The irreducible representations of \(\mathfrak{sl}_3\) are given by a pair of non-negative integers \((m,n)\), called the Dynkin label of the highest weight vector which labels the irreducible representation. The webapp generates the weight system for a particular label, which is a finite set of vectors. The number gives the multiplicity of each vector.
The convex hull of each weight diagram is a hexagon. For the irreducible representations where one of the Dynkin labels is zero (which is not accessible in this app), one obtains a triangle which can be viewed as a degenerate hexagon.
The adjoint representation of \(\mathfrak{sl}_3\) is irreducible and labelled by \((1,1)\).
They are interesting in particle theory as quarks and anti-quarks are modelled as associated vector fields of an \(\mathrm{SU}(3)\) principal bundle, the \(\mathrm{SU}(3)\) being part of the Standard Model gauge group. Specifically quarks are an associated vector field in the \((1,0)\) representation, and antiquarks in the \((0,1)\).
Some hadrons are composed of three quarks. The way three quarks can be combined is modelled by the tensor product of three copies of the \((1,0)\) representation. This has a Clebsch–Gordan decomposition into a direct sum of representations: \[(1,0)^{\otimes 3} = (3,0) \oplus (1,1) \oplus (1,1) \oplus (0,0),\] of respective dimensions \(10, 8, 8, 1\). Many known hadrons fit together to form the so-called decuplet and octets of hadrons.