In mathematics, the Bruhat order (also called the strong order, strong Bruhat order, Chevalley order, Bruhat–Chevalley order, or Chevalley–Bruhat order) is a
partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
on the elements 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 ...
, that corresponds to the inclusion order on
Schubert varieties.
History
The Bruhat order on the
Schubert varieties of a
flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a sm ...
or a
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
was first studied by , and the analogue for more general
semisimple algebraic groups was studied by
Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a found ...
in an unpublished manuscript from 1958, not published until 1994. started the combinatorial study of the Bruhat order on the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
, and introduced the name "Bruhat order" because of the relation to the
Bruhat decomposition introduced by
François Bruhat.
The left and right weak Bruhat orderings were studied by .
Definition
If is a
Coxeter system with generators , then the Bruhat order is a partial order on the group . The definition of Bruhat order relies on several other definitions: first, ''
reduced word In group theory, a word is any written product of group elements and their inverses. For example, if ''x'', ''y'' and ''z'' are elements of a group ''G'', then ''xy'', ''z''−1''xzz'' and ''y''−1''zxx''−1''yz''−1 are words in the set . Two ...
'' for an element of is a minimum-length expression of as a product of elements of , and the ''length'' of is the length of its reduced words. Then the (strong) Bruhat order is defined by if some substring of some (or every) reduced word for is a reduced word for . (Here a substring is not necessarily a consecutive substring.)
There are two other related partial orders:
*the weak left (Bruhat) order is defined by if some final substring of some reduced word for is a reduced word for , and
*the weak right (Bruhat) order is defined by if some initial substring of some reduced word for is a reduced word for .
For more on the weak orders, see the article
Weak order of permutations.
Bruhat graph
The Bruhat graph is a directed graph related to the (strong) Bruhat order. The vertex set is the set of elements of the Coxeter group and the edge set consists of directed edges whenever for some reflection and . One may view the graph as an edge-labeled directed graph with edge labels coming from the set of reflections. (One could also define the Bruhat graph using multiplication on the right; as graphs, the resulting objects are isomorphic, but the edge labelings are different.)
The strong Bruhat order on the symmetric group (permutations) has
Möbius function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...
given by
,
and thus this poset is
Eulerian, meaning its Möbius function is produced by the rank function on the poset.
See also
*
Kazhdan–Lusztig polynomial
Notes
References
*
*
*
*
*{{Citation , last1=Verma , first1=Daya-Nand , title=Structure of certain induced representations of complex semisimple Lie algebras , doi=10.1090/S0002-9904-1968-11921-4 , mr=0218417 , year=1968 , journal=
Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...
, issn=0002-9904 , volume=74 , pages=160–166, doi-access=free
Coxeter groups
Order theory
Bruhat family