McKay quivers of the groups G(r, m, n)
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Date
2018
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University of New Brunswick
Abstract
The following thesis proves a simple combinatorial description of the McKay quivers of the infinite family of complex reflection groups typically denoted G(r, m, n). These are the groups sometimes referred to as the finite imprimitive reflection groups, see [4].
Our approach is combinatorial in nature - identifying the irreducible representations of the groups through the use of specific configurations of Young tableaux, and the action of the groups thereon.
This project is motivated by a desire to understand the skew group algebra: it can be shown, see Theorem 1.8 in [12], that the skew group algebra is Morita equivalent to the path algebra generated by the McKay quiver of the group. Hence, understanding the McKay quiver is a first step to understanding the path algebra and a straight-forward combinatorial description of the McKay graph is developed herein. To fully describe the skew group algebra, one must also understand the relations on the composition of arrows. This problem is not addressed here, though the author is hopeful that the techniques from this thesis could be extended to that end.
A similar description of the McKay quiver has already been developed for the classes G(1, 1, n) = Sn, the permutation groups on n letters, in Chapter 6 of [3].
The first three chapters of this thesis are preliminary in nature: a brief overview of the relevant parts of representation theory; a description of Young tableaux and some of their properties; and an exegesis of a paper by Ariki and Koike [1], which explicitly describes the action of the groups G(r, 1, n) and their irreducible representations.
Chapter 4 describes the McKay quivers for the groups G(r, 1, n) and is new work. Chapter 5 recalls some more advanced theorems from representation theory and describes the irreducible representations of G(r, m, n). As such, it is not new work - though the description of the irreducibles is novel (as far as the author knows). Chapter 6 describes the McKay quivers of the groups G(r, m, n).