Minuscule reverse plane partitions via quiver representations (1904.02754v1)

Abstract: The HiLLMan--Grassl correspondence is a well-known bijection between multisets of rim hooks of a partition shape $\lambda$ and reverse plane partitions of $\lambda$. We use the tools of quiver representations to generalize HiLLMan--Grassl in type $A$ and to define an analogue in all minuscule types. This is an extended abstract prepared for FPSAC 2019 based on arXiv:1812.08345, emphasizing its combinatorial aspects.

• The paper generalizes the Hillman–Grassl correspondence by formulating a uniform bijection for reverse plane partitions on minuscule posets via quiver representation theory.
• The paper employs quiver representations and the Auslander–Reiten translation to classify isomorphism classes, thereby linking combinatorial structures with algebraic methods.
• The paper introduces toggling operations that reveal periodic promotion dynamics, ensuring configuration invariance when applied a number of times equal to the Coxeter number.

An Analysis of "Minuscule Reverse Plane Partitions via Quiver Representations"

The paper "Minuscule Reverse Plane Partitions via Quiver Representations" by Alexander Garver, Rebecca Patrias, and Hugh Thomas explores the intersection of combinatorics and representation theory by expanding classical correspondences to a more general algebraic setting. The research primarily aims to generalize the HiLLMan--Grassl correspondence using quiver representations, thereby formulating analogous results in minuscule types.

The investigation commences by reviewing reverse plane partitions (RPPs) of a Young diagram, which are essential combinatorial structures with variables such as symmetric functions and representation theory. HiLLMan and Grassl initially connected these partitions with multisets of rim hooks through a transformative bijection. This correspondence has influenced subsequent studies, and the current work innovates by extending these results to minuscule posets via the theory of quiver representations.

Summary of Main Contributions

1. Generalization of the HiLLMan–Grassl Correspondence: Using the framework of quiver representations, the paper offers a comprehensive proof of the generating function for RPPs within minuscule posets. This step represents a critical advance from the established type $A$ scenarios (rectangular forms) to other simply-laced Dynkin types, as demonstrated through a uniform bijective method.
2. Quiver Representation Theory Application: Quivers, as directed graphs, provide the groundwork for the algebraic formalism employed. Particularly, the category of finite-dimensional representations of these quivers is leveraged to identify isomorphism classes corresponding to RPPs on minuscule posets. The approach is facilitated by the Austrlander–Reiten translation, harnessing the natural $\tau$-orbits in the representation category.
3. Implementation of Promotion and Toggling Operations: By incorporating a series of piecewise-linear maps known as toggles, the paper establishes a promotion dynamic indicating periodicity in minuscule RPPs. This mathematical operation ensures that when promotion is applied a number of times equivalent to the Coxeter number, the configuration is invariant—an assertion supported by the Coxeter group's algebraic properties.
4. Computational Examples in Type $A$: The research offers detailed computational examples, showcasing the novel bijection and explicating established correspondences such as HiLLMan--Grassl and Pak’s, within different quiver orientations. This not only confirms the theoretical underpinnings but also enhances the comprehension of the methods for practical types, demonstrating flexibility and informativity.

Implications and Future Work

The paper maneuvers intricate relationships between quiver theory and RPP combinatorics, which might initiate further inquiries into other algebraic structures and expanded ranges of poset types. The alignment of quiver representation categories with combinatorial configurations unlocks new avenues for mathematical exploration, possibly influencing computational techniques or novel algebraic proofs across broader mathematical contexts. The findings propose a potential template for examining other correspondence types and enhancing understanding within enumerative combinatorics and algebraic representation theory.

Through this formal and calculated approach, the research integrates distinct mathematical domains and paves the way for converting theoretical insights into algebraic and computational utilities. Further explorations could delve into non-simply-laced Dynkin types or investigate potential applications in adjacent fields such as geometric representation theory or statistical mechanics within mathematical physics.

In conclusion, the work accomplished in "Minuscule Reverse Plane Partitions via Quiver Representations" augments the foundational knowledge of reverse plane partitions and quivers, thus presenting a significant step forward in combinatorial representation theory.