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In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
on the elements of a Coxeter group, that corresponds to the inclusion order on
Schubert varieties In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using linea ...
.


History

The Bruhat order on the
Schubert varieties In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using linea ...
of a flag manifold or a Grassmannian was first studied by , and the analogue for more general
semisimple algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a dire ...
s was studied by . 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 ...
, 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 (''W'', ''S'') is a Coxeter system with generators ''S'', then the Bruhat order is a partial order on the group ''W''. Recall that a reduced word for an element ''w'' of ''W'' is a minimal length expression of ''w'' as a product of elements of ''S'', and the length ''ℓ''(''w'') of ''w'' is the length of a reduced word. *The (strong) Bruhat order is defined by ''u'' ≤ ''v'' if some substring of some (or every) reduced word for ''v'' is a reduced word for ''u''. (Note that here a substring is not necessarily a consecutive substring.) *The weak left (Bruhat) order is defined by ''u'' ≤''L'' ''v'' if some final substring of some reduced word for ''v'' is a reduced word for ''u''. *The weak right (Bruhat) order is defined by ''u'' ≤''R'' ''v'' if some initial substring of some reduced word for ''v'' is a reduced word for ''u''. 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 (''u'', ''v'') whenever ''u'' = ''tv'' for some reflection ''t'' and ''ℓ''(''u'') < ''ℓ''(''v''). 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 given by \mu(\pi,\sigma)=(-1)^, and thus this poset is Eulerian, meaning its Möbius function is produced by the rank function on the poset.


See also

* Kazhdan–Lusztig polynomial


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