arXiv subjects

Swapnil Garg

Publications and source records attributed to Swapnil Garg.

4 recordsLinked to original sources

arXiv2026

GIT for root stacks and 3d mirror symmetry

Using window theory, Bodzenta--Donovan showed that the derived category of a root stack $\sqrt[n]{X/D}$ has a $2n$-periodic $2$-term semiorthogonal decomposition. We define a categorical generalization of the root stack construction and interpret this as a pullback on the B-side of the 3d mirror symmetry equivalence of Gammage--Hilburn--Mazel-Gee. We analyze this pullback using categorical representation theoretic results of Ben-Zvi--Francis--Nadler and Ben-Zvi--Nadler--Preygel applied to SODs. We also show that the pullback is equivalent to a pushforward of perverse schobers on the A-side, allowing us to deduce periodicity from a simple decomposition of an A-side Lagrangian skeleton. Additionally, we adapt the construction of Bodzenta--Donovan to a new GIT problem, which yields an embedding of Coh($\sqrt[m]{X/D}$) into Coh($\sqrt[n]{X/D}$) for $m<n$ coprime, and prove $2n$-periodicity of the resulting $2$-term SOD.

math.RT↗

arXiv2022

New Results on Nyldon Words and Nyldon-like Sets

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with "nondecreasing" replaced by "nonincreasing." Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also show that the Nyldon words form a right Lazard set, and equivalently, a right Hall set. In this paper, we provide a new proof of unique factorization into Nyldon words related to Hall set theory and resolve several questions of Charlier et al. In particular, we prove that Nyldon words of a fixed length form a circular code, we prove a result on factorizing powers of words into Nyldon words, and we investigate the Lazard procedure for generating Nyldon words. We show that these results generalize to a new class of Hall sets, of which Nyldon words are an example, that we name "Nyldon-like sets."

math.CO↗

arXiv2022

Classical and consecutive pattern avoidance in rooted forests

Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern $σ\in\mathcal{S}_k$ if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as $σ$. For each permutation $σ\in\mathcal{S}_{k-2}$, we construct a bijection between $n$-vertex forests avoiding $(σ)(k-1)k:=σ(1)\cdotsσ(k-2)(k-1)k$ and $n$-vertex forests avoiding $(σ)k(k-1):=σ(1)\cdotsσ(k-2)k(k-1)$, giving a common generalization of results of West on permutations and Anders--Archer on forests. We further define a new object, the forest-Young diagram, which we use to extend the notion of shape-Wilf equivalence to forests. In particular, this allows us to generalize the above result to a bijection between forests avoiding $\{(σ_1)k(k-1), (σ_2)k(k-1), \dots, (σ_\ell)k(k-1)\}$ and forests avoiding $\{(σ_1)(k-1)k, (σ_2)(k-1)k, \dots, (σ_\ell)(k-1)k\}$ for $σ_1, \dots, σ_\ell \in \mathcal{S}_{k-2}$. Furthermore, we give recurrences enumerating the forests avoiding $\{123\cdots k\}$, $\{213\}$, and other sets of patterns. Finally, we extend the Goulden--Jackson cluster method to study consecutive pattern avoidance in rooted trees as defined by Anders and Archer. Using the generalized cluster method, we prove that if two length-$k$ patterns are strong-c-forest-Wilf equivalent, then up to complementation, the two patterns must start with the same number. We also prove the surprising result that the patterns $1324$ and $1423$ are strong-c-forest-Wilf equivalent, even though they are not c-Wilf equivalent with respect to permutations.

math.CO↗

arXiv2021

Antipowers in Uniform Morphic Words and the Fibonacci Word

Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$, there exists a constant $c$ such that for any $k$ and any index $i$, a $k$-antipower with block length at most $ck$ starts at the $i$th position of $w$. They prove their conjecture in the case of binary words, and we extend their result to alphabets of arbitrary finite size and characterize those words for which the result does not hold. We also prove their conjecture in the specific case of the Fibonacci word.

math.CO↗