[Paper Review] Birman-Murakami-Wenzl type algebras for arbitrary Coxeter systems

Zhi Chen|arXiv (Cornell University)|Aug 16, 2017

Advanced Combinatorial Mathematics14 references3 citations

TL;DR

This paper constructs Birman-Murakami-Wenzl (BMW) type algebras $B_{I_{2}(k)}(m,l)$ for arbitrary dihedral Coxeter systems $I_{2}(k)$, generalizing the original BMW algebras beyond simply-laced types. It establishes semisimplicity, cellularity, and a degree-3 annihilating polynomial for Artin group generators, and identifies a $k$-dimensional Lawrence-Krammer-type representation that conjecturally matches Marin’s generalized monodromy representation.

ABSTRACT

In this paper we first present a Birman-Murakami-Wenzl type algebra for every Coxeter system of rank 2 (corresponding to dihedral groups). We prove they have semisimple for generic parameters, and having natural cellular structures. And classcify their irreducible representations. Among them there is one serving as a generalization of the Lawrence-Krammer representation with quite neat shape and the "correct" dimension. We conjecture they are isomorphic to the generalized Lawrence-Krammer representaions defined by I.Marin as monodromy of certain KZ connections. We prove these representations are irreducible for generic parameters, and find a quite neat invariant bilinear form on them. Based on above constructions for rank 2, we introduce a Birman-Murakami-Wenzl type algebra for an arbitrary Coxeter system. For every Coxeter system, the introduced algebra is a quotient of group algebra of the Artin group (associated with this Coxeter system), having the corresponding Hecke algebra as a quotient. The simple generators of the Artin group have degree 3 annihiating polynomials in this algebra.

Motivation & Objective

• To construct a generalization of the Birman-Murakami-Wenzl algebra for non-simply-laced Coxeter systems, particularly dihedral types $I_{2}(k)$.
• To resolve the challenge of incorporating multiple edges (non-simply-laced relations) in BMW-type algebras, which previously prevented generalization beyond ADE types.
• To define an algebra that supports a Lawrence-Krammer-type representation of the Artin-Tits group $A_{I_{2}(k)}$ with explicit, neat matrix form.
• To conjecture that this representation is isomorphic to Marin’s generalized Lawrence-Krammer representation defined via monodromy of flat connections.
• To provide a systematic method to determine the structure constants of the algebra without explicit integration of flat connections, by leveraging algebraic constraints like cellularity and involution.

Proposed method

• Define a new BMW-type algebra $B_{I_{2}(k)}(m,l)$ for dihedral Coxeter systems $I_{2}(k)$ using a set of generators and relations derived from Brauer-type algebras and deformation constraints.
• Use algebraic testing and symmetry considerations to guess the structure constants in the defining relations, particularly focusing on ensuring the existence of an involution and cellular structure.
• Construct a $k$-dimensional representation of the Artin-Tits group $A_{I_{2}(k)}$ within $B_{I_{2}(k)}(m,l)$, with a clean matrix form and an invariant bilinear form.
• Prove that the algebra satisfies key algebraic properties: semisimplicity, cellularity, and the existence of a natural involution.
• Demonstrate that the Artin group generators satisfy a degree-3 annihilating polynomial in the algebra, and that the algebra surjects onto the corresponding Hecke algebra.
• Use operator-theoretic analysis of iterated actions of $X_0X_1$ on a basis to verify the key braid-like relations, showing that the central element $ riangle$ acts as scalar multiplication.

Experimental results

Research questions

• RQ1 Can a Birman-Murakami-Wenzl-type algebra be constructed for arbitrary Coxeter systems, particularly non-simply-laced ones like $I_{2}(k)$ for $k eq 3$?
• RQ2 What are the correct structure constants for such an algebra that ensure semisimplicity, cellularity, and compatibility with the Artin group action?
• RQ3 Does the resulting algebra support a Lawrence-Krammer-type representation of the Artin-Tits group $A_{I_{2}(k)}$ with a neat matrix form?
• RQ4 Is this representation isomorphic to Marin’s generalized Lawrence-Krammer representation defined via monodromy of a flat connection?
• RQ5 Can the monodromy of Marin’s flat connection be computed algebraically without explicit integration, by deriving the correct algebraic structure?

Key findings

• The algebra $B_{I_{2}(k)}(m,l)$ is semisimple and admits a cellular structure, satisfying key algebraic conditions of the original BMW algebra.
• The algebra supports a $k$-dimensional irreducible representation of the Artin-Tits group $A_{I_{2}(k)}$ with a neat matrix form and an invariant bilinear form.
• For odd $k = 2n+1$, the representation is $2n+1$-dimensional; for even $k = 2n$, there are two $n$-dimensional irreducible representations.
• The Artin group generators in the algebra satisfy a degree-3 annihilating polynomial, a hallmark of BMW-type algebras.
• The algebra surjects onto the corresponding Hecke algebra $H_{I_{2}(k)}(v)$, and is a quotient of the group algebra $bQ[m,l^{ imes}]A_{I_{2}(k)}$.
• The central element $ riangle$ acts as scalar multiplication by $l_0 v_0^{-1}$ on the representation space, confirming consistency with braid group relations.