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Whenever you see people using Young tableaux to discuss representations of $GL_n$ they include an apologetic "we'll only consider polynomial representations, i.e. not $det^{-1}$". That is to say, they …

The complex irreps of a finite group come in three types: self-dual by a symmetric form, self-dual by a symplectic form, and not self-dual at all. In the first two cases, the character is real-valued, …

Yes. In combinatorics this is known as Robinson-Schensted-Knuth vs. just Robinson-Schensted. (Properly speaking the latter is about a yet smaller duality, $\mathbb C[S_n] = \bigoplus_{\lambda\vdash n} …

What I'm writing here seems more like a contribution to a big-list than an "answer", but since you've already chosen one anyway... Say you're interested in which irreps $V_\nu$ occur in $V_\lambda \ …

Note that your statement is only true in rational cohomology. For example, $H^\ast(SO(5)/T)$ is not generated in degree $2$ (though it is rationally). The easiest proof I know starts from equivariant …

2-color your Dynkin diagram, black and white. Let $w$ be the product of the white simple reflections, $b$ the product of the black. Note that $w$ and $b$ are well-defined, as the reflections you're mu …

I would say there are three basic reasons for / proofs of positivity. Geometry. [Kleiman 1973] proves that the number one's trying to compute is the number of points in a transverse intersection of …

What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P \hookrightarrow \mathbb P(V_\omega)$ where $V …

While I appreciate Dave Ben-Zvi's half-densities answer, I'm going to put forth the contrary opinion that it's largely a bookkeeping artifact. The most familiar place that $\rho$ shows up is in the W …

Let $\lambda$ be a dominant integral weight of $\mathfrak g$, a finite-dimensional reductive Lie algebra over $\mathbb C$. Let $M(w\cdot \lambda)$ denote the Verma module with high weight $w\cdot \lam …

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra over $\mathbb C$. Theorem 1. The highest root is perpendicular to all but one simple root, except in the case ${\mathfrak g}={\mathfrak sl …

There are a number of better statements (i.e. yes, this is "well-known"). To begin with, wouldn't you rather have the character of this representation, instead of just its dimension? You can get that …

If $\lambda,\mu,\nu$ are considered as vectors in $\mathbb Z^n$, then the set of triples forms a convex cone, given by a finite (for each $n$) list of inequalities. (The corresponding statement does n …

This is only about the "big question". The symmetric $n\times n$ matrices appear naturally as the big cell on the Lagrangian Grassmannian. Any projectively normal embedding (e.g. first include into t …

I often find it more useful to say $SU(3)$ is a $T^2$ bundle over the manifold of flags in ${\mathbb C}^3$ (itself a ${\mathbb CP}^1$-bundle over ${\mathbb CP}^2$). Partly this is because $T^2$'s homo …