Weight Diagrams Research Articles

On simple lie groups of rank 3

Properties of simple Lie algebras of rank three, are investigated in view of physical applications; more precisely, dimensions, weight diagrams, decompositions with respect to regular subalgebras and decomposition of products of representations are given for the lowest order irreducible representations; we also explain somme techniques to deal with these problems even in the case of higher rank algebras.

Weak interaction current and the unitary group

A unified description is presented of the decay modes of mesons having DELTA I = 1/2, 3/2 or DELTA S = plus or minus DELTA Q rules. The weight diagram of D/sup 10/(3,0) in SU(3) is given. (R.E.U.)

A discrete subgroup for higher-symmetry models

It is shown that in a higher-symmetry model based on a compact semi-simple Lie group, the strong interactions are invariant under a discrete subgroup. The associated symmetry transformations are related to the transformations induced on the weight diagrams by elements of the Weyl group. Such symmetries lead to simple predictions of certain relationships among the electromagnetic form factors of the hadrons.

Weyl reflections in the unitary symmetry theory of strong interactions

Recent work has clearly demonstrated the fact that useful predictions in theNe’eman, Gell-Mann unitary symmetry theory of strong interactions follow from consideration of invariance under the Weyl reflections (generalized charge symmetry operations) ofSU3. Here we describe a fairly rapid and general algebraic method for obtaining the effect of the Weyl reflections on the basis vectors of an arbitrary irreducible representation (IR) ofSU3. The important feature of the method is that it applies to those basis vectors of the IR, which belong to the nonsimple weights of the IR and which can therefore not be treated by inspection of the weight diagram of the IR. Results are given for certain IR’s ofSU3 relevant to the Ne’eman-Gell-Mann theory — the 8, 27 and 35 component IRs (1.1) (2.2) and (4.1) of SU3.

Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley–Lieb algebra with infinitely many generators and defining parameter 0 on the category $\mathcal{F}_n$ by certain translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m \vert n)$. We discover two natural highest weight structures. Using the Temperley–Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of irreducibles in standard and costandard modules and classify the blocks of $\mathcal{F}_n$. We also prove the surprising fact that indecomposable projective modules in this category are multiplicity-free.