimension, branching, and product rules for these are given. Finally, the connection between color (super) algebras and generalized particle statistics is elucidated.