In mathematics, the nil-Coxeter algebra, introduced by Fomin & Stanley (1994), is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent.

Definition

The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by u1, u2, u3, ... with the relations

u i 2 = 0 , u i u j = u j u i if | i − j | > 1 , u i u j u i = u j u i u j if | i − j | = 1. {\displaystyle {\begin{aligned}u_{i}^{2}&=0,\\u_{i}u_{j}&=u_{j}u_{i}&&{\text{ if }}|i-j|>1,\\u_{i}u_{j}u_{i}&=u_{j}u_{i}u_{j}&&{\text{ if }}|i-j|=1.\end{aligned}}}

These are just the relations for the infinite braid group, together with the relations u2 i = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u2 i = 0 to the relations of the corresponding generalized braid group.