We introduce a family of "Rascal posets" $R(k,n-k)$ containing (n choose k) elements. The undirected Hasse diagram of $R(k,n-k)$ is isomorphic to Young's lattice on integer partitions in a rectangle, but the orientation is different. We show that $R(k,n-k)$ is a graded poset of height n-k containing $k^{n-k}$ maximal chains and we derive an explicit formula for rank-selected multichains. The poset $R(k,n-k)$ has two interesting geometric interpretations. First, its order complex is the type C alcove triangulation of a dilated simplex. Second, it is the poset of bounded regions in the cyclic hyperplane arrangement. This is related to work of Karp and Williams on the $m=1$ amplituhedron.

Volume polynomials are a class of Lorentzian polynomials coming from algebraic geometry. They measure the self intersection number of linear combinations of positive divisors in projective varieties. Their coefficients satisfy additional inequalities, and the support of a (realizable) volume polynomial is the set of bases of an algebraic polymatroid. We show that Aluffi's covolume polynomials are precisely the polynomial differential operators preserving volume polynomials. This provides us with a sufficiency criterion for linear operators preserving volume polynomials. We furthermore explore applications to algebraic matroids. This is joint work with Huh, Michałek, Süß and Wang.

We study the centralizer of a parabolic subalgebra in the Hecke algebra associated with an arbitrary (possibly infinite) Coxeter group. While the center and cocenter have been extensively studied in the finite and affine cases, much less is known in the indefinite setting. We describe a basis for the centralizer, generalizing known results about the center. Our approach combines algebraic techniques with geometric tools from the Davis complex, a CAT(0)-space associated to the Coxeter group. As part of the construction, we classify finite partial conjugacy classes in infinite Coxeter groups and define a variant of the class polynomial adapted to the centralizer.

The representation theory of quivers, well-understood over algebraically closed fields, presents deeper challenges over general fields K. The established approach in this setting, developed by Dlab and Ringel, utilizes the framework of K-species. Separately, a celebrated result by Gelfand connects the representation theory of Lie groups to quivers, establishing an equivalence between a block of Harish-Chandra modules for SL(2,R) and representations of the Gelfand quiver. This talk presents a new framework designed to unify and generalize these concepts in a rational setting. We introduce the notion of a K-rational structure on a quiver, which endows a quiver with a compatible action of a Galois group. We link this concept to a refinement of K-species, which we term étale K-species. There is a categorical anti-equivalence between K-rational quivers and étale K-species, which extends to an equivalence of their respective representation categories. This framework also provides a canonical notion of base change for these objects. As the primary application, we use this machinery to generalize Gelfand's equivalence to a Q-rational setting. We define a Q-rational structure on the Gelfand quiver and construct a functor from the category of Q-rational Harish-Chandra modules to the category of nilpotent Q-rational quiver representations. A key technical tool, which we call unipotent stabilization, is necessary to construct this functor and prove that it is an equivalence.

The representation theory of quivers, well-understood over algebraically closed fields, presents deeper challenges over general fields K. The established approach in this setting, developed by Dlab and Ringel, utilizes the framework of K-species. Separately, a celebrated result by Gelfand connects the representation theory of Lie groups to quivers, establishing an equivalence between a block of Harish-Chandra modules for SL(2,R) and representations of the Gelfand quiver. This talk presents a new framework designed to unify and generalize these concepts in a rational setting. We introduce the notion of a K-rational structure on a quiver, which endows a quiver with a compatible action of a Galois group. We link this concept to a refinement of K-species, which we term étale K-species. There is a categorical anti-equivalence between K-rational quivers and étale K-species, which extends to an equivalence of their respective representation categories. This framework also provides a canonical notion of base change for these objects. As the primary application, we use this machinery to generalize Gelfand's equivalence to a Q-rational setting. We define a Q-rational structure on the Gelfand quiver and construct a functor from the category of Q-rational Harish-Chandra modules to the category of nilpotent Q-rational quiver representations. A key technical tool, which we call unipotent stabilization, is necessary to construct this functor and prove that it is an equivalence.