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There are several rings-with-bases to get straight here. I'll explain that, then describe three serious connections (not just Ehresmann's Lesieur's proof as recounted in the OP). The wrong one is $Rep …

In Hall's Marriage Theorem, we have a set $B$ of brides and $G$ of grooms, where each bride $b$ has an acceptable set $A_b \subseteq G$ of grooms. A matching $m:B\to G$ is an injection such that $m(b) …

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 …

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\la …

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 …

I think it's because we have well-developed techniques with which to prove that this condition holds, and when those fail, people don't put that much effort into trying to describe the (more difficult …

Motivated by algebraic geometry, I've come up with a purely combinatorial definition within the theory of matroids. The question is: is this concept known? If you like matroids but not algebraic geo …

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: …

No locking, even if you restrict to translations. Scale the whole arrangement up by a factor of $c$, then scale each polyhedron down by $c$ around its center of mass. Neither step introduces collision …

Naively, there can be no reasonable way of distinguishing the red nodes from green in the case $A_{even}$, as the Dynkin diagram automorphism switches them. Less naively, there is indeed a way of dis …

Let S := the nonnegative integer solutions to {$a_1 + ... + a_n = n$}, and center := (1,1,1,...,1). Call a vector v generic if v.s = v.center <=> s = center. Then each generic v defines a positive sys …

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 …

There are conjectural ones in the Berenstein-Zelevinsky paper referenced in that one. They have another paper with a general theorem, Tensor product multiplicities, canonical bases and totally positiv …

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 …

This is a rather specialized example, but dear to my heart. Consider the set of "Richardson subvarieties" of the flag manifold $GL_n/B$, intersections of Schubert and opposite Schubert varieties. The …