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Arrangements of hyperplanes, higher braid groups and higher Bruhat orders. (English) Zbl 0759.20002

Algebraic number theory – in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values \(L\)-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 289-308 (1989).
[For the entire collection see Zbl 0721.00006.]
Discriminantal arrangements of hyperplanes are defined in section 1 and discussed in connection with higher braid groups and their nilpotent completions. Combinatorics on discriminantal arrangements give rise to the definition of posets \(B(n,k)\), which turn out to be generalizations of Bruhat orders – \(B(n,1)\) is the symmetric group \(S_ n\) in its weak Bruhat order – (section 2). In the last section \(S_ n\) is given the structure of \((n-1)\)-category associated to the weak Bruhat order. Some results are already announced in earlier papers [compare Funkts. Anal. Prilozh. 20, No. 2, 74-75 (1986; Zbl 0646.20014) and Group theoretical methods in physics 1985, 151-165 (1986; Zbl 0699.58069)].

MSC:

20B30 Symmetric groups
06A11 Algebraic aspects of posets
05E25 Group actions on posets, etc. (MSC2000)
20C30 Representations of finite symmetric groups
20F36 Braid groups; Artin groups
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
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