Representing Symmetries

[Rapping] [Verse 1] In the world of math, we break it down to a base, Abstract algebra, we need to see its face, Linear transformations, making them concrete, Matrices and modules where theories meet. [Rapping] [Verse 2] Vectors and fields, the space we need to choose, Real or complex, n-dimensional views, Groups, algebras, Lie, we analyze their state, Invertible matrices, the forms they create. [Singing] [Chorus] Representation theory, make it clear, Abstract objects, bring them near, Matrices and vectors, feel the beat, Concrete algebra, can't be beat. [Rapping] [Verse 3] Mappings and actions, how they interact, Group or algebra, the way they act, Homomorphisms, linear and clean, Group elements to maps, in this scene. [Singing] [Chorus] Representation theory, make it clear, Abstract objects, bring them near, Matrices and vectors, feel the beat, Concrete algebra, can't be beat. [Rapping] [Verse 4] Irreducible reps, subspaces in play, If stable under action, subrepresentation we say, Trivial to full, no in-between scene, It's reducible if there's more to glean. [Rapping] [Verse 5] Schur's lemma tells, in a crisp clear line, Equivariant maps are scalar, by design, Zero or isomorphisms, no middle ground, In a closed field, this truth is found. [Rapping] [Verse 6] Characters and weights, they play their role, Symmetries and structures, making math whole, Theorems and lemmas, in reps they shine, Guiding the way through algebraic design. [Singing] [Chorus] Representation theory, make it clear, Abstract objects, bring them near, Matrices and vectors, feel the beat, Concrete algebra, can't be beat. [Singing] [Chorus] Representation theory, make it clear, Abstract objects, bring them near, Matrices and vectors, feel the beat, Concrete algebra, can't be beat.