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K. N. Raghavan

Publications and source records attributed to K. N. Raghavan.

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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arXiv2026

On ${\pi}$-systems of symmetrizable Kac-Moody algebras

Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $\pi$-systems, which are subsets of real roots, the pairwise differences of whose elements are not roots. Such systems arise as simple systems of regular subalgebras of $\mathfrack{g}$, and were originally studied by Dynkin, Morita and Naito. We show that the binary relation introduced by Morita defines a partial order on the set of $\mathfrack{g}$ of finite, untwisted affine or hyperbolic type. We also formulate general principles for constructing $\pi$-systems as well as for finding forbidden diagrams that cannot occur as Dynkin diagrams of $\pi$-systems of a given $\mathfrack{g}$. Among other applications, we use this to determine the set of maximal hyperbolic Dynkin diagrams in ranks $3$-$10$ relative to the Morita partial order.

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arXiv2025

The saturation property for refined Littlewood-Richardson coefficients

Given dominant integral weights $λ, μ, ν$ of a finite-dimensional simple Lie algebra $\mathfrak{g}$ and an element $w$ of its Weyl group, the refined tensor product multiplicity $c_{λμ}^ν(w)$ is the multiplicity of the irreducible $\mathfrak{g}$-module $V(ν)$ in the so-called Kostant--Kumar submodule $K(λ, w, μ)$ of the tensor product $V(λ) \otimes V(μ)$. 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 $c_{λμ}^ν(w)$ 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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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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arXiv2024

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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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

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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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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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

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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arXiv2013

Borel-de Siebenthal discrete series and associated holomorphic discrete series

Let G_0 be a simply connected noncompact real simple Lie group with maximal compact subgroup K_0. Assume that rank(G_0) = rank(K_0) so that G_0 has discrete series representations. If G_0/K_0 is Hermitian symmetric, there exists a relatively simple discrete series of G_0, called holomorphic discrete series. Now assume that G_0/K_0 is not Hermitian symmetric. In this case, we can define Borel-de Siebenthal discrete series of G_0 analogous to holomorphic discrete series. We consider a certain circle subgroup of K_0 whose centralizer L_0 is such that K_0/L_0 is an irreducible compact Hermitian symmetric space. Let (K_0)* be the dual of K_0 with respect to L_0. Then (K_0)*/L_0 is an irreducible non-compact Hermitian symmetric space dual to K_0/L_0. To each Borel-de Siebenthal discrete series of G_0, we can associate a holomorphic discrete series of (K_0)*. In this article, we address occurrence of common L_0-types between these two discrete series under certain conditions.

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arXiv2011

RSK bases and Kazhdan-Lusztig cells

From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, 'RSK bases' are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory of the symmetric group are discussed.

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arXiv2009

Frobenius splitting of certain rings of invariants

Two classical rings of invariants are shown to be Frobenius split: for the special linear group acting on the direct sum of several copies of the defining representation and several copies of the dual of the defining representation; and for the special orthogonal group acting on several copies of the defining representation.

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arXiv2008

Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians

We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.

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arXiv2007

Hilbert functions of points on Schubert varieties in Orthogonal Grassmannians

A solution is given to the following problem: how to compute the multiplicity, or more generally the Hilbert function, at a point on a Schubert variety in an orthogonal Grassmannian. Standard monomial theory is applied to translate the problem from geometry to combinatorics. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for the Grassmannian and the symplectic Grassmannian. As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. Taking the Schubert variety to be of a special kind and the point to be the "identity coset," our problem specializes to a problem about Pfaffian ideals treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a "generic singularity."

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arXiv2006

Standard monomial bases, moduli of vector bundles, and invariant theory

Consider the diagonal action of the special orthogonal group on the direct sum of a finite number of copies of the standard representation--the underlying field is assumed to be algebraically closed and of characteristic not equal to two. We construct a "standard monomial" basis for the ring of polynomial invariants for this action. We then deduce, by a deformation argument, our main result that this ring of polynomial invariants is Cohen-Macaulay. We give three applications of this result: (1) the first and second fundamental theorems of invariant theory for the above action; (2) Cohen-Macaulayness of the moduli space of equivalence classes of semi-stable vector bundles of rank two and degree zero on a smooth projective curve of genus at least three (for this application, characteristic three is also excluded); (3) a basis in terms of traces for the ring of polynomial invariants for the diagonal adjoint action of the special linear group SL(2) on a finite number of copies of its Lie algebra sl(2).

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