arXiv subjects

Shiyun Wang

Publications and source records attributed to Shiyun Wang.

9 recordsLinked to original sources

arXiv2026

Varieties of chain complexes and mixed dimer covers

A quiver representation consists of a collection of vector spaces along with a set of arrows, which are linear maps between these spaces. In this work, we study quiver representations in equioriented type $A$ which are also chain complexes; that is, in which consecutive arrows compose to zero. We show that orbits of these representations under a change of basis action are in bijection with mixed dimer covers of a $2 \times n$ grid graph. The latter object can be endowed with a partial order which is a distributive lattice, and we show that the degeneration order on the orbits of chain complexes is a coarsening of this partial order. In addition, we use recent matrix formulae of Claussen and Ovenhouse to enumerate these orbits. This also computes the Kostant partition function applied to height-restricted, type $A$ roots. When the dimension vector is uniform, we discuss a correspondence with paths of a beam of light bouncing between glass plates and give an explicit generating function.

math.CO↗

arXiv2025

Kirillov-Reshetikhin Dual Equivalence Graphs

Let $U$ be a tensor product of highest weight modules of $GL_n(\mathbb C)$ corresponding to multiples of fundamental weights (i.e. rectangles). We consider three ways to stratify $U^{\otimes k}$ into components: using isotypic components of the cyclic action on tensor factors, using a generalization of the charge statistic, and using certain generalizations of Assaf's dual equivalence graphs. We conjecture that all three ways coincide, and we prove that the latter two ways coincide. The Kirillov-Reshetikhin dual equivalence graphs (KR DEGs) we introduce for this purpose are defined on $0$-weight spaces of tensor products of Kirillov-Reshetikhin crystals. They generalize Kazhdan-Lusztig dual equivalence graphs (KL DEGs) that previously appeared in the study of Kazhdan-Lusztig cells in affine type A. While the tensor products of Kirillov-Reshetikhin crystals are connected as affine crystals, the KR DEGs in general are not.

math.CO↗

arXiv2024

The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order

For integral weights $\lambda$ and $\mu$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $\mu$ in the irreducible representation with highest weight $\lambda$, which we denote by $m(\lambda,\mu)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(\lambda,\mu)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(\lambda,\mu)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tilde{\alpha},\mu)$, where $\tilde{\alpha}$ is the highest root and $\mu$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tilde{\alpha},\mu)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $\mu=-(\alpha_i+\alpha_{i+1}+\cdots+\alpha_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$.

math.RT↗

arXiv2024

Collaborative Participatory Research with LLM Agents in South Asia: An Empirically-Grounded Methodological Initiative and Agenda from Field Evidence in Sri Lanka

The integration of artificial intelligence into development research methodologies presents unprecedented opportunities for addressing persistent challenges in participatory research, particularly in linguistically diverse regions like South Asia. Drawing from an empirical implementation in Sri Lanka's Sinhala-speaking communities, this paper presents an empirically grounded methodological framework designed to transform participatory development research, situated in the challenging multilingual context of Sri Lanka's flood-prone Nilwala River Basin. Moving beyond conventional translation and data collection tools, this framework deploys a multi-agent system architecture that redefines how data collection, analysis, and community engagement are conducted in linguistically and culturally diverse research settings. This structured agent-based approach enables participatory research that is both scalable and responsive, ensuring that community perspectives remain integral to research outcomes. Field experiences reveal the immense potential of LLM-based systems in addressing long-standing issues in development research across resource-limited regions, offering both quantitative efficiencies and qualitative improvements in inclusivity. At a broader methodological level, this research agenda advocates for AI-driven participatory research tools that maintain ethical considerations, cultural respect, and operational efficiency, highlighting strategic pathways for deploying AI systems that reinforce community agency and equitable knowledge generation, potentially informing broader research agendas across the Global South.

cs.CY↗

arXiv2024

The Stanley-Stembridge Conjecture for $\bf 2 + 1 +1$-avoiding unit interval orders: a diagrammatic proof

A natural unit interval order is a naturally labelled partially ordered set that avoids patterns ${\bf 3} + {\bf 1}$ and $\bf 2 + 2$. To each natural unit interval order one can associate a symmetric function. The Stanley-Stembridge conjecture states that each such symmetric function is positive in the basis of complete homogenous symmetric functions. This conjecture has connections to cohomology rings of Hessenberg varieties, and to Kazhdan-Lusztig theory. We use a diagrammatic technique to re-prove the special case of the conjecture for unit interval orders additionally avoiding pattern $\bf 2 + 1 + 1$. Originally this special case is due to Gebhard and Sagan.

math.CO↗

arXiv2023

The $e$-positivity of the chromatic symmetric functions and the inverse Kostka matrix

We expand the chromatic symmetric functions for Dyck paths of bounce number three in the elementary symmetric function basis using a combinatorial interpretation of the inverse of the Kostka matrix studied in Eğecioğlu-Remmel (1990). We prove that certain coefficients in this expansion are positive. We establish the $e$-positivity of an extended class of chromatic symmetric functions for Dyck paths of bounce number three beyond the "hook-shape" case of Cho-Huh (2019).

math.CO↗

arXiv2022

Pieri rules for skew dual immaculate functions

In this paper we give Pieri rules for skew dual immaculate functions and their recently discovered row-strict counterparts. We establish our rules using a right-action analogue of the skew Littlewood-Richardson rule for Hopf algebras of Lam-Lauve-Sottile. We also obtain Pieri rules for row-strict (dual) immaculate functions.

math.CO↗

arXiv2022

Row-strict dual immaculate functions

We define a new basis of quasisymmetric functions, the row-strict dual immaculate functions, as the generating function of a particular set of tableaux. We establish that this definition gives a function that can also be obtained by applying the $\psi$ involution to the dual immaculate functions of Berg, Bergeron, Saliola, Serrano, and Zabrocki (2014) and establish numerous combinatorial properties for our functions. We give an equivalent formulation of our functions via Bernstein-like operators, in a similar fashion to Berg et. al (2014). We conclude the paper by defining skew dual immaculate functions and hook dual immaculate functions and establishing combinatorial properties for them.

math.CO↗

arXiv2022

0-Hecke modules for row-strict dual immaculate functions

We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution $\psi$ on the ring of quasisymmetric functions. We give an explicit description of the effect of $\psi$ on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing that \emph{all} the possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets.

math.CO↗