arXiv subjects

Souvik Pal

Publications and source records attributed to Souvik Pal.

10 recordsLinked to original sources

arXiv2026

Minimal Filling pair of non orientable surfaces

For $g\ge 3$, let $N_g$ denote the non-orientable surface of genus $g$. In this article, we establish the existence of filling pairs on $N_g$ that intersect minimally by construction using the theory of fat graphs. The mapping class group $\mathrm{Mod}(N_g)$ acts on the set of all such filling pairs. We count $\mathrm{Mod}(N_g)$-orbits of this action by providing both lower and upper bounds. Furthermore, we show that both bounds grow super-exponentially with $g$ using graph cohomology. Also, we investigate the lengths of minimally intersecting filling pairs on hyperbolic non-orientable surfaces $X$ in moduli space $\mathcal{M}_g$ of $N_g$. We define a function $\mathcal{F}_g:\mathcal{M}_g\to\mathbb{R}_{>0}$, where for $X\in \mathcal{M}_g$, the function $\mathcal{F}_g(X)$ is the shortest total length of a minimally intersecting filling pair on $X$. We determine its minimum $m_g$ and show that the set of minimizers is in bijection with the $\mathrm{Mod}(N_g)$-orbits of minimally intersecting filling pairs. We further extend $\mathcal{F}_g$ to $\mathcal{Y}_g$, defined by minimizing the length over all filling pairs, and show that $\mathcal{Y}_g$ attains the same minimum value as $\mathcal{F}_g$.

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arXiv2026

Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix $A$ that is filled by negative integers. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathrm{g}$ and the classical cone of dominant integral weights $P^+\subset \mathfrak{h}^*$. The non-integrable simple highest weight $\mathfrak{g}$-modules $L(\mu)$'s widely studied were those by Naito ([Trans. Amer. Soc., 1995]), for $\mu$'s dot-linked to $P^+$-translates of sums $- \sum_{j\in J}\alpha_j$ of mutually orthogonal and imaginary simple roots $\alpha_j$'s. Recently, we computed weights of all highest weight $\mathfrak{g}$-modules $V$'s (over all BKM LA's), and character of $L(\rho)$ for Weyl vector $\rho$. These needed a family of ``integrable'' $L(\mu)$'s for $\mu$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $\mu(\alpha_i^{\vee})\leq 0$ for $\mu\in P^{\pm}$ are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(\mu)$ contains ``Chevalley-Serre relations'' $f_i^{\frac{2}{A_{ii}}{\mu(\alpha_i^{\vee})}+1}L(\mu)_{\mu}=0$, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers $M(\mu)$'s of $L(\mu)$'s for $\mu\in P^{\pm}$. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a uniform equality, of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all $V$'s when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on results of Kac and Kazhdan in crucial unique solution case.

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arXiv2025

Characterization of Erd\"os matrices by their zero entries

An Erd\"os matrix $E$ is a bistochastic matrix whose sum of squares of entries (Frobenius norm squared) equals its maxtrace (maximum of all the $\sigma$-traces for permutations $\sigma$'s). We characterize all Erd\"os $E$ by the patterns of their zero entries; showing that each such skeleton has at most one $E$. We present an algorithm to find all $n\times n$ Erd\"os matrices, which finds them up to $n\leqslant 5$ quickly and also size $n=6$. We further show some presently known RCDS matrices to be Erd\"os.

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arXiv2025

Weights and characters of highest weight modules

Let $\mathfrak{g}=\mathfrak{g}(A)$ be any Borcherds-Kac-Moody $\mathbb{C}$-Lie algebra (BKM LA) for BKM-Cartan matrix $A$, with Cartan subalgebra $\mathfrak{h}$. Let $V$ denote a highest weight $\mathfrak{g}$-module, with top weight $\lambda\in \mathfrak{h}^*$ (not necessarily in the domninant integral cone $P^+$). The non-integrable simples $V= L(\lambda)$ by Naito ([Trans. Amer. Soc., 1995]) are widely studied beyond integrable simple $L(\nu)s,\ \nu \in P^+$. We introduce and study: 1) A weight cone $P^{\pm}=\big\{\mu\in \mathfrak{h}^*\ \big|\ \mu(\alpha_i^{\vee})\in \frac{A_{ii}}{2}\mathbb{Z}_{\geq 0}\text{ for all simple co-roots }\alpha_i^{\vee}\big\}$; note Weyl vector $\rho\in P^{\pm}\setminus P^+$. 2) The resulting (novel) non-integrable simple $L(\lambda)s, \ \lambda \in P^{\pm}\setminus P^{+}$; their Chevalley-Serre (CS) type relations (which are, in fact, complementary to those of integrable $L(\nu)$s); 3) Higher length CS type relations in any highest weight module under the name ``holes". Using these, we obtain explicitly and uniformly, (notably) Weyl-orbit typed formulas for weight-sets of: all simples $L(\lambda)$s ($\forall$ $\lambda\in \mathfrak{h}^*$) and all quotients of parabolic Verma modules along imaginary directions. This generalizes and extends in one stroke, such formulas over Kac-Moody (KM) $\mathfrak{g}$, of all $L(\lambda)$ by Khare ([Trans. Amer. Math. Soc. 2017]), and Dhillon and Khare ([Adv. Math., 2017], and also of all $V$ by Khare and Teja recently; which used parabolic and higher order Verma modules. We obtain Weyl-Kac-Borcherds type character formulas for $L(\lambda) \text{ for } \lambda\in P^{\pm}$, over negative rank-2 $\mathfrak{g}$'s; by exploring Verma module embeddings. We obtain character of every highest weight module $V$ for $\lambda=\rho$ in negative $A$-type cases.

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arXiv2025

Rank-one geometry and mixed complexes in representations of Cartan type Lie algebras on a torus

In this paper, we develop a unified theory of reducibility and indecomposability for Shen-Larsson modules over the Witt, special and Hamiltonian type Lie algebras on a torus. Our approach is based on a rank-one mechanism governing irreducible submodules, Loewy filtrations, rank reduction, uniseriality and mixed complex structures. We first provide a uniform intrinsic characterization of the trivial and fundamental representations of $gl_N, sl_N, sp_{2n}$ in terms of quadratic relations satisfied by rank-one elements of these matrix Lie algebras and utilize it to determine the irreducibility of Shen-Larsson modules over $W_N, S_N, H_{2n}$. Using the rank-one operators arising from these relations, we then construct rank-reducing operators corresponding to distinguished lattice directions and apply them to show that the submodule structure of the reducible Shen-Larsson modules over $W_N, S_N, H_{2n}$ attached to the fundamental representations of $gl_N, sl_N, sp_{2n}$ respectively are generically uniserial. In the Hamiltonian case, we show that the submodules of these reducible Shen-Larsson modules come from kernels and images of differentials of the de Rham and Koszul-type complexes. These differentials anti-commute and thus endow the tensor field modules with a mixed complex structure, which also admit a natural interpretation formally analogous to the de Rham differential and co-differential type operator appearing in symplectic Hodge theory. In particular, we provide complete answers to the questions recently posed by Pei-Sheng-Tang-Zhao [J. Inst. Math. Jussieu 2023] concerning the structure of Shen-Larsson modules over $H_{2n}$.

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arXiv2023

Quasi-finite modules over affine and extended affine Lie algebras

In this paper, we consider irreducible quasi-finite (or equivalently weakly integrable) modules, with non-trivial action of the core, over the extended affine Lie algebras (EALAs) whose centerless cores are multiloop algebras. The centerless cores of all but one family of EALAs having nullity greater than 1 are known to admit such multiloop realizations. For any such (untwisted) EALA, we show that the irreducible quasi-finite modules are either integrable with the center of the underlying core acting trivially, or restricted generalized highest weight (GHW) modules. We further prove that in the nullity 2 case, these irreducible restricted GHW modules turn out to be highest weight type modules, thereby classifying the irreducible quasi-finite modules over all such EALAs. In particular, we obtain the classification of irreducible quasi-finite modules over toroidal Lie algebras, minimal EALAs and toroidal EALAs of nullity 2. Along the way, we completely classify the irreducible weakly integrable modules over affine Kac-Moody algebras (studied by Rao-Futorny [Trans. Amer. Math. Soc. 2009] for non-zero level modules). Our results generalize the well-known work of Chari [Invent. Math. 1986] and Chari-Pressley [Math. Ann. 1986] concerning the classification of irreducible integrable modules over (nullity 1) affine Kac-Moody algebras.

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arXiv2022

Error Correction in ASR using Sequence-to-Sequence Models

Post-editing in Automatic Speech Recognition (ASR) entails automatically correcting common and systematic errors produced by the ASR system. The outputs of an ASR system are largely prone to phonetic and spelling errors. In this paper, we propose to use a powerful pre-trained sequence-to-sequence model, BART, further adaptively trained to serve as a denoising model, to correct errors of such types. The adaptive training is performed on an augmented dataset obtained by synthetically inducing errors as well as by incorporating actual errors from an existing ASR system. We also propose a simple approach to rescore the outputs using word level alignments. Experimental results on accented speech data demonstrate that our strategy effectively rectifies a significant number of ASR errors and produces improved WER results when compared against a competitive baseline. We also highlight a negative result obtained on the related grammatical error correction task in Hindi language showing the limitation in capturing wider context by our proposed model.

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arXiv2022

Classification of irreducible Harish-Chandra modules over full toroidal Lie algebras and higher-dimensional Virasoro algebras

In this paper, we classify the irreducible Harish-Chandra modules over the full toroidal Lie algebra, which is a natural higher-dimensional analogue of the affine-Virasoro algebra. In particular, we complete the classification of irreducible bounded modules, which were studied by Billig for non-zero level modules [Int. Math. Res. Not. 2006]. As a by-product, we also obtain the classification of irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, which was introduced by Rao-Moody [Comm. Math. Phys. 1994], thereby generalizing the well-known result of O. Mathieu [Invent. Math. 1992] for the classical Virasoro algebra. More precisely, we show that any irreducible Harish-Chandra module over the higher-dimensional Virasoro algebra turns out to be either a quotient of a module of tensor fields on a torus or a highest weight type module up to a twist of an automorphism, as conjectured by Eswara Rao in 2004.

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arXiv2021

Classification of level zero irreducible integrable modules for twisted full toroidal Lie algebras

In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus $LT$ which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial $LT$-action and where the centre of the underlying Lie algebra acts trivially.

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arXiv2021

Integrable Modules For Graded Lie Tori With Finite Dimensional Weight Spaces

An important problem in the representation theory of affine and toroidal Lie algebras is to classify all possible irreducible integrable modules with finite dimensional weight spaces. Recently the irreducible integrable modules having finite dimensional weight spaces with non-trivial central action have been classified for a more general class of Lie algebras, namely the graded Lie tori. In this paper, we classify all the irreducible integrable modules with finite dimensional weight spaces for this graded Lie tori where the central elements act trivially. Thus we ultimately obtain all the simple objects in the category of integrable modules with finite dimensional weight spaces for the graded Lie tori.

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