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 u₁, u₂, u₃, ... with the relations

• u2
  i = 0
• u_iu_j = u_ju_i if |i − j| > 1
• u_iu_ju_i = u_ju_iu_j if |i − j| = 1

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.

References

• Fomin, Sergey; Stanley, Richard P. (1994), "Schubert polynomials and the nil-Coxeter algebra", Advances in Mathematics 103 (2): 196–207, doi:10.1006/aima.1994.1009, ISSN 0001-8708, MR 1265793