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 groups.
We introduce the notion of reflections for selfinjective algebras from the point of view of torsion theories induced by two-term tilting complexes. As an application, we determine the transformations of Brauer trees associated with reflections. In particular, we provide a way to transform every Brauer tree into a Brauer line.
We introduce Brauer algebras associated to complex reflection groups of type $G(m,p,n)$, and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.
The Brauer-Chen algebra is a generalization of the algebra of Brauer diagrams to arbitrary complex reflection groups, that admits a natural monodromic deformation. We determine the generic representation theory of the first non trivial quotient of this algebra. We also define natural extensions of this algebra and prove that they similarly admit natural monodromic deformations.
There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra in characteristic zero. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra in characteristic zero can be reduced to a study of the block containing the weight 0. We define parabolic Kazhd…
We introduce a Brauer type algebra $B_G (Υ) $ associated with every pseudo reflection group and every Coxeter 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 ({\it J. Algebra}, {\bf 280} (2005), 107-153). We also prove $B_G (Υ)$ has a cellular structure and be semisimple for generic parameters when $G$ is a dihedral group or the type $H_3$ Coxeter group. Moreover, in the process of construction, we introduce a further generalization of Lawrence-Krammer repr…
A complete system of pairwise orthogonal minimal idempotents for Birman-Murakami-Wenzl algebras is obtained by a consecutive evaluation of a rational function in several variables on sequences of quantum contents of up-down tableaux. A by-product of the construction is a one-parameter family of fusion procedures for Hecke algebras. Classical limits to two different fusion procedures for Brauer algebras are described.
We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. In particular, this provides a new fusion procedure for the symmetric group involving an arbitrary parameter. The R-matrices are solutions of the Yang--Baxter equation associated with the classical Lie algebras g_N of types B, C and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra B(g_N) to U(g_N) and show that the fusion procedure provi…
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.
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.
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.
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.
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…
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.
We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the elements of G(de,e,n) over some generating set. Using these geodesic normal forms, we construct intervals in G(e,e,n) that give rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we study some of their properties and compute their second integral homology groups. Inspired by th…
We define Lie subalgebras of the group algebra of a finite pseudo-reflection group that are involved in the definition of the Cherednik KZ-systems, and determine their structure. We provide applications for computing the Zariski closure of the image of generalized (pure) braid group B inside the representations of the corresponding Hecke algebras. We also get unitarizability results for the representations of B originating from Hecke algebras for suitable parameters, and relate our Lie algebras with the topological closure of B in these compact cases.
The paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group $G$, a subgroup $K$), we construct categories, whose morphisms are two-dimensional surfaces tiled by polygons and colored in a certain way. A product of morphisms is a gluing of combinatorial bordisms. For a unitary representation of $G$ we assign a functor from the category of bordisms to the category of Hilbert spaces and bounded operators. The construction has numerous variations, instead of surfaces there arise also one-dimensional objects of Brauer diagram type, …
