Hyeonmi Lee

Rigged configuration descriptions of the crystals B(∞) and B(λ) for special linear Lie algebras

The rigged configuration realization RC(∞) of the crystal B(∞) was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements of RC(∞) explicitly for the An-type case. Two separate descriptions of RC(∞) are obtained. These lead naturally to isomorphisms RC(∞)≅T(∞) and RC(∞)≅T¯(∞), i.e., those with the marginally large tableau and marginally large reverse tableau realizations of B(∞), that may be computed explicitly. We also present two descriptions of the irreducible highest weight crystal B(λ) in terms of rigged configurations. These are obtained by combining our two descriptions of RC(∞), the two mentioned isomorphisms, and two existing realizations of B(λ) that were based on T(∞) and T¯(∞).

ANALYSIS OF POSSIBLE PRE-COMPUTATION AIDED DLP SOLVING ALGORITHMS

【A trapdoor discrete logarithm group is a cryptographic primitive with many applications, and an algorithm that allows discrete logarithm problems to be solved faster using a pre-computed table increases the practicality of using this primitive. Currently, the distinguished point method and one extension to this algorithm are the only pre-computation aided discrete logarithm problem solving algorithms appearing in the related literature. This work investigates the possibility of adopting other pre-computation matrix structures that were originally designed for used with cryptanalytic time memory tradeoff algorithms to work as pre-computation aided discrete logarithm problem solving algorithms. We find that the classical Hellman matrix structure leads to an algorithm that has performance advantages over the two existing algorithms.】

Crystal ℬ(λ) as a Subset of the Tableau Description of ℬ(∞) for the Classical Lie Algebra Types

We study the crystal base of the negative part of a quantum group. An explicit realization of the crystal is given in terms of Young tableaux for types

CRYSTAL B(λ) IN B(∞) FOR G 2 TYPE LIE ALGEBRA

A_n

Crystal ℬ(∞) for Quantum Affine Algebras: A Young Wall Description

,

The crystal

B_n

\mathcal {B}(\infty )

,

and extended Nakajima monomials

C_n

Young tableaux and crystal B(∞) for finite simple Lie algebras☆

,

Young tableaux and crystal B(∞) for the exceptional Lie algebra types

D_n