Towers of powers and bruhat order

Abstract

An A m e r i c a n M a t h e m a t i c a l M o n t h l y article by B. W. Brunson [3] deals with an interesting partial order on the symmetric group, ~ , , which arises f rom comparing permutat ions of i terated exponentials. In this note, we consider the relationship between this partial order and a well-known partial order on Sen known as Bruhat order (cf. [2]). Brunson essentially shows that his partial order is stronger than the dual of Bruhat order, i.e. if o T in his order. For n ~< 4, Brunson 's order turns out to be identical to Bruhat order. It is then natural to hope that this be true for general n. We show that this is not the case by giving a simple counter-example for n = 5. In fact, our counter-example also disproves Conjectures A and B in [3], and shows that Brunson's order does not satisfy the Jo rdan -Bedek ind chain condition (c.f. [1]). Our counter-example follows from Theorem 1 below which establishes, by elementary means, a fundamental proper ty of i terated exponentials. At about the same time that this paper was first written, J. R. Stembridge had independently shown that Brunson's order and Bruhat order differ by giving an illuminating characterization of Brunson's order (see [5]). For another t rea tment of Brunson 's order, see [4]. We begin by defining Brunson 's order, which is obtained by considering i terated exponentials