arXiv subjects

Evuilynn Nguyen

Publications and source records attributed to Evuilynn Nguyen.

3 recordsLinked to original sources

arXiv2026

Uncrowding the 5-Vertex Model: RSK and Crystal Structures

While the uncrowding algorithm on set-valued tableaux has long been instrumental in proving the Schur positivity of stable symmetric Grothendieck polynomials, lattice models have emerged as a modern framework for investigating symmetric functions, in particular symmetric Grothendieck polynomials. In this work, we synthesize these combinatorial and lattice-theoretic approaches by defining both the Robinson--Schensted--Knuth (RSK) correspondence and the uncrowding operation directly on a 5-vertex model of Motegi and Sakai and its subsequent reinterpretation by Buciumas, Scrimshaw, and Weber. Our lattice-based RSK formulation yields a powerful new result: the direct construction of the associated crystal structure on the states of the 5-vertex model.

math.CO↗

arXiv2024

The immersion poset on partitions

We introduce the immersion poset $(\mathcal{P}(n), \leqslant_I)$ on partitions, defined by $\lambda \leqslant_I \mu$ if and only if $s_\mu(x_1, \ldots, x_N) - s_\lambda(x_1, \ldots, x_N)$ is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of $GL_N(\mathbb{C})$ form an immersion pair, as defined by Prasad and Raghunathan (2022). We develop injections $\mathsf{SSYT}(\lambda, \nu) \hookrightarrow \mathsf{SSYT}(\mu, \nu)$ on semistandard Young tableaux given constraints on the shape of $\lambda$, and present results on immersion relations among hook and two column partitions. The standard immersion poset $(\mathcal{P}(n), \leqslant_{std})$ is a refinement of the immersion poset, defined by $\lambda \leqslant_{std} \mu$ if and only if $\lambda \leqslant_D \mu$ in dominance order and $f^\lambda \leqslant f^\mu$, where $f^\nu$ is the number of standard Young tableaux of shape $\nu$. We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions $p_{A_\mu}$ on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram (2018).

math.CO↗

arXiv2020

The kernel of newform Dedekind sums

Newform Dedekind sums are a class of crossed homomorphisms that arise from newform Eisenstein series. We initiate a study of the kernel of these newform Dedekind sums. Our results can be loosely described as showing that these kernels are neither "too big" nor "too small." We conclude with an observation about the Galois action on Dedekind sums that allows for significant computational efficiency in the numerical calculation of Dedekind sums.

math.NT↗