scholarly journals A factorisation theorem for the coinvariant algebra of a unitary reflection group

2020 ◽  
Vol 114 (6) ◽  
pp. 631-639
Author(s):  
G. I. Lehrer
Keyword(s):  
Reflection Group ◽  
Unitary Reflection ◽  
Unitary Reflection Group ◽  
Coinvariant Algebra

scholarly journals On the centralizer algebra of the unitary reflection group G(m,p,n)

1997 ◽  
Vol 148 ◽  
pp. 113-126 ◽  
Author(s):  
Kenichiro Tanabe
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Symmetric Group ◽  
Reflection Group ◽  
Unitary Reflection ◽  
Unitary Reflection Group ◽  
Group Case ◽  
Centralizer Algebra

AbstractThe imprimitive unitary reflection group G(m, p, n) acts on the vector space V =Cn naturally. The symmetric group Sk acts on ⊗kV by permuting the tensor product factors. We show that the algebra of all matrices on ⊗kV commuting with G(m, p, n) is generated by Sk and three other elements. This is a generalization of Jones’s results for the symmetric group case [J].

CHARACTERIZATION FOR THE MODULAR PARTY ALGEBRA

2008 ◽  
Vol 17 (08) ◽  
pp. 939-960 ◽  
Author(s):  
M. KOSUDA
Keyword(s):  
Positive Integer ◽  
Endomorphism Ring ◽  
Reflection Group ◽  
Generators And Relations ◽  
Unitary Reflection ◽  
Unitary Reflection Group

In this paper, we give a characterization for the modular party algebra Pn,r(Q) by generators and relations. By specializing the parameter Q to a positive integer k, this algebra becomes the centralizer of the unitary reflection group G(r, 1, k) in the endomorphism ring of V⊗n under the condition that k ≥ n.

scholarly journals Canonical Systems of Basic Invariants for Unitary Reflection Groups

2016 ◽  
Vol 59 (3) ◽  
pp. 617-623 ◽  
Author(s):  
Norihiro Nakashima ◽  
Hiroaki Terao ◽  
Shuhei Tsujie
Keyword(s):  
Explicit Formula ◽  
Reflection Group ◽  
Canonical System ◽  
Reflection Groups ◽  
Canonical Systems ◽  
The Third ◽  
Unitary Reflection ◽  
Unitary Reflection Group

AbstractIt is known that there exists a canonical system for every finite real reflection group. In a previous paper, the first and the third authors obtained an explicit formula for a canonical system. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

scholarly journals Generalized Coinvariant Algebras for $G(r,1,n)$ in the Stanley-Reisner setting

10.37236/8109 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Daniël Kroes
Keyword(s):  
Polynomial Ring ◽  
Reflection Group ◽  
Roots Of Unity ◽  
Complex Reflection ◽  
Complex Reflection Group ◽  
Positive Integers ◽  
Coinvariant Algebra

Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n \times n$ monomial matrices whose entries are $r^{\textrm{th}}$ roots of unity and let $0 \leq k \leq n$ be an integer. Recently, Haglund, Rhoades and Shimozono ($r=1$) and Chan and Rhoades ($r>1$) introduced quotients $R_{n,k}$ (for $r>1$) and $S_{n,k}$ (for $r \geq 1$) of the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ in $n$ variables, which for $k=n$ reduce to the classical coinvariant algebra attached to $G_n$. When $n=k$ and $r=1$, Garsia and Stanton exhibited a quotient of $\mathbb{C}[\mathbf{y}_S]$ isomorphic to the coinvariant algebra, where $\mathbb{C}[\mathbf{y}_S]$ is the polynomial ring in $2^n-1$ variables whose variables are indexed by nonempty subsets $S \subseteq [n]$. In this paper, we will define analogous quotients that are isomorphic to $R_{n,k}$ and $S_{n,k}$.

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