Representations of Ariki–Koike Algebras and Gröbner–Shirshov Bases

Abstract

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.

Gröbner–Shirshov bases and their calculation

“…Applications of Theorem 8 appeared in [125][126][127]. Take two sets X and Y and consider the free left k X -module Mod k X Y with k X -basis Y .…”

Cyclotomic Hecke Algebras of G(r, p, n)

“…The summary on the theory of Gröbner-Shirshov pairs is given in [10,11,15] with some computational examples. An algorithmic issue for finding Gröbner-Shirshov bases was dealt with in [9].…”

“…We consider the bijection ζ : CZ(λ) → ST(λ) between the labeling schemes of the bases of S λ [11,Example 4.6]. Then, corresponding to the shift operator on the set ST(λ) of standard tableaux of shape λ in [2, Section 2], we define a shift operator on the set CZ(λ) of cozy tableaux t = (t (k,l ) ) of shape λ by |λ|i (ζ(t)) .…”

“…Lee [10,11]. In this paper, we want to obtain linear bases for Hecke algebras of complex reflection groups and their irreducible modules.…”

Composition-diamond lemma for modules

“…They gave applications of this lemma for irreducible modules over sl n (k) [16], Specht modules over Hecke algebras and Ariki-Koike algebras in [17] and [18]. Some years later, E. S. Chibrikov [11] suggested a new Composition-Diamond lemma for modules that treat any module as a factor module of "double-free" module, a free module mod k X Y over a free algebra k X .…”