title: Nil-Coxeter algebra
text: # Nil-Coxeter algebra
  
  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 $u_1,$ $u_2,$ $u_3,$ ... with the relations
  
      ${\begin{aligned}u_{i}^{2}&=0,\\u_{i}u_{j}&=u_{j}u_{i}&&{\text{ if }}|i-j|>1,\\u_{i}u_{j}u_{i}&=u_{j}u_{i}u_{j}&&{\text{ if }}|i-j|=1.\end{aligned}}$
  
  These are just the relations for the infinite braid group, together with the relations $u^2$ $_i$ = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations $u^2$ $_i$ = 0 to the relations of the corresponding generalized braid group.
  
  