In this paper, we determine the modular invariants of finite modular pseudo-reflection subgroups of the finite general linear group $ \text{GL}_n(q) $ acting on the tensor product of the symmetric algebra $ S^{\bullet}(V) $ and the exterior algebra $ \wedge^{\bullet}(V) $ of the natural $ \text{GL}_n(q) $-module $ V $. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ \text{GL}_n(q) $ which is a generalization of Weyl group of Cartan type Lie algebra.
As a sequel of \cite{Ou}, in this shot note, we investigate block basis for coinvariants of finite modular pseudo-reflection groups. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ GL_n(q) $ which generalizes the Weyl groups of restricted Cartan type Lie algebra.
We present a way to associate an algebra $B_G (Υ) $ with every pseudo reflection group $G$. When $G$ is a Coxeter group of simply-laced type we show $B_G (Υ)$ is isomorphic to the generalized Brauer algebra of simply-laced type introduced by Cohen,Gijsbers and Wales[10]. We prove $B_G (Υ)$ has a cellular structure and be semisimple for generic parameters when $G$ is a rank 2 Coxeter group. In the process of construction we introduce a Cherednik type connection for BMW algebras and a generalization of Lawrence-Krammer representation to complex braid groups associated with all pseudo reflection …
We show that the Eigenspace Representations for $\mathbb{R}^{n}$ semidirect with a finite pseudo-reflection group $K$, which satisfy some generic property are equivalent to the induced representations from $\mathbb{R}^{n}$ to $\mathbb{R}^{n} \rtimes K$, which satisfy the same property by Mackey little group method.And the proof of the equivalence is by using matrix coefficients and invariant theory.As a consequence, these eigenspace representations are irreducible.
The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the "philosophy of deformations" both as a conceptual and computational tool, and suggest possible connections to Lusztig's "philosophy of generations". On the geometric side one can understand deformations in terms of localisation in equivariant cohomology. Recently Treumann and Leslie-Lonergan have added Smith theory, which provides a useful tool when considering mod p coefficients. In this context, we make contact with some remarkable work of Hazi. Using recent work…
In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Broué,Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They ar…
First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field. Special case of the result is a finite presentation of the algebra of multisymmetric polynomials. Reducedness of the invariant commuting scheme is proved as a by-product. The algebra of multisymmetric polynomials over an arbitrary base ring is revisited.
For each subgroup of GL_2(F_p) or order divisible by p, generated by (pseudo-)reflections, we compute the ideals of stable and generalized invariants. These groups and these ideals are related to the cohomology of compact Lie groups, Kac-Moody groups and p-compact groups.
We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "top-down" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the s…
This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic $0$. Our most general result determines the top component in total degree, which we prove for all Shephard--Todd groups $G(m, p, n)$ with $m \neq p$ or $m=1$. Our strongest result gives tight bi-degree bounds and is proven for all $G(m, 1, n)$, which includes the Weyl groups of types $A$ and $B$/$C$. For symmetric groups (i.e. type $A$), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund--…
In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of reflections in the universal Coxeter group is defined in association with a mutation of a quiver. A matrix representation of these reflections is determined by a linear ordering on the set of vertices of the quiver. It was conjectured that there exists an ordering (called a pseudo-acyclic ordering in this paper) such that whenever two mutation sequences of a quiver lead to the same labeled seed, the representations of the associated reflections also coincide. In this paper, we prove this conjecture for every quiver mutation-equ…
How far can the elementary description of centralizers of parabolic subalgebras of Hecke algebras of finite real reflection groups be generalized to the complex reflection group case? In this paper we begin to answer this question by establishing results in two directions. First, under conditions closely analogous to those existing for the real case, we give explicit relations between coefficients in an element centralizing a generator. Second, we introduce a tool for dealing with a major challenge of the complex case -- the ``instability'' of certain double cosets -- through the definition an…
We show that the projectivized complex reflection group $Γ$ of the unique $(1+i)$-modular Hermitian $\mathbb{Z}[i]$-module of signature $(9,1)$ is a new arithmetic reflection group in $PU(9,1)$. We find $32$ complex reflections of order four generating $Γ$. The mirrors of these $32$ reflections form the vertices of a sort of Coxeter-Dynkin diagram $D$ for $Γ$ that encode Coxeter-type generators and relations for $Γ$. The vertices of $D$ can be indexed by sixteen points and sixteen affine hyperplanes in $\mathbb{F}_2^4$. The edges of $D$ are determined by the finite geometry of these points and…
Let $\ell$ be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an $\ell$-Sylow subgroup. The classification assists in describing the $\ell$-Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an $\ell$-Sylow subgroup is $G$ itself, the reflection subgroup within th…
We give a modular branching rule for certain wreath products as a generalization of Kleshchev's modular branching rule for the symmetric groups. Our result contains a modular branching rule for the complex reflection groups $G(m,1,n)$ (which are often called the generalized symmetric groups) in splitting fields for $\mathbb{Z}/m\mathbb{Z}$. Especially for $m=2$ (which is the case of the Weyl groups of type $B$), we can give a modular branching rule in any field. Our proof is elementary in that it is essentially a combination of Frobenius reciprocity, Mackey theorem, Clifford's theory and Klesh…
In this note we characterize those unitary one-parameter groups U^c which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary representation $U$ of the circle group. These are precisely the ones for which there exists an anti-unitary involution $J$ commuting with $U^c$. This provides an interesting link with the modular data arising in Tomita--Takesaki theory. Introducing the concept of a positive definite function with values in the space of sesquilinear forms, we further establish a link b…
After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W, through a `periodic' family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply |A| >= n(n+1)/2 for n = dim V, and relate the W-equivariance of the corresponding map with the period of our family.
In this article we study rational Cherednik algebras at $t = 1$ in positive characteristic. We study a finite dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series $G(m,d,n)$, we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.
The aim of this note is to study a generalization of theorems by James and Fayers on the modular representations of the symmetric group and its Hecke algebra to the case of the complex reflection groups of type $G(l,1,n)$ and the associated Ariki-Koike algebra.
