arXiv subjects

R. Venkatesh

Publications and source records attributed to R. Venkatesh.

34 records · Page 2Linked to original sources

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.

math.RT↗

arXiv2020

Oxygen Vacancy-Induced Topological Hall effect in a Nonmagnetic Band Insulator

The discovery of skyrmions has sparked tremendous interests about topologically nontrivial spin textures in recent times. The signature of noncoplanar nature of magnetic moments can be observed as topological Hall effect (THE) in electrical measurement. Realization of such nontrivial spin textures in new materials and through new routes is an ongoing endeavour due to their huge potential for future ultra-dense low-power memory applications. In this work, we report oxygen vacancy (OV) induced THE and anomalous Hall effect (AHE) in a 5$d^0$ system KTaO$_3$. The observation of weak antilocalization behavior and THE in the same temperature range strongly implies the crucial role of spin-orbit coupling (SOC) behind the origin of THE. Ab initio calculations reveal the formation of the magnetic moment on Ta atoms around the OV and Rashba-type spin texturing of conduction electrons. In the presence of Rashba SOC, the local moments around vacancy can form bound magnetic polarons (BMP) with noncollinear spin texture, resulting THE. Scaling analysis between transverse and longitudinal resistance establishes skew scattering driven AHE in present case. Our study opens a route to realize topological phenomena through defect engineering.

cond-mat.mtrl-sci↗

arXiv2019

Superconductivity in V$_{1-x}$Zr$_x$ alloys]{Evolution of high field superconductivity and high critical current density in the as-cast V$_{1-x}$Zr$_x$ alloys

We report here the structural, electrical and magnetic properties of as-cast V$_{1-x}$Zr$_x$ alloys ($x$ =0 - 0.4) at low temperatures. We observe that all the alloys undergo successive peritectic and eutectic reactions during cooling from the melt which leads to the formation of five phases, namely, a body centred cubic $β$-V phase, two phases with slightly different compositions having face centred cubic ZrV$_2$ structure, a hexagonal closed packed $α$-Zr phase, and the $β$-Zr precipitates. The amount of each phase is found to be dependent on the concentration of zirconium in vanadium. The $β$-V and ZrV$_2$ phases show superconductivity below 5.3~K and 8.5~K respectively. We show that the critical current density is large for V-rich V$_{1-x}$Zr$_x$ alloys with $x >$ 0.1. The grain boundaries generated from the eutectic reaction, and the point defects formed due to the variation in the composition are found to be responsible for the pinning of flux lines in low and high magnetic fields respectively. Our studies reveal that the choice of the composition and the heat treatment which leads to eutectic reaction are important in improving the critical current density in this alloy system.

cond-mat.supr-con↗

arXiv2018

Maximal closed subroot systems of real affine root systems

We completely classify and give explicit descriptions of the maximal closed subroot systems of real affine root systems. As an application we describe a procedure to get the classification of all regular subalgebras of affine Kac Moody algebras in terms of their root systems.

math.RA↗

arXiv2018

On tensor products of irreducible integrable representations

We consider integrable category $\mathcal{O}$ representations of Borcherds--Kac--Moody algebras whose Cartan matrix is finite dimensional, and determine the necessary and sufficient conditions for which the tensor product of irreducible representations from this category is isomorphic to another. This result generalizes a fundamental result of C. S. Rajan on unique factorization of tensor products of finite dimensional irreducible representations of finite dimensional simple Lie algebras over complex numbers.

math.RT↗

arXiv2018

Borel-de Siebenthal theory for affine reflection systems

We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.

math.RA↗

arXiv2016

Root multiplicities for Borcherds algebras and graph coloring

We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. Using this connection we give a second interpretation, namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph (depending on the root). Finally, using the combinatorics of Lyndon words we construct a basis for the root spaces corresponding to these roots and determine the Hilbert series in the case when all simple roots are imaginary. As an application we give a Lie theoretic proof of Stanley's reciprocity theorem of chromatic polynomials.

math.CO↗

arXiv2015

Twisted Demazure modules, fusion product decomposition and twisted Q--systems

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an $|R^+|$-tuple of partitions $\bxi=(ξ^α)_{α\in R^+}$ satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of $\bxi$ these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the notion of fusion products for twisted modules, first defined in \cite{FL99} for untwisted modules, and use the simplified presentation to prove a fusion product decomposition of twisted Demazure modules. As a consequence we prove that twisted Demazure modules can be obtained by taking the associated graded modules of (untwisted) Demazure modules for simply-laced affine algebras. Furthermore we give a semi-infinite fusion product construction for the irreducible representations of twisted affine algebras. Finally, we prove that the twisted $Q$-sytem defined in \cite{HKOTT02} extends to a non-canonical short exact sequence of fusion products of twisted Demazure modules.

math.RT↗

arXiv2015

Chromatic polynomials of graphs from Kac-Moody algebras

We give a new interpretation of the chromatic polynomial of a simple graph G in terms of the Kac-Moody Lie algebra with Dynkin diagram G. We show that the chromatic polynomial is essentially the q-Kostant partition function of this Lie algebra evaluated on the sum of the simple roots. Applying the Peterson recurrence formula for root multiplicities, we obtain a new realization of the chromatic polynomial as a weighted sum of paths in the bond lattice of G.

math.RT↗

arXiv2015

Fusion Product Structure of Demazure Modules

Let g be a finite-dimensional complex simple Lie algebra. Fix a non-negative integer l, we consider the set of dominant weights λ of g such that lΛ_0+λ is a dominant weight for the corresponding untwisted affine Kac-Moody algebra. For these special family of dominant weights, we show that the fusion product of an irreducible g-module V(λ) and a finite number of special family of g-stable Demazure modules of level l (considered in [15] and [16]), for the current algebra g[t] associated to g, again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the g[t]-module structure of the irreducible module V(lΛ_0 + λ) as a semi-infinite fusion product of finite dimensional g[t]-modules as conjectured in [16]. As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see [7]).

math.RT↗

arXiv2014

A Steinberg type decomposition theorem for higher level Demazure modules

We study Demazure modules which occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a module is isomorphic to the fusion product of "prime" \ Demazure modules, where the prime factors are indexed by dominant integral weights which are either a multiple of $\ell$ or take value less than $\ell$ on all simple coroots. Our proof depends on a technical result which we prove in all the classical cases and $G_2$. Calculations with mathematica show that this result is correct for small values of the level. Using our result, we show that there exist generalizations of $Q$--systems to pairs of weights where one of the weights is not necessarily rectangular and is of a different level. Our results also allow us to compare the multiplicities of an irreducible representation occuring in the tensor product of certian pairs of irreducible representations, i.e., we establish a version of Schur positvity for such pairs of irreducible modules for a simple Lie algebra.

math.RT↗

arXiv2014

Demazure modules, Fusion products and Q--systems

In this paper, we introduce a family of indecomposable finite--dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by a tuple of partitions one for each positive root of the simple Lie algebra. We assume that the partitions satisfy a natural compatibility condition. In the case when the partitions are all rectangular, for instance, we prove that these are precisely the Demazure modules which are stable under the action of the simple Lie algebra. As a consequence we see that the defining relations of these Demazure modules can be greatly simplified. We use this simplified presentation to relate our results to the fusion products (defined by Feigin and Loktev) of representations of the current algebra. We prove that the Q--system can be actually thought of as a canonical short exact of the fusion products of representations associated to certain special partitions. Finally, in the last section we deal with the case of sl(2) and prove that the modules we define are just fusion products of irreducible representations of the associated current algebra. We also give monomial bases for these modules.

math.RT↗

arXiv2012

Unique factorization of tensor products for Kac-Moody algebras

We consider integrable, category O-modules of indecomposable symmetrizable Kac-Moody algebras. We prove that unique factorization of tensor products of irreducible modules holds in this category, upto twisting by one dimensional modules. This generalizes a fundamental theorem of Rajan for finite dimensional simple Lie algebras over C. Our proof is new even for the finite dimensional case, and uses an interplay of representation theory and combinatorics to analyze the Kac-Weyl character formula.

math.RT↗

arXiv2007

Complex ferromagnetic state and magnetocaloric effect in single crystalline Nd_{0.7}Sr_{0.3}MnO_{3}

The magnetocaloric effect in single crystalline Nd_{0.7}Sr_{0.3}MnO_{3} is investigated by measuring the field-induced adiabatic change in temperature which reveals a single negative peak around 130 K well below the Curie temperature (T_C=203 K). In order to understand this unusual magnetocaloric effect, we invoke the reported {55}^Mn spin-echo nuclear magnetic resonance, electron magnetic resonance and polarized Raman scattering measurements on Nd_{0.7}Sr_{0.3}MnO_{3}. We show that this effect is a manifestation of a competition between the double exchange mechanism and correlations arising from coupled spin and lattice degrees of freedom which results in a complex ferromagnetic state. The critical behavior of Nd_{0.7}Sr_{0.3}MnO_{3} near Curie temperature is investigated to study the influence of the coupled degrees of freedom. We find a complicated behavior at low fields in which the order of the transition could not be fixed and a second-order-like behavior at high fields.

cond-mat.mtrl-sci↗

arXiv2005

Towards a Group Theoretic Quantum Encryption Scheme Based on Generalized Hidden Subgroup Problem

This paper introduces a completely new approach to encryption based on group theoretic quantum framework. Quantum cryptography has essentially focused only on key distribution and proceeded with classical encryption algorithm with the generated key. Here, we present a first step towards a quantum encryption scheme based on the solution for the hidden subgroup problem. The shared secret key K from QKD evolves as a generator for a subgroup H of a group G, in combination of the plain text data modeled as group elements. The key K helps in regeneration of the plain data on the receiver's side based on subgroup reconstruction. This paper models all quantum computations using group representations. A non-constructive proof is attempted towards the security of the encryption scheme. We also address the issues involved in a such a venture into the realms of Quantum data encryption.

cs.DM↗

arXiv2005

A QKD Protocol Extendable to Support Entanglement and Reduce Unauthorized Information Gain by Randomizing the Bases Lists with Key Values and Invalidate Explicit Privacy Amplification

This paper suggests an improvement to the BB84 scheme in Quantum key distribution. The original scheme has its weakness in letting quantifiably more information gain to an eavesdropper during public announcement of unencrypted bases lists. The security of the secret key comes at the expense of the final key length. We aim at exploiting the randomness of preparation (measurement) basis and the bit values encoded (observed), so as to randomize the bases lists before they are communicated over the public channel. A proof of security is given for our scheme and proven that our protocol results in lesser information gain by Eve in comparison with BB84 and its other extensions. Moreover, an analysis is made on the feasibility of our proposal as such and to support entanglement based QKD. The performance of our protocol is compared in terms of the upper and lower bounds on the tolerable bit error rate. We also quantify the information gain (by Eve) mathematically using the familiar approach of the concept of Shannon entropy. The paper models the attack by Eve in terms of interference in a multi-access quantum channel. Besides, this paper also hints at the invalidation of a separate privacy amplification step in the "prepare-and-measure" protocols in general.

quant-ph↗