arXiv subjects

Tanusree Khandai

Publications and source records attributed to Tanusree Khandai.

8 recordsLinked to original sources

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.

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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 $A_2$.

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arXiv2023

Demazure Filtrations of Tensor Product Modules and Character Formula

We study the structure of the finite-dimensional representations of $\mathfrak{sl}_2[t]$, the current Lie algebra type of $A_1$, which are obtained by taking tensor products of special Demazure modules. We show that these representations admit a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for $\mathfrak{sl}_2[t]$. Furthermore, we derive an explicit expression for graded character of the tensor product of a local Weyl module with an irreducible $\mathfrak{sl}_2[t]$ module. In conjunction with the results of \cite{MR3210603}, our findings provide evidence for the conjecture in \cite{9} that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.

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arXiv2018

Spectral Characters of a class of Integrable Representations of Toroidal Lie Algebras

In this paper we study the subcategory of finite-length objects of the category of positive level integrable representations of a toroidal Lie algebra. The main goal is to characterize the blocks of the category. In the cases when the underlying finite type Lie algebra associated with the toroidal Lie algebra is simply-laced, we are able to give a parametrization for the blocks.

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arXiv2016

Irreducible Integrable Representations of Toroidal Lie Algebras

The irreducible integrable representations with finite-dimensional weight spaces of toroidal Lie algebras on which the center acts non-trivially were classified by S.Eswara Rao. In this paper we give a compact proof of the results that lead to the classification theorem. Further we establish a necessary and sufficient condition under which two such irreducible modules are isomorphic.

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arXiv2011

Local Weyl modules for equivariant map algebras with free abelian group actions

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defining twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological properties and a tensor product property) extend to the more general setting considered in the current paper.

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arXiv2011

Weyl modules for multiloop algebras

Global and local Weyl modules for the untwisted multiloop Lie algebras were defined by Chari, the first and the second author via homological properties. In this paper we extended the ideas to give a categorical definition of the Weyl modules for twisted multiloop algebras. Our methods led us to describe an identification of the finite--dimensional highest weight modules for twisted multiloop algebras with suitably chosen finite--dimensional highest weight modules for untwisted multiloop algebras.

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arXiv2009

A categorical approach to Weyl modules

Global and local Weyl Modules were introduced via generators and relations in the context of affine Lie algebras in a work by the first author and Pressley and were motivated by representations of quantum affine algebras. A more general case was considered by Feigin and Loktev by replacing the polynomial ring with the coordinate ring of an algebraic variety. We show that there is a natural definition of the local and global modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing previous results. Finally an analysis of the fundamental Weyl modules proves that the functors are not left exact in general, even for coordinate rings of affine varieties.

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