A practical construction algorithm for the projection and transition operators onto the irreducible representations of compact Lie groups on mixed tensor product spaces

01.10.2019 bis 30.09.2021

The first and immediate goal of this project is to find a computationally efficient construction algorithm for Hermitian projection operators corresponding to the irreducible representations of the special unitary group SU(N) on a mixed tensor product space containing both factors of V (carrying
the fundamental representation of SU(N)) as well as its dual space V*. Concurrently, we wish to find a general construction algorithm for the transition operators between equivalent irreducible projectors. We desire the operators in question to be written as partially contracted products of
symmetrizers and antisymmetrizers in analogy to the standard Young operators projecting onto the irreducible representations of SU(N) on tensor product spaces containing factors of V only. In physics parlance, a tensor product space containing V and V* represents the Fock space
component of quarks and antiquarks, and the multiplets onto the irreducible representations on such spaces furnish calculations in particle physics, in particular quantum chromodynamics. To further the representation theory of compact Lie groups as a whole, we also wish to study the irreducible multiplets of other groups: In analogy to what has been done for the irreducible representations of SU(N) on mixed tensor product spaces, we aim to find a general, compact construction algorithm for the irreducible multiplets of the special orthogonal group SO(N). In fact, what we learned from the case of SU(N) promises to be extendable to the irreducible
representations of SO(N) since the commonalities shared by these two groups is particularly apparent in the birdtrack formalism (this is explained in more detail in the attached research proposal). Hence, SO(N) presents a natural continuation of the study conducted for SU(N). Lastly, the birdtrack formalism promises advances in finding a simpler formula for the partition
function p(n), which counts the number of integer partitions of a natural number n, than is currently known. Using the Robinson-Schensted correspondence between permutations and Young tableaux, we aim to exploit topological properties of the birdtracks to advance research in this area.