Representation theory

Rep­re­sen­ta­tion the­ory (which physi­cists and chemists some­times just call Group the­ory) is the study of how groups \(G\) act on vec­tor spaces \(V\). A cen­tral idea in math­e­mat­ics is that, be­cause lin­ear alge­bra is so easy (com­pared to other parts of math), we should always try to ex­tract lin­ear alge­bra from any situ­a­tion we can, and rep­re­sen­ta­tion the­ory is a stan­dard tool to ap­ply when ex­tract­ing lin­ear alge­bra from a situ­a­tion in­volv­ing a Group ac­tion.

In physics, vec­tor spaces ap­pear nat­u­rally as Hilbert spaces of quan­tum states as­so­ci­ated to phys­i­cal sys­tems, and rep­re­sen­ta­tions of groups on these Hilbert spaces ap­pear nat­u­rally as sym­me­tries of the sys­tem. The study of these rep­re­sen­ta­tions is a fun­da­men­tal or­ga­niz­ing prin­ci­ple of mod­ern physics; for ex­am­ple, it is be­hind Wigner’s clas­sifi­ca­tion of par­ti­cle types.