Hecke algebras, Specht modules and Gröbner–Shirshov bases

Abstract

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. 

Invariant tensors for the spin representation of (7)

“…This theory originates from the diamond lemma in Newman (1942) and our account is based on (Sims, 1994, Chapter 2). This theory has been applied to the Hecke and Temperley-Lieb algebras in Kang et al (2002). Our approach differs from these standard approaches in that we take A P (n) to be a quotient of the free algebra on a commutation monoid (instead of a free monoid) and we only require a reduction order to be a partial order (instead of a linear order).…”

Representations of Ariki–Koike Algebras and Gröbner–Shirshov Bases

“…; T ðkþ1;0Þ À1 are the same as the dening relations of H j ðkÞ j ðqÞ given in [9]. Also we get the Specht module S ðkÞ q over H j ðkÞ j ðqÞ dened in [9].…”

“…Moreover, we show that the standard monomials are in 1-1 correspondence with the cozy tableaux of shape . The results of this paper can be considered not only as a generalization but also as a renement of the results on the nite Hecke algebras of type A obtained in [9]. In some of the proofs in this paper, we quote the results and notation of [9] without presenting them thoroughly.…”

“…The results of this paper can be considered not only as a generalization but also as a renement of the results on the nite Hecke algebras of type A obtained in [9]. In some of the proofs in this paper, we quote the results and notation of [9] without presenting them thoroughly. The readers may refer to [9] for more details.…”

“…In some of the proofs in this paper, we quote the results and notation of [9] without presenting them thoroughly. The readers may refer to [9] for more details.…”