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First, one can resolve the Schubert varieties using Bott-Samelson manifolds, and discover that any two resolutions give the same class upon pushforward. (This good situation ends with K-theory, i.e. i …

Let $A$ be the directed adjacency matrix of an acyclic directed graph, with variables as its nonzero entries (one for each edge). The $(a,b)$ entry of the matrix $(I-A)^{-1}$ is the sum over all paths …

I believe most of what you want is in https://arxiv.org/abs/math/0308101 , especially the polynomiality you're looking for. Note that that was first proven in [H. Derksen, J. Weyman] "On the Littlewoo …

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 …

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 …

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 …

The Schubert classes on $G/P$ are the classes of the Schubert varieties, which are the closures of the Schubert cells, each of which contains a unique $T$-fixed point. The $T$-fixed points on $G/P$ ar …

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 …

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries $$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\ -1 &\text{i …

I apologize for asking questions that seem likely to be answered in Conway & Sloane's "Sphere Packings, Lattices, and Groups" if I knew where to look. Let $L$ be the unique* even unimodular lattice o …

In general, $V_{k\rho} \cong \bigotimes\limits_{\beta\in \Delta_+} (\mathbb C_{k\beta/2} \oplus \mathbb C_{(k-2)\beta/2} \oplus \ldots \oplus \mathbb C_{-k\beta/2})$ as $T$-representations, provable v …

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 …

The proof I know is more representation-theoretic than combinatorial; the condition $\lambda \unrhd \mu$ is replaced by the type-independent $\mu \in hull(W\cdot \lambda) \cap (\lambda +$ root lattice …

Maschke tells you that $\mathbb C[G] \cong \bigoplus_V V^*\otimes V$, where $V$ runs over the irreps of a finite group $G$. Apply this to $G= S_n$ and take dimensions, and you get $n! = \sum_{\lambda …

Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its Ehrhart polynomial, which is also the Hilbert polynomial of the corresponding proje …