Modified Ariki-Koike Algebra and Yokonuma-Hecke like Relations

IF 0.5 4区 数学 Q3 MATHEMATICS
Myungho Kim, Sungsoon Kim
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引用次数: 0

Abstract

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji’s algebra) \(\mathcal {H}_{n,r}\) over an integral domain R associated with a set of parameters \(q,u_1,\ldots ,u_r\) in R. It turns out that the algebra \(\mathcal {H}_{n,r}\) has a set of generators \(t_1,\ldots ,t_n\) and \(g_1,\ldots g_{n-1}\) subject to some defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of \(\mathcal {H}_{n,r}\) which is independent of the choice of \(u_1,\ldots u_r\). As applications of the presentations, we find an explicit and direct isomorphism between the modified Ariki-Koike algebras with different choices of parameters \((u_1,\ldots ,u_r)\). We also find an explicit trace form on the algebra \(\mathcal {H}_{n,r}\) which is symmetrizing provided the parameters \(u_1,\ldots , u_r\) are invertible in R. We show that the symmetric group \(\mathfrak {S}(r)\) acts on the algebra \(\mathcal H_{n,r}\), and find a basis and a set of generators of the fixed subalgebra \(\mathcal H_{n,r}^{\mathfrak {S}(r)}\).

修正的有熊小池代数和横沼-赫克相似关系
我们发现了在积分域 R 上的、与 R 中的参数集 (q,u_1,\ldots ,u_r/)相关的修正有熊小池代数(也称为庄司代数)的新表述。事实证明,代数(mathcal {H}_{n,r})有一组生成器(t_1,\ldots ,t_n\)和(g_1,\ldots g_{n-1}\),它们的定义关系类似于横沼-贝克(Yokonuma-Hecke)代数的关系。我们还得到了与\(u_1,\ldots u_r\) 的选择无关的\(mathcal {H}_{n,r}\) 的呈现。作为这些呈现的应用,我们发现在参数选择不同的修正有熊科(Ariki-Koike)代数之间存在明确而直接的同构关系(\((u_1,\ldots ,u_r)\)。只要参数 \(u_1,\ldots , u_r\) 在 R 中是可逆的,我们就能在代数 \(\mathcal {H}_{n,r}\) 上找到一个明确的迹形式,它是对称的。我们证明了对称群 \(\mathfrak {S}(r)\) 作用于代数 \(\mathcal H_{n,r}\) ,并找到了固定子代数 \(\mathcal H_{n,r}^{mathfrak {S}(r)}\) 的一个基和一组发电机。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
61
审稿时长
6-12 weeks
期刊介绍: Algebras and Representation Theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including Lie algebras and superalgebras, rings of differential operators, group rings and algebras, C*-algebras and Hopf algebras, with particular emphasis on quantum groups. The journal contains high level, significant and original research papers, as well as expository survey papers written by specialists who present the state-of-the-art of well-defined subjects or subdomains. Occasionally, special issues on specific subjects are published as well, the latter allowing specialists and non-specialists to quickly get acquainted with new developments and topics within the field of rings, algebras and their applications.
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