# Representations of reflection monoids

Albaity, Majed (2018) Representations of reflection monoids. PhD thesis, University of York.

Reflection groups have been investigated since the nineteenth century and now have fundamental relevance for various strands of mathematics, including Lie groups, Lie algebra and Weyl groups. Recently, Everitt and Fountain [19] formulated the notion of reflection monoids, a generalization of the idea of reflection groups to the semigroup theory. In particular, they introduced the family of Boolean reflection monoids of types $\mathscr A_{n-1}$ and $\mathscr B_{n}$, where type $\mathscr A_{n-1}$ is isomorphic to the symmetric inverse monoid, and type $\mathscr B_{n}$ is isomorphic to the monoid of partial signed permutations. The last quarter of a century has witnessed a resurgence in the interest in representations of semigroups. The principal approach to identifying these representations is the Clifford-Munn correspondence, the underlying idea of which is that irreducible representations are in one-to-one correspondence with irreducible representations of the maximal subgroups. For type $\mathscr A_{n-1}$ Boolean reflection monoids, the maximal subgroups are symmetric groups $S_ {k}\ (k\leq n)$, while for type $\mathscr B_{n}$, the maximal subgroups are signed permutation groups $B_ {k}\ (k\leq n).$ These signed permutation groups and their properties can be found scattered in the literature. In this thesis, we collect and reformulate these properties in a form that is analogous to the symmetric group $S_{n}.$ We employ the Clifford-Munn correspondence to provide an explicit account of the ordinary irreducible representations of types $\mathscr A_{n-1}$ and $\mathscr B_{n}$ Boolean reflection monoids, utilizing combinatorial objects called Young tableaux. Preliminary to this, we also prove, for an arbitrary finite inverse monoid, that induced and reduced representations of an irreducible representation, derived from employing the Clifford-Munn correspondence, are themselves irreducible.