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My bashful, nameless, colleague asked me: When you identify opposite faces of a square, then depending on where you twist or not, you get a torus, Klein bottle, or projective plane. What spaces c …

Let $R$ be a commutative ring, with whatever hypotheses let you answer the question -- e.g. Noetherian, local, finitely generated over $\mathbb C$. Let $I$ be the ideal defining the singular locus in …

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 …

This paper showed two century-old classification results, each of very undergrad-comprehensible things, were the same; pretty amazing! http://arxiv.org/abs/1308.0751 "Sums of squares and varieties o …

The complete list is quite short -- at least, up to changing real form. I'll give the compact group version, up to isogeny. A_1 = B_1 = C_1: $SU(2) \cong Spin(3) \cong U(1,{\mathbb H})$ For the firs …

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be). Define the greedy or Demazure product of $R$ as follows: …

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 …

Let's pull back to irreps of $GL_1({\mathbb R}) \times SL_n({\mathbb R})$. By Schur's lemma, the first factor acts by scalars, so the representation is of the form $V \otimes W$ where $V$ is a chara …

The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen i …

If one iterates the multiplicity-free branching rule from $\operatorname{GL}(n)$ representations (finite-dim, over $\mathbb C$) to $\operatorname{GL}(n-1)$ all the way down to $\operatorname{GL}(0)$, …

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure o …

A partial permutation matrix $\pi$ is one with at most one 1 in any row and column (the rest 0s). Given one, we can cross out to the East and South (but not Southeast) of each 1. Some boxes get crosse …

If $G$ is compact, then as explained in comments $G/ad$ is $T/W$. If further $G$ is simply-connected, hence a product of simple factors, then $T/W$ is a corresponding product of simplices. The point …

Hello again. Yes, it's true. The more general statement you want is, let $X$ be projective with a $K$-equivariant ample line bundle ${\mathcal O}(1)$. For each $n$, let $\mu_n$ be a measure on ${\ma …

I won't fight hard about "important", but here's a theorem that was definitely combinatorial before it was geometric. Consider the basis of $K(G/P)$ consisting of $K$-classes of structure sheaves of …