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Rohit Joshi

Publications and source records attributed to Rohit Joshi.

11 recordsLinked to original sources
arXiv2026

Characteristic Classes Of Representations Of Lie Groups

An irreducible representation of a reductive Lie algebra, when restricted to a Cartan subalgebra, decomposes into weights with multiplicity. The first part of this paper outlines a procedure to compute symmetric polynomials (e.g., power sums) of this multiset of weights, as functions of the highest weight. Next, let G be a connected reductive complex algebraic group with maximal torus T. We express the restrictions of the Chern classes of irreducible representations of G to T, as polynomial functions in the highest weight. We do the same for Stiefel-Whitney classes of orthogonal representations.

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arXiv2025

Divisibility of Character Values of Representations of Coxeter Groups

Let $d$ be a positive integer. We study the proportion of irreducible characters of infinite families of irreducible Coxeter groups whose values evaluated on a fixed element $g$ are divisible by $d$. For Coxeter groups of types $A_n, B_n$ and $D_n$, the proportion tends to $1$ as $n$ approaches infinity. For Dihedral groups, which are Coxeter groups of type $I_2(n)$, we compute the limit of the proportion.

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arXiv2023

Stiefel-Whitney Classes Of Representations Of Dihedral Groups

We compute the Stiefel-Whitney Classes for representations of dihedral groups $D_m$ in terms of character values of order two elements. We also provide criteria to identify representations V which lift to the double covers of the orthogonal group O(V ) and those with non-trivial mod 2 Euler class.

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arXiv2023

Central Extensions and Cohomology

Let G be a group which is topologically a CW-complex, BG a classifying space for G, and A a discrete abelian group. To a central extension of G by A, we associate a cohomology class in $H^2(BG;A)$. We prove this association is injective, and bijective in many cases. A homomorphism of such groups lifts to a central extension iff the pullback of the associated cohomology class vanishes.

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arXiv2021

Stiefel Whitney Classes for Real representations of $\mathrm{GL}_2(\mathbb{F}_q)$

We compute the total Stiefel Whitney class for a real representation $π$ of $\mathrm{GL}_2(\mathbb{F}_q)$, where $q$ is odd. The obstruction class of $π$ is defined to be the Stiefel Whitney class of lowest positive degree that does not vanish. We provide an expression for the obstruction class of $π$ in terms of its character values if $\detπ=1$.

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arXiv2021

On Opportunistic Selection of Common Randomness and LLR generation for Algebraic Group Secret-Key Generation

It is well known that physical-layer key generation methods enable wireless devices to harvest symmetric keys by accessing the randomness offered by the wireless channels. Although two-user key generation is well understood, group secret-key (GSK) generation, wherein more than two nodes in a network generate secret-keys, still poses open problems. Recently, Manish Rao et al., have proposed the Algebraic Symmetrically Quantized GSK (A-SQGSK) protocol for a network of three nodes wherein the nodes share quantized versions of the channel realizations over algebraic rings, and then harvest a GSK. Although A-SQGSK protocol guarantees confidentiality of common randomness to an eavesdropper, we observe that the key-rate of the protocol is poor since only one channel in the network is used to harvest GSK. Identifying this limitation, in this paper, we propose an opportunistic selection method wherein more than one wireless channel is used to harvest GSKs without compromising the confidentiality feature, thereby resulting in remarkable improvements in the key-rate. Furthermore, we also propose a log-likelihood ratio (LLR) generation method for the common randomness observed at various nodes, so that the soft-values are applied to execute LDPC codes based reconciliation to reduce the bit mismatches among the nodes.

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arXiv2020

Spinoriality of orthogonal representations of reductive groups

Let G be a connected reductive group over a field of characteristic zero, and consider an orthogonal representation of G. We give a simple criterion for whether the representation lifts to the spin group, in terms of the highest weights of the irreducible constituents of the representation.

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arXiv2020

Group Secret-Key Generation using Algebraic Rings in Wireless Networks

It is well known that physical-layer Group Secret-Key (GSK) generation techniques allow multiple nodes of a wireless network to synthesize a common secret-key, which can be subsequently used to keep their group messages confidential. As one of its salient features, the wireless nodes involved in physical-layer GSK generation extract randomness from a subset of their wireless channels, referred as the common source of randomness (CSR). Unlike two-user key generation, in GSK generation, some nodes must act as facilitators by broadcasting quantized versions of the linear combinations of the channel realizations, so as to assist all the nodes to observe a CSR. However, we note that broadcasting linear combination of channel realizations incurs non-zero leakage of the CSR to an eavesdropper, and moreover, quantizing the linear combination also reduces the overall key-rate. Identifying these issues, we propose a practical GSK generation protocol, referred to as Algebraic Symmetrically Quantized GSK (A-SQGSK) protocol, in a network of three nodes, wherein due to quantization of symbols at the facilitator, the other two nodes also quantize their channel realizations, and use them appropriately over algebraic rings to generate the keys. First, we prove that the A-SQGSK protocol incurs zero leakage to an eavesdropper. Subsequently, on the CSR provided by the A-SQGSK protocol, we propose a consensus algorithm among the three nodes, called the Entropy-Maximization Error-Minimization (EM-EM) algorithm, which maximizes the entropy of the secret-key subject to an upper-bound on the mismatch-rate. We use extensive analysis and simulation results to lay out guidelines to jointly choose the parameters of the A-SQGSK protocol and the EM-EM algorithm.

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arXiv2020

Spinorial Representations of Orthogonal Groups

Let $G$ be a real compact Lie group, such that $G=G^0\rtimes C_2$, with $G^0$ simple. Here $G^0$ is the connected component of $G$ containing the identity and $C_2$ is the cyclic group of order $2$. We give a criterion for whether an orthogonal representation $π: G \to \mathrm{O}(V)$ lifts to $\mathrm{Pin}(V)$ in terms of the highest weights of $π$. We also calculate the first and second Stiefel-Whitney classes of the representations of the Orthogonal groups.

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