Gröbner–Shirshov Bases for Irreducible sln+1-Modules

Abstract

We determine the Gröbner-Shirshov bases for finite-dimensional irreducible representations of the special linear Lie algebra sl n+1 and construct explicit monomial bases for these representations. We also show that each of these monomial bases is in 1-1 correspondence with the set of semistandard Young tableaux of a given shape.

Representations of Ariki–Koike Algebras and Gröbner–Shirshov Bases

“…In this section, we briey recall the Gr€ o obner--Shirshov basis theory for the representations of associative algebras which was developed in [7,8].…”

“…We construct the Specht module S as a quotient of H r;1;n , obtaining a presentation given by generators and relations. One of the main ingredients of our approach is the Gr€ o obner --Shirshov basis theory for the representations of associative algebras developed in [7,8], where one can nd the motivation and applications as well as the exposition of the theory. This approach naturally enables us to construct a linear basis of S consisting of standard monomials.…”

“…In the rst section, we briey explain the Gr€ o obner--Shirshov basis theory for the representations of associative algebras developed in [7,8]. In x 2, we recall the denition of the Ariki --Koike algebra H r;1;n and determine its Gr€ o obner --Shirshov basis, which yields a monomial basis of H r;1;n .…”

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 .…”

Gröbner-Shirshov Bases: Some New Results

“…Kang and K.-H. Lee in [36] and [37]. According to their approach, a Gröbner-Shirshov basis of a cyclic module M over an algebra A is a pair (S, T ), where S is the set of the defining relations of A, A = k X|S , and T is the defining relations for the A-module A M = A M(e|T ).…”