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

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

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.

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