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
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 refl ...
, 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 smoot ...
or a
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
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 direc ...
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 th ...
, and introduced the name "Bruhat order" because of the relation to the
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle ...
introduced by
François Bruhat
François Georges René Bruhat (; 8 April 1929 – 17 July 2007) was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him. ...
.
The left and right weak Bruhat orderings were studied by .
Definition
If (''W'', ''S'') is a
Coxeter system
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 ...
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
In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural order.
Definitions
Inversion
Let \pi be a permutation.
There is an inversion of \pi between i and j if i \pi(j) ...
.
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
,
and thus this poset is Eulerian, meaning its Möbius function is produced by the rank function on the poset.
See also
*
Kazhdan–Lusztig polynomial In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial P_(q) is a member of a family of integral polynomials introduced by . They are indexed by pairs of elements ''y'', ''w'' of a Coxeter group ''W'', which can in part ...
References
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*{{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. I ...
, issn=0002-9904 , volume=74 , pages=160–166, doi-access=free
Coxeter groups
Order theory