10
votes
1answer
628 views

Why are they called Specht Modules?

I know that the simple modules of $\mathbb{C}S_n$ are called Specht Modules, and they are named after the German Mathematician Wilhelm Specht because he studied them, but I think these modules were ...
6
votes
1answer
391 views

Origin of the term “weight” in representation theory

In representation theory, there are the related concepts of weights and roots. Since both are kinds of generalised eigenvalues, and eigenvalues are roots of e.g. the characteristic polynomial, the ...
12
votes
3answers
1k views

History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask: Question: What was the motivation and historical context for ...
4
votes
1answer
356 views

Where does the name “Reynolds operator” come from?

I always found it strange that, in the context of invariant and representation theory, averaging over the group is called the "Reynolds operator". As far as I know the work of Reynolds was in fluid ...
1
vote
1answer
231 views

What is the name of the following theorem: dimension of complex irreducible representation divides order of group

Who proved it? When? See also: Irreducible Degrees and the Order of a Finite Group http://planetmath.org/proofthatdimensionofcomplexirreduciblerepresentationdividesorderofgroup Why would dim ...
8
votes
0answers
419 views

Reference/quote request: “All of combinatorics is the representation theory of $S_n$”

I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like "All of combinatorics is essentially [or can be reduced to?] the representation ...
6
votes
1answer
217 views

Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem?

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define ...
10
votes
1answer
486 views

Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`

The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently ...
2
votes
2answers
492 views

Why limit of discrete series representation?

In what sense is the limit of discrete series representation of $SL(2, \mathbb{R})$ a limit of discrete series representations? Where does the name origin from?
5
votes
1answer
450 views

Where does the definition of “Tower of Algebras” come from?

A tower of algebras is a sequence of algebras $$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$ with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...
17
votes
3answers
2k views

Why did Gabriel invent the term “quiver”?

A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he ...
23
votes
4answers
2k views

Overview of the interplay of Harmonic Analysis and Number Theory

I'm kind of disappointed that the question here was never sharpened. The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
21
votes
3answers
999 views

What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra?

This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras: 37602. Like some others who started graduate study in ...
15
votes
3answers
3k views

The only great book that Bourbaki ever wrote?

OK, the title is opinionated and contentious, but I have a definite question. I know that the title refers to the Bourbaki volume Groupes et Alg├Ębres de Lie (Chapters 4-6), published in 1968, but ...
28
votes
2answers
2k views

Why are parabolic subgroups called “parabolic subgroups”?

Over the years, I have heard two different proposed answers to this question. It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a ...