In this paper, we study the structure of Specht modules over Hecke algebras using the Gröbner-Shirshov basis theory for the representations of associative algebras. The Gröbner-Shirshov basis theory enables us to construct Specht modules in terms of generators and relations. Given a Specht module S λ q , we determine the Gröbner-Shirshov pair (R q , R λ q ) and the monomial basis G(λ) consisting of standard monomials. We show that the monomials in G(λ) can be parameterized by the cozy tableaux. Using the division algorithm together with the monomial basis G(λ), we obtain a recursive algorithm of computing the Gram matrices. We discuss its applications to several interesting examples including Temperley-Lieb algebras. 

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We investigate a recently proposed algorithm solving the strong discrete logarithm problem with auxiliary inputs, and show that this algorithm in general is not more efficient than ordinary discrete-logarithm-solving algorithms such as Pollard’s rho method, by analyzing a lower bound on the sum of digits of integers.

In this paper, we investigate the structure of Ariki–Koike algebras and their Specht modules using Gröbner–Shirshov basis theory and combinatorics of Young tableaux. For a multipartition λ, we find a presentation of the Specht module Sλ given by generators and relations, and determine its Gröbner–Shirshov pair. As a consequence, we obtain a linear basis of Sλ consisting of standard monomials with respect to the Gröbner–Shirshov pair. We show that this monomial basis can be canonically identified with the set of cozy tableaux of shape λ. 2000 Mathematics Subject Classification 16Gxx, 05Exx.

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Abstract. For a finite group G to be used in the MOR public key cryptosystem, it is necessary that the discrete logarithm problem(DLP) over the inner automorphism group Inn(G) of G must be computationally hard to solve. In this paper, under the assumption that the special conjugacy problem of G is easy, we show that the complexity of the MOR system over G is about log |G| times larger than that of DLP over G in a generic sense. We also introduce a group-theoretic method, called the group extension, to analyze the MOR cryptosystem. When G is considered as a group extension of H by a simple abelian group, we show that DLP over Inn(G) can be 'reduced' to DLP over Inn(H). On the other hand, we show that the reduction from DLP over Inn(G) to DLP over G is also possible for some groups. For example, when G is a nilpotent group, we obtain such a reduction by the central commutator attack.