arXiv subjects

Sankaran Viswanath

Publications and source records attributed to Sankaran Viswanath.

At least 19 recordsLinked to original sources

arXiv2026

Kostant--Kumar modules: presentation and multiplicities

Kostant--Kumar modules $K(\lambda,w,\mu)$ are submodules of a tensor product $V(\lambda)\otimes V(\mu)$ of irreducible highest weight modules over a symmetrizable Kac--Moody algebra, indexed by Weyl group elements $w$; their decomposition numbers $c^\nu_{\lambda\mu}(w)$ refine ordinary tensor product multiplicities. We study them module-theoretically. We show that $c^\nu_{\lambda\mu}(w)$ is computed by a natural quotient of the Kostant--Parthasarathy--Ranga Rao--Varadarajan multiplicity space, via orthogonal projection onto a Demazure module. For $\mathfrak{g}$ finite-dimensional semisimple or symmetric Kac--Moody, we present $K(\lambda,w,\mu)$ by generators and relations, extending the presentation of Demazure modules due to Joseph, Polo and Mathieu. We apply the presentation to obtain upper bounds on $c^\nu_{\lambda\mu}(w)$ and to study Schur positivity.

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arXiv2025

The affine Brylinski filtration and $\mathscr{W}$-algebras

The Brylinski-Kostant filtration on a representation of a finite-dimensional semisimple Lie algebra has interpretations in terms of the algebra, geometry and combinatorics of the representation. Its extension to affine Lie algebras was first studied by Slofstra. Recent work of the present authors constructed a Poincar\'{e}-Birkhoff-Witt type basis for the dominant weight spaces of the basic representation of affine Lie algebras of type $A$, which is compatible with the affine Brylinski filtration. In this paper, we overcome the constraint of type dependence, and furnish a new, uniform proof which holds for all simply-laced affine Lie algebras.

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arXiv2024

$q$-Whittaker polynomials: bases, branching and direct limits

We study $q$-Whittaker polynomials and their monomial expansions given by the fermionic formula, the inv statistic of Haglund-Haiman-Loehr and the quinv statistic of Ayyer-Mandelshtam-Martin. The combinatorial models underlying these expansions are partition overlaid patterns and column strict fillings. The former model is closely tied to representations of the affine Lie algebra $\widehat{\mathfrak{sl}_n}$ and admits projections, branching maps and direct limits that mirror these structures in the Chari-Loktev basis of local Weyl modules. We formulate novel versions of these notions in the column strict fillings model and establish their main properties. We construct weight-preserving bijections between the models which are compatible with projection, branching and direct limits. We also establish connections to the coloured lattice paths formalism for $q$-Whittaker polynomials due to Wheeler and collaborators.

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arXiv2024

Crystals for Kostant-Kumar modules of $\widehat{\mathfrak{sl}_2}$

We consider the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and the Kostant-Kumar submodules of tensor products of its level 1 highest weight integrable representations. We construct crystals for these submodules in terms of the charged partitions model and describe their decomposition into irreducibles.

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arXiv2023

Unique Factorization For Tensor Products of Parabolic Verma Modules

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak{h}$. We prove a unique factorization property for tensor products of parabolic Verma modules. More generally, we prove unique factorization for products of characters of parabolic Verma modules when restricted to certain subalgebras of $\mathfrak{h}$. These include fixed point subalgebras of $\mathfrak{h}$ under subgroups of diagram automorphisms of $\mathfrak{g}$ and twisted graph automorphisms in the affine case.

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arXiv2023

Monomial expansions for $q$-Whittaker and modified Hall-Littlewood polynomials

We consider the monomial expansion of the $q$-Whittaker polynomials given by the fermionic formula and via the inv and quinv statistics. We construct bijections between the parametrizing sets of these three models which preserve the $x$- and $q$-weights, and which are compatible with natural projection and branching maps. We apply this to the limit construction of local Weyl modules and obtain a new character formula for the basic representation of $\widehat{\mathfrak{sl}_n}$. Finally, we indicate how our main results generalize to the modified Hall-Littlewood case.

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arXiv2023

Saturation for Flagged Skew Littlewood-Richardson Coefficients

We define and study a generalization of the Littlewood-Richardson (LR) coefficients, which we call the flagged skew LR coefficients. These subsume several previously studied extensions of the LR coefficients. We establish the saturation property for these coefficients, generalizing work of Knutson-Tao and Kushwaha-Raghavan-Viswanath.

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arXiv2023

Simple Procedures for Left and Right Keys of Semi-Standard Young Tableaux

We give simple procedures to obtain the left and right keys of a semi-standard Young tableau. Keys derive their interest from the fact that they encode the characters of Demazure and opposite Demazure modules for the general and special linear groups. Given the importance of keys, there are indeed several procedures available in the literature to determine them. In comparison, our procedures are new (to the best of our knowledge) and especially simple. Having said that, we hasten to add that there is nothing new in any individual ingredient that goes into our procedures. These ingredients are all routine, straightforward, and (in any case) occur in the literature. But they never quite seem to have been put together as done here. Our procedures end up repeatedly performing the Deodhar lifts, maximal lifts for the left key and minimal lifts for right key. Together with the well known fact that keys can be obtained by such repeated lifts, this justifies the procedures. The relevance of Deodhar lifts to combinatorial models for Demazure characters is well known in Standard Monomial Theory. Right and left keys appear respectively as initial and final directions of Lakshmibai-Seshadri paths in Littelmanns Path Model Theory.

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arXiv2022

The saturation property for refined Littlewood-Richardson coefficients

Given dominant integral weights $\lambda, \mu, \nu$ of a finite-dimensional simple Lie algebra $\mathfrak{g}$ and an element $w$ of its Weyl group, the refined tensor product multiplicity $c_{\lambda \mu}^\nu(w)$ is the multiplicity of the irreducible $\mathfrak{g}$-module $V(\nu)$ in the so-called Kostant--Kumar submodule $K(\lambda, w, \mu)$ of the tensor product $V(\lambda) \otimes V(\mu)$. We derive properties of these coefficients in general type, including a Brauer--Klimyk type formula and restriction theorems. In type $A$, we obtain a hive model for the $c_{\lambda \mu}^\nu(w)$ and prove that the saturation and strong semigroup properties hold if the permutation $w$ is $312$-avoiding, $231$-avoiding, or a commuting product of such elements. This generalizes the classical Knutson--Tao saturation theorem.

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arXiv2021

The Brylinski filtration for affine Kac-Moody algebras and representations of $\mathcal{W}$-algebras

We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra $\mathfrak{g}$, a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of integrable highest weight modules of $\mathfrak{g}$ has Hilbert series coinciding with Lusztig's $t$-analogue of weight multiplicities. For the level 1 vacuum module $L(Λ_0)$ of affine Kac-Moody algebras of type $A$, we show that the Brylinski filtration may be most naturally understood in terms of (vertex algebra) representations of the corresponding $\mathcal{W}$-algebra. We show that the dominant weight spaces together form an irreducible Verma module of $\mathcal{W}$ and that the natural PBW basis of this module is compatible with the Brylinski filtration, thereby determining explicitly the subspaces of the filtration. Our basis is the analogue for the principal vertex operator realization of $L(Λ_0)$, of Feigin-Frenkel's basis of $\mathcal{W}$.

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arXiv2021

$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

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arXiv2021

A note on the fusion product decomposition of Demazure modules

We settle the fusion product decomposition theorem for higher-level affine Demazure modules for the cases $E^{(1)}_{6, 7, 8}, F^{(1)}_4$ and $E^{(2)}_{6}$, thus completing the main theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). We obtain a new combinatorial proof for the key fact, that was used in Chari et al. (op cit.), to prove this decomposition theorem. We give a case free uniform proof for this key fact.

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arXiv2020

${\boldsymbolπ}$-systems of symmetrizable Kac-Moody algebras

As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of a $π$-system. This is a subset of the roots such that pairwise differences of its elements are not roots. These arise as simple systems of regular semisimple subalgebras. Morita and Naito generalized this notion to all symmetrizable Kac-Moody algebras. In this work, we systematically develop the theory of $π$-systems of symmetrizable Kac-Moody algebras and establish their fundamental properties. We study the orbits of the Weyl group on $π$-systems, and completely determine the number of orbits in many cases of interest in physics. In particular, we show that there is a unique $π$-system of type $HA_1^{(1)}$ (the Feingold-Frenkel algebra) in $E_{10}$ (the rank 10 hyperbolic algebra) up to Weyl group action and negation.

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arXiv2019

A study of Kostant-Kumar modules via Littelmann paths

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a symmetrizable Kac-Moody algebra. Our main result is an identification of a path model for any KK module as a subset of the well known path model for the tensor product consisting of concatenations of Lakshmibai-Seshadri paths. The technical results about extremal elements in Coxeter groups that we formulate and prove en route and the technique of their proofs should be of independent interest. We also discuss the existence of PRV components and generalised PRV components in KK modules. Specialising to the case of the special linear Lie algebra, we record a decomposition rule for KK modules in terms of Littlewood-Richardson tableaux. In this connection, we present a new procedure to determine the permutation that is the initial element of the minimal standard lift of a semi-standard Young tableau. The appendix, necessitated by the derivation of the tableau decomposition rule, deals with standard concatenations of Lakshmibai-Seshadri paths of arbitrary shapes, of which semi-standard Young tableaux form a very special case.

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arXiv2019

A relationship between Gelfand-Tsetlin bases and Chari-Loktev bases for irreducible finite dimensional representations of special linear Lie algebras

We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of (integral Gelfand-Tsetlin) patterns with a fixed bounding sequence determined by the highest weight of the representation. We define the "row-wise dominance" partial order on this set of patterns, and prove that the transition matrix between the two bases is triangular with respect to this partial order. We write down explicit expressions for the diagonal elements of the transition matrix.

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arXiv2017

On Chari-Loktev bases for local Weyl modules in type $A$

This paper is a study of the bases introduced by Chari-Loktev for local Weyl modules of the current algebra associated to a special linear Lie algebra. Partition overlaid patterns, POPs for short---whose introduction is one of the aims of this paper---form convenient parametrizing sets of these bases. They play a role analogous to that played by (Gelfand-Tsetlin) patterns in the representation theory of the special linear Lie algebra. The notion of a POP leads naturally to the notion of area of a pattern. We observe that there is a unique pattern of maximal area among all those with a given bounding sequence and given weight. We give a combinatorial proof of this and discuss its representation theoretic relevance. We then state a conjecture about the "stability", i.e., compatibility in the long range, of Chari-Loktev bases with respect to inclusions of local Weyl modules. In order to state the conjecture, we establish a certain bijection between colored partitions and POPs, which may be of interest in itself.

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arXiv2015

The $t$-analogs of string functions for $A_1^{(1)}$ and Hecke indefinite modular forms

We study generating functions for Lusztig's $t$-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra $A_1^{(1)}$. At $t=1$, these reduce to the {\em string functions} of $A_1^{(1)}$, which were shown by Kac and Peterson to be related to certain Hecke indefinite modular forms. Using their methods, we obtain a description of the general $t$-string function; we show that its values can be realized as radial averages of a certain extension of the Hecke indefinite modular form.

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arXiv2015

Stability of the Chari-Pressley-Loktev bases for local Weyl modules of $sl_2[t]$

We prove stability of the Chari-Pressley-Loktev bases for natural inclusions of local Weyl modules of the current algebra $sl_2[t]$. These modules being known to be Demazure submodules in the level 1 representations of the affine Lie algebra $\widehat{sl_2}$, we obtain, by passage to the direct limit, bases for the level 1 representations themselves.

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