[Paper Review] Reducibility of the Cohen-Wales representation of the Artin group of type $D_n$

Claire Levaillant|arXiv (Cornell University)|Mar 29, 2011

Geometric and Algebraic Topology27 references3 citations

TL;DR

This paper establishes a reducibility criterion for the Cohen–Wales representation of the Artin group of type $D_n$ by constructing a knot-theoretic linear representation of the CGW algebra of type $D_n$. Using computer-assisted methods in Mathematica, it proves that the representation is generically irreducible but becomes reducible precisely when the parameters $t$ and $r$ satisfy $t \in \{r^{4n-4}, r^{2n-4}, -r^{2n-2}, 1, r^4, -1\}$, under specified constraints on $r$. This resolves a key structural question about the faithfulness and decomposition of this representation.

ABSTRACT

Using knot theory, we construct a linear representation of the CGW algebra of type $D_n$. This representation has degree $n^2-n$, the number of positive roots of a root system of type $D_n$. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type $D_n$, this representation is equivalent to the faithful linear representation of Cohen-Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type $D_n$. Our proof is computer-assisted using Mathematica.

Motivation & Objective

• To determine the conditions under which the Cohen–Wales representation of the Artin group of type $D_n$ becomes reducible.
• To construct a linear representation of the CGW algebra of type $D_n$ using knot-theoretic methods.
• To establish a precise reducibility criterion for the Cohen–Wales representation in terms of its parameters $t$ and $r$.
• To investigate the semisimplicity of the CGW algebra of type $D_n$ and provide a conjecture based on the representation's reducibility.

Proposed method

• Construct a linear representation of the CGW algebra of type $D_n$ using techniques from knot theory.
• Define the representation space as spanned by vectors indexed by the $n^2 - n$ positive roots of the $D_n$ root system.
• Use a computer-assisted approach with Mathematica to verify irreducibility and reducibility conditions.
• Relate the constructed representation to the Cohen–Wales representation via equivalence under specific parameter transformations.
• Analyze the decomposition of the representation space using Specht modules and branching rules for the Hecke algebra $\mathcal{H}(D_n)$.
• Apply results from [5] to confirm equivalence between the constructed representation and the original Cohen–Wales representation.

Experimental results

Research questions

• RQ1 Under what conditions on the parameters $t$ and $r$ does the Cohen–Wales representation of the Artin group of type $D_n$ become reducible?
• RQ2 Can a knot-theoretic construction yield a faithful linear representation of the CGW algebra of type $D_n$?
• RQ3 Is the Cohen–Wales representation irreducible for generic values of $t$ and $r$, and if not, what are the exceptions?
• RQ4 Does the reducibility of the representation imply non-semisimplicity of the CGW algebra of type $D_n$?
• RQ5 What is the precise structure of the invariant subspaces in the Cohen–Wales representation when $t$ lies in the exceptional set?

Key findings

• The Cohen–Wales representation of the Artin group of type $D_n$ is generically irreducible, but becomes reducible when $t \in \{r^{4n-4}, r^{2n-4}, -r^{2n-2}, 1, r^4, -1\}$, under the given constraints on $r$.
• The constructed representation of the CGW algebra of type $D_n$ is equivalent to the Cohen–Wales representation as a representation of the Artin group.
• The representation is shown to be reducible for the specified parameter values, and this reducibility is confirmed via computer-assisted calculations in Mathematica.
• The Specht modules $S^{(0),(4,3)}$ and its conjugate do not occur as submodules in the Cohen–Wales space $V_7$, which supports the reducibility criterion.
• The $rac{n(n-1)}{2}$-dimensional subspace of $V_n$ spanned by the $t_{ij}$ vectors is irreducible, which is essential for the main reducibility result.
• The CGW algebra of type $D_n$ is not semisimple when $t$ lies in the exceptional set, as the representation fails to be completely reducible.