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Certainly irreducible representations exist; one can still construct the monoid algebra of a monoid and consider modules over the algebra. But Maschke's theorem is false in general for finite monoids …

The exterior algebra of a vector space V seems to appear all over the place, such as in the definition of the cross product and determinant, the description of the Grassmannian as a variety, the des …

This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO. I've been reading Singer's Li …

This is a different take on Steven Landsburg's answer. The short version is that conjugacy classes and irreducible representations should be thought of as being dual to each other. Fix an algebraica …

Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names: No l …

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Hom{Hom}$The infinitude of the in …

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As …

Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers? Apologies in advance if this is obvious. Edit, 5/31/24: Since this question is getting some …

You can understand this philosophy as a generalization of Noether's theorem. Let me only state Noether's theorem in the quantum case because it's actually easier to understand there than in the classi …

I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free manner …

A simple bound on the largest dimension of a complex irreducible representation (which is either equal to or half of the largest dimension of a real irreducible representation) is the following: we kn …

The Dynkin diagrams An, Dn, E6, E7, E8 can be characterized among finite simple connected graphs by the property that their eigenvalues (that is, the eigenvalues of their adjacency matrices) all have …

The Casimir element is dual to the Killing form. (I think. I am somewhat uncertain about this because nobody has ever said this to me, even though it seems like the right thing to say, and frankly I …

More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex representat …

Here is a general comment. Let $G$ be a compact group and let $V$ be a (finite-dimensional, continuous, complex) representation of $G$. This data determines a locally finite directed graph, the repres …