In mathematics, the nil-Coxeter algebra, introduced by , is an algebra similar to the group algebra of a

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

except that the generators are

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

Definition

The nil-Coxeter algebra for the infinite

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

is the algebra generated by ''u''₁, ''u''₂, ''u''₃, ... with the relations :

\begin
u_i^2 & = 0, \\
u_i u_j & = u_j u_i & & \text , i-j,  > 1, \\
u_i u_j u_i & = u_j u_i u_j & & \text , i-j, =1.
\end

These are just the relations for the infinite

braid group In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...

, together with the relations ''u'' = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations ''u'' = 0 to the relations of the corresponding generalized braid group.

References

*{{Citation , last1=Fomin , first1=Sergey , authorlink1=Sergey Fomin , last2=Stanley , first2=Richard P. , authorlink2=Richard P. Stanley , title=Schubert polynomials and the nil-Coxeter algebra , doi=10.1006/aima.1994.1009 , doi-access=free , mr=1265793 , year=1994 , journal=

Advances in Mathematics ''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed ...

, issn=0001-8708 , volume=103 , issue=2 , pages=196–207 Representation theory