scholarly journals Slightly Degenerate Categories and $\mathbb{Z}$-Modular Data

2019 ◽  
Author(s):  
Abel Lacabanne
Keyword(s):  
Reflection Group ◽  
Fusion Category ◽  
Complex Reflection ◽  
Complex Reflection Group ◽  
Modular Data

Abstract Given a slightly degenerate braided pivotal fusion category $\mathscr{C}$, we explain how it naturally gives rise to a $\mathbb{Z}$-modular data. We do not restrict to spherical categories and work with pivotal categories. Finally, we give an interpretation in this framework of the Bonnafé–Rouquier categorification of the $\mathbb{Z}$-modular datum associated to nontrivial family of the cyclic complex 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 $S_{n,k}$.

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

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 438
Author(s):  
Jeong-Yup Lee ◽  
Dong-il Lee ◽  
SungSoon Kim
Keyword(s):  
Reflection Group ◽  
Type B ◽  
Reflection Groups ◽  
Combinatorial Interpretation ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Standard Monomials ◽  
Fully Commutative Elements ◽  
The One

We construct a Gröbner-Shirshov basis of the Temperley-Lieb algebra T ( d , n ) of the complex reflection group G ( d , 1 , n ) , inducing the standard monomials expressed by the generators { E i } of T ( d , n ) . This result generalizes the one for the Coxeter group of type B n in the paper by Kim and Lee We also give a combinatorial interpretation of the standard monomials of T ( d , n ) , relating to the fully commutative elements of the complex reflection group G ( d , 1 , n ) . More generally, the Temperley-Lieb algebra T ( d , r , n ) of the complex reflection group G ( d , r , n ) is defined and its dimension is computed.

scholarly journals Irreducible Modular Representations of the Reflection GroupG(m,1,n)

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
José O. Araujo ◽  
Tim Bratten ◽  
Cesar L. Maiarú
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Reflection Group ◽  
Geometric Construction ◽  
Weyl Groups ◽  
Modular Representations ◽  
Complex Reflection ◽  
Complex Reflection Group ◽  
One Year ◽  
Combinatorial Methods

In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of typeBn. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of typesAnandBn. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of typeG(m,1,n).

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

Automorphism Groups of the Imprimitive Complex Reflection Groups II

2011 ◽  
Vol 18 (02) ◽  
pp. 315-326
Author(s):  
Li Wang
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Automorphism Groups ◽  
Reflection Group ◽  
Reflection Groups ◽  
Group Of Automorphisms ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Scalar Multiple

We prove that the automorphism group Aut (m,p,n) of an imprimitive complex reflection group G(m,p,n) is the product of a normal subgroup T(m,p,n) by a subgroup R(m,p,n), where R(m,p,n) is the group of automorphisms that preserve reflections and T(m,p,n) consists of automorphisms that map every element of G(m,p,n) to a scalar multiple of itself.

scholarly journals ON THE MILNOR MONODROMY OF THE IRREDUCIBLE COMPLEX REFLECTION ARRANGEMENTS

2017 ◽  
Vol 18 (06) ◽  
pp. 1215-1231 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Hyperplane Arrangement ◽  
Reflection Group ◽  
Milnor Fiber ◽  
Complex Reflection ◽  
Complex Reflection Group

Using recent results by Măcinic, Papadima and Popescu, and a refinement of an older construction of ours, we determine the monodromy action on $H^{1}(F(G),\mathbb{C})$ , where $F(G)$ denotes the Milnor fiber of a hyperplane arrangement associated to an irreducible complex reflection group $G$ .

scholarly journals Major Indices and Perfect Bases for Complex Reflection Groups

10.37236/785 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Robert Shwartz ◽  
Ron M. Adin ◽  
Yuval Roichman
Keyword(s):  
Direct Product ◽  
Hilbert Series ◽  
Reflection Group ◽  
Reflection Groups ◽  
Cyclic Subgroups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Mild Conditions ◽  
Major Index ◽  
Complex Reflection Group

It is shown that, under mild conditions, a complex reflection group $G(r,p,n)$ may be decomposed into a set-wise direct product of cyclic subgroups. This property is then used to extend the notion of major index and a corresponding Hilbert series identity to these and other closely related groups.

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