Partial symmetry, reflection monoids and Coxeter groups

Abstract

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.

Factorizable inverse monoids

“…Citations of [1] then slowed for more than a decade, though classes of inverse semigroups important for other reasons have turned out to be factorizable, notably the Renner monoids ( [30] Prop. 11.1; [32], section 8.1) and latterly the re ‡ection monoids of Everitt and Fountain [8]. Appropriate generalisations of the concept were developed: signi…cantly, Lawson [17] identi…ed the appropriate generalisation from monoids to semigroups as almost factorizable semigroups, which had been used in McAlister [25]; for an account, see section 7.1 of [18].…”

“…: The subgroup h w j w 2 i of GL (V ) is a re ‡ection group, the Weyl group W associated with ; and it has an action on V inherited from GL (V ) : A system of subspaces for this action is a set H of subspaces of V closed under intersection and the action of W = W ; clearly the lower semilattice hH i generated by H is one such (but not the only one). The restriction monoid of H with respect to the action of W is called a re ‡ection monoid [8].…”

A symmetrical presentation for the singular part of the symmetric inverse monoid

“…00, XX The singular part of the symmetric inverse monoid 3 of the transposition (i, j) to both n \ {i} and n \ {j}. Thus, we are presenting I n \ S n as a reflection monoid ; see [19]. The presentation �E | (E1-E7)� we obtain here has several advantages over the presentation…”

Partial mirror symmetry, lattice presentations and algebraic monoids

“…'Numbers measure size, groups measure symmetry' and inverse monoids measure partial symmetry. In [6], we initiated the formal study of partial mirror symmetry via the theory of what we call reflection monoids. The aim is threefold: (i) to wrap up a reflection group and a naturally associated combinatorial object into a single algebraic entity having nice properties, (ii) to unify various unrelated (until now) parts of the theory of inverse monoids under one umbrella and (iii) to provide workers interested in partial symmetry with the appropriate tools to study the phenomenon systematically.…”

“…Sections 3 and 4 work out explicit presentations for the two main families of reflection monoids that were introduced in [6]: the Boolean monoids and the Coxeter arrangement monoids.…”