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

Abstract

In this note, we find a monomial basis of the cyclotomic Hecke algebra H r, p,n of G(r, p, n) and show that the Ariki-Koike algebra H r,n is a free module over H r, p,n , using the Gröbner-Shirshov basis theory. For each irreducible representation of H r, p,n , we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.

Some Cases on Cozy Partitions

“…Also, Kang et al established an explicit isomorphic between cozy and standard tableaux of shape 𝜔. These relations are commonly used for connecting to Hecke algebras and Specht modules; (see [2], [3], [4], [5], [6], and [7]). To the best of our knowledge, there is no rule for counting the number of cozy 𝜔-tableaux.…”

“…( 1) has the following form 𝑐𝑜𝑧𝑦 (4,3) = (𝑐𝑜𝑧𝑦 (3,2) + 𝑐𝑜𝑧𝑦 (4,2) ). 𝑐𝑜𝑧𝑦 (5,3) = (𝑐𝑜𝑧𝑦 (3,2) + 𝑐𝑜𝑧𝑦 (4,2) + 𝑐𝑜𝑧𝑦 (5,2) ). 𝑐𝑜𝑧𝑦 (6,3) = (𝑐𝑜𝑧𝑦 (3,2) + 𝑐𝑜𝑧𝑦 (4,2) + 𝑐𝑜𝑧𝑦 (5,2) + 𝑐𝑜𝑧𝑦 (6,2) ).…”

“…4(𝑐𝑜𝑧𝑦 (4,2) ) = 4(𝑐𝑜𝑧𝑦 (2,1) + 𝑐𝑜𝑧𝑦 (3,1) + 𝑐𝑜𝑧𝑦 (4,1) ). 3(𝑐𝑜𝑧𝑦 (5,2) ) = 3(𝑐𝑜𝑧𝑦 (2,1) + 𝑐𝑜𝑧𝑦 (3,1) + 𝑐𝑜𝑧𝑦 (4,1) + 𝑐𝑜𝑧𝑦 (5,1) ). 2(𝑐𝑜𝑧𝑦 (6,2) ) = 2(𝑐𝑜𝑧𝑦 (2,1) + 𝑐𝑜𝑧𝑦 (3,1) + 𝑐𝑜𝑧𝑦 (4,1) + 𝑐𝑜𝑧𝑦 (5,1) + 𝑐𝑜𝑧𝑦 (6,1) ).…”

Gröbner-Shirshov Bases for Temperley-Lieb Algebras of Complex Reflection Groups

“…The Gröbner-Shirshov bases for Coxeter groups of classical and exceptional types were completely determined by Bokut, Lee et al in [3,16,18,19,25]. The cases for Hecke algebras and Temperley-Lieb algebras of type A as well as for Ariki-Koike algebras were calculated by Lee et al in [11,12,17]. This paper consists of two principal parts as follows : 1) In the first part of this paper, extending the result for type B n in [13], we construct a Gröbner-Shirshov basis for the Temperley-Lieb algebra T (d, n) of the complex reflection group of type G(d, 1, n) and compute the dimension of T (d, n), by enumerating the standard monomails which are in bijection with the fully commutative elements.…”