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Shushma Rani

Publications and source records attributed to Shushma Rani.

8 recordsLinked to original sources
arXiv2026

Asymmetric Catalan and Shi Hyperplane Arrangements

In this paper, we study the regions of asymmetric extensions of the Catalan and Shi arrangements. We first determine the characteristic polynomials using the finite field method, thereby obtaining explicit formulas for the number of regions. We then introduce two new classes of combinatorial objects: $\mathbf{c}$-labelled Dyck paths and $\mathbf{c}$-building functions, which generalize classical Dyck paths and parking functions, respectively. We prove that the regions of the asymmetric $\mathbf{c}$-Catalan arrangement are in natural bijection with $\mathbf{c}$-labelled Dyck paths, while those of the asymmetric $\mathbf{c}$-Shi arrangement are in natural bijection with $\mathbf{c}$-building functions. This resolves a problem proposed by Theo Douvropoulos during the 2022 Oberwolfach Workshop on Enumerative Combinatorics. In particular, our results extend Stanley's celebrated correspondence between $k$-parking functions and the regions of the $k$-Shi arrangement by introducing new combinatorial models and developing entirely different proof techniques.

math.CO↗
arXiv2025

Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements

We introduce the concepts of marked multi-colorings, marked chromatic polynomials, and marked (multivariate) independence series for hypergraphs. We show that the coefficients of the q-th power of the marked independence series of a hypergraph coincide with its marked chromatic polynomials in q, thereby generalizing a corresponding result for graphs established in Chaithra et al. 2025 (arXiv:2503.11230). These notions are then naturally extended to subspace arrangements. In particular, we prove that the number of marked multi q-colorings of a subspace arrangement is a polynomial in q. We also define the (marked) independence series for subspace arrangements and prove that the (-q)-th power of the independence series of a hyperplane arrangement has non-negative coefficients. We further conjecture that the (-q)-th power of the independence series of a hypergraph has non-negative coefficients if and only if all its edges have even cardinality.

math.CO↗
arXiv2025

Graded representations of current Lie superalgebras $\mathfrak{sl}(1|2)[t]$

This paper is the study of finite-dimensional graded representations of current lie superalgebras $\mathfrak{sl}(1|2)[t]$. We define the notion of super POPs, a combinatorial tool to provide another parametrization of the basis of the local Weyl module given in [2]. We derive the graded character formula of local Weyl module for $\mathfrak{sl}(1|2)[t]$. Furthermore, we construct a short exact sequence of Chari-Venkatesh modules for $\mathfrak{sl}(1|2)[t]$. As a consequence, we prove that Chari-Venkatesh modules are isomorphic to the fusion of generalized Kac modules.

math.RT↗
arXiv2024

Filtration of tensor product of local Weyl modules for $\mathfrak{sl}_{n+1}[t]$

In this paper, we consider the tensor product of local Weyl modules for $\mathfrak{sl}_{n+1}[t]$ whose highest weights are multiples of the first and $n^{th}$ fundamental weights. We determine the graded character of these tensor product modules in terms of the graded character of local Weyl modules and prove that these modules admit a filtration whose successive quotients are either truncated Weyl modules or fusion products of Demazure modules. Furthermore, we establish that the truncated Weyl modules appearing as quotients in the filtration of tensor products of local Weyl modules of $\mathfrak{sl}_3[t]$ are indeed isomorphic to fusion products of irreducible $\mathfrak{sl}_3[t]$-modules which establish the independence of a family of fusion product modules of $\mathfrak{sl}_3[t]$ from the set of its evaluation parameters.

math.RT↗
arXiv2024

On Commuting automorphisms of nilpotent Lie algebras

We investigate the commuting automorphisms of nilpotent Lie algebras $L$ with coclass $\leq 3$. Our examination exposes the conditions under which the set of commuting automorphisms of $L$ forms a subgroup within its automorphism group.

math.RA↗
arXiv2024

Graded Character Formula for Fusion Products of Irreducible Modules and Littlewood-Richardson Coefficients in Type $A_2$

We investigate a specific class of CV modules for $\mathfrak{sl}_3$ and establish an exact sequence for these modules. Utilizing dimension arguments, we demonstrate that this module is isomorphic to the fusion product of irreducible modules, thereby offering a new proof of the conjecture regarding the independence of fusion products from parameters. By analyzing the filtration of the kernel within the exact sequence, we derive the graded character formula for fusion product modules. Moreover, we leverage the graded character to deduce the algebraic characterization of the Littlewood-Richardson (LR) coefficients and present an alternative proof of the saturation theorem in type $A_2$.

math.RT↗
arXiv2021

A study on free roots of Borcherds-Kac-Moody Lie Superalgebras

Let $\mathfrak g$ be a Borcherds-Kac-Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak g$ is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph $G$. By Chevalley relations we get a triangular decomposition $\mathfrak g = \mathfrak{ n_+} \oplus \mathfrak h \oplus \mathfrak n_{-}$ and each roots space $\mathfrak{ g_α}$ is either contained in $\mathfrak{n_+}$ or $\mathfrak{ n_{-}}$. In particular, each $\mathfrak{ g_α}$ involves only the relations (2) and (3). In this paper, we are interested in the root spaces of $\mathfrak g$ which are independent of the Serre relations. We call these roots free roots$^{1}$ of $\mathfrak g$. Since these root spaces involve only commutation relations coming from the graph $G$ we can study them combinatorially. We use heaps of pieces to study these roots and prove many combinatorial properties. We construct two different bases for these root spaces of $\mathfrak g$: One by extending the Lalonde's Lyndon heap basis of free partially commutative Lie algebras to the case of free partially commutative Lie superalgebras and the other by extending the basis given in \cite{akv17} for the free root spaces of Borcherds algebras to the case of BKM superalgebras. This is done by studying the combinatorial properties of super Lyndon heaps. We also discuss a few other combinatorial properties of free roots.

math.CO↗