Some Infinite-Dimensional Representations of Certain Coxeter Groups

Abstract

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs.

Appendix: A Sketched Proof of Theorem 1.1

This proof is given by an anonymous referee of a previous version of this paper. The proof uses the following fact.

Lemma 6.1

[5, Corollary 2] Suppose (W, S) is an infinite non-affine irreducible Coxeter group of finite rank. Then there exists a subgroup $Y^\prime \subseteq W$ of finite index and a surjective homomorphism $\varphi ^\prime : Y^\prime \twoheadrightarrow F^\prime$, where $F^\prime$ is a non-abelian free group.

Now we can prove Theorem 1.1, and we only need to prove the “only if” part.

Suppose (W, S) is infinite and non-affine. Let $Y^\prime$, $\varphi ^\prime$, $F^\prime$ be as in Lemma 6.1. Let Y be the intersection of all conjugates of $Y^\prime$ in W. Then Y is a normal subgroup of W of finite index. Since Y is of finite index in $Y^\prime$, the image $\varphi ^\prime (Y)$ is a subgroup of $F^\prime$ of finite index. Therefore, $\varphi ^\prime (Y)$ is also a non-abelian free group (see, for example, [6, Theorem 85.1]). Let $X_1, X_2$ be two free generators of $\varphi ^\prime (Y)$, and let $F = \langle X_1, X_2 \rangle$. Then F is a free group of rank two, and we have a surjection $\varphi ^\prime (Y) \twoheadrightarrow F$. By composing this surjection with the map $\varphi ^\prime |_Y$, we obtain a surjective homomorphism $\varphi : Y \twoheadrightarrow F$.

Let V be an infinite-dimensional irreducible representation of F (for example, $V := \bigoplus _{n \in \mathbb {Z}} \mathbb {C} \alpha _{n}$, and let $X_1 \cdot \alpha _n = \alpha _{n+1}$, $X_2 \cdot \alpha _n = 2^n \alpha _{n+1}$, as in the proof of Theorem 3.2). By pulling back this F-representation along $\varphi$, V becomes an irreducible representation of Y.

We consider the induced representation $V_Y^W: = \mathbb {C}[W] \otimes _{\mathbb {C}[Y]} V$ of W, and we choose $w_1, \dots , w_n$ as coset representatives for Y in W. Then $V_Y^W = \bigoplus _{1 \le i \le n} w_i \otimes V$ as a vector space. Since Y is a normal subgroup of W, each summand $w_i \otimes V$ is an irreducible representation of Y. Thus, $V_Y^W$ is a semisimple Noetherian and Artinian $\mathbb {C}[Y]$-module, and hence a Noetherian and Artinian $\mathbb {C}[W]$-module. Consequently, there exists an irreducible W-representation $M \subseteq V_Y^W$ as a subrepresentation.

If we view M as a $\mathbb {C}[Y]$-module, then, by semi-simplicity of the $\mathbb {C}[Y]$-module $V_Y^W$ and Schur’s lemma, M is isomorphic to a direct sum of some $\mathbb {C}[Y]$-modules $w_i \otimes V$. In particular, M is infinite dimensional. Theorem 1.1 is proved.