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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 …

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 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 …

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 …

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: …

Instead of thinking of the double Schubert polynomial $S_u$ as representing $[\overline{B_- uB}/B] \in H^*_T(GL_n/B)$, equivalently think of it as representing $[\overline{B_- uB}] \in H^*_{T\times B} …

I assume you mean that $T$ acts preserving each fiber. Then the flatness says that the multigraded Hilbert polynomial is constant. As the Duistermaat-Heckman measure is the leading-order behavior of t …

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 …

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 …