arXiv subjects

Abhik Pal

Publications and source records attributed to Abhik Pal.

4 recordsLinked to original sources

arXiv2026

Cohomology of Flag Superschemes and Syzygies of Compositional Varieties

We study the coherent cohomology of partial flag supervarieties and calculate the sheaf cohomology of the structure sheaves of four new infinite families of flag supervarieties. We introduce compositional varieties, a generalization of determinantal varieties obtained by imposing rank conditions on compositions of a pair of maps. We study geometric properties of compositional varieties and, in cases of interest, compute their $\mathrm{Tor}$-groups. For each of the four families of flag supervarieties, we show that the graded sheaf cohomology is isomorphic to the tensor product of the singular cohomology ring of an ordinary partial flag variety and the graded $\mathrm{Tor}$-groups of a compositional variety.

math.AG↗

arXiv2026

Syzygies of Isotropic Kalman Varieties

Let $L$ be a subspace of a complex vector space $V$ and fix $s \leq \dim{L}$. The (type A) Kalman variety consists of all endomorphisms of $V$ that have an $s$-dimensional invariant subspace in $L$. We introduce a generalization where $V$ and $L$ are symplectic vector spaces. We fix an isotropic subspace $W \subseteq V$ satisfying $W^\perp = W \oplus L$. The isotropic (type C) Kalman variety consists of symplectic morphisms of $V$ that have an invariant coisotropic subspace of a prescribed dimension inside $W^\perp$. We are mainly interested in studying the Lagrangian case. In type C, we prove analogues of results known for type A Kalman varieties; in particular, we determine the defining equations, compute geometric invariants, and analyze their singularities. We conjecture the existence of a long exact sequence relating the structure sheaves. Based on the results in the symplectic case, we describe Kalman variety analogues with respect to endomorphisms of odd orthogonal (type B) and even orthogonal (type D) vector spaces.

math.AG↗

arXiv2025

Lie Superalgebras and Generalized Kazhdan-Lusztig Polynomials

We present `liesuperalg` a SageMath package for representation-theoretic calculations involving Lie superalgebras in Type A. Our package introduces functionality to calculate invariants of weights and produce the associated cup diagrams. We expose functionality to calculate characters of irreducible representations, work with combinatorics of generalized Kazhdan-Lusztig polynomials, and determine composition factor multiplicities of indecomposable modules. Our package implements an algorithm to decompose arbitrary modules in terms of irreducible ones in the Grothendeick group of Lie superalgebra representations.

math.RT↗

arXiv2021

Beginners' Quest to Formalize Mathematics: A Feasibility Study in Isabelle

How difficult are interactive theorem provers to use? We respond by reviewing the formalization of Hilbert's tenth problem in Isabelle/HOL carried out by an undergraduate research group at Jacobs University Bremen. We argue that, as demonstrated by our example, proof assistants are feasible for beginners to formalize mathematics. With the aim to make the field more accessible, we also survey hurdles that arise when learning an interactive theorem prover. Broadly, we advocate for an increased adoption of interactive theorem provers in mathematical research and curricula.

cs.LO↗