Root system
A finite set of vectors in a vector space
over
, possessing the following properties: 1)
does not contain the null vector, and it generates
; 2) for every
there exists an element
of the space
dual to
such that
and such that the endomorphism
of
maps
into itself; 3)
for all
.
Sets of vectors with the properties listed above first arose in the theory of semi-simple complex Lie algebras, as weight systems of the adjoint representation of a maximal torus in such an algebra (see Weight of a representation of a Lie algebra; Lie algebra, semi-simple). Later it was noticed that such systems of vectors appear naturally in many other branches of mathematics, such as algebraic geometry [4], [7], the theory of singularities [7] and the theory of integer-valued quadratic forms [5]. Certain problems in number theory have also turned out to be connected with root systems [6].
General properties of root systems.
The endomorphism is a reflection relative to
and is uniquely determined by the properties 1) and 2). The set of fixed points of
is
, and
. The elements of
are called the roots of the root system
. Its rank is
. A root system
is said to be reduced if, for any
, the vector
is the only root collinear with
. The set
is a root system in
, and
for all
; it is called the dual (or inverse) of
. The finite group
generated by all automorphisms of
which map
onto itself is called the automorphism group of the root system
. The subgroup
of
generated by the reflections
,
, is called the Weyl group of
. If
is a direct sum of subspaces
,
, and
is a root system in
, then
is a root system in
, called the direct sum of the root systems
. A non-empty root system
is said to be irreducible if it is not the direct sum of two non-empty root systems. Every root systems is the direct sum of some set of irreducible root systems, and this decomposition is unique up to the order of terms.
The connected components of the set are open simplicial cones, known as the chambers of the root system
in
(see Chamber). The Weyl group acts in a simply transitive manner on the set of all chambers. The closure
of any chamber
is a fundamental domain of the discrete group
. Let
be the walls of a chamber
. For each wall
there exists a unique root
such that
and
lies on the same side of
as
. The family of roots
forms a basis in
, called the basis of the root system defined by the chamber
. One also says that
is the set of simple roots defined by
. The group
is generated by the reflections
,
, and moreover
, where
is the order of
, is a system of defining relations for
, so that
is a Coxeter group. The group
is the semi-direct product of the subgroup of all elements in
that leave the set
invariant, with
.
A choice of a chamber defines an order relation on
(compatible with the vector-space structure), with respect to which the positive elements are precisely the linear combinations of the simple roots
with non-negative coefficients. Any root is either positive or negative, and all its coordinates with respect to the basis
are integers. The subgroup
of
generated by the root system
is a lattice (i.e. a discrete subgroup of rank
, cf. Lattice in a Lie group) invariant under the Weyl group
. Its elements are called the radical weights of the root system
. The Weyl groups of root systems are precisely those discrete linear groups generated by reflections that leave no non-zero vectors fixed and possess an invariant lattice. If
is regarded as a group of translations of the space
, then the semi-direct product
of
and
is known as the affine Weyl group of
.
is a discrete group of transformations of
generated by reflections in the hyperplanes
![]() |
where ,
. The quotient space of
by
is compact; if
is irreducible, then a fundamental domain for
is a simplex.
One can choose a positive-definite symmetric bilinear form on
which is invariant under
(the choice is not unique). This form gives
the structure of a Euclidean space in which the elements of
are orthogonal transformations, while the reflections
,
, have the following form for all
:
![]() |
Using the form , one can identify the spaces
and
, and then
; condition 3) in the definition of a root system means then that
for all
.
Thanks to the form one can speak of metric relations between roots, in particular of the angle between roots and of the length of a root. It turns out that the magnitude of the angle is independent of the choice of
, while if a root system
is irreducible, then this is also true for the ratio of the lengths of two roots.
Classification of root systems.
Let be some fixed basis of a reduced root system
and let
. The matrix
,
, is called the Cartan matrix of the root system
; this matrix has
, and
(
) may be
, or
. Up to permutation of indices, the Cartan matrix is independent of the choice of the basis. Two root systems with identical Cartan matrices are isomorphic.
With any root system one usually associates its Coxeter graph, the vertices of which are the basis elements , with vertices
and
joined by one, two or three edges or not joined at all, respectively, according as the product
equals
, or
. A root system is irreducible if and only if its Coxeter graph is connected. The Coxeter graph defines only the angles between pairs of basic roots; it does not determine the Cartan matrix (though it does determine the Weyl group): There exist dual non-isomorphic root systems with the same Coxeter graph. However, the Cartan matrix (and together with it the root system) is completely determined by the directed Coxeter graph, also known as the Dynkin diagram or simple root diagram of the root system. The direction is defined by the rule: If the simple roots
and
are not orthogonal and are different in length, the two or three edges joining the
-th and
-th vertices are assigned the inequality sign
, directed towards the vertex corresponding to the root of smaller length. In some cases, above each vertex of the Coxeter graph one writes a number proportional to the squared length of the corresponding root (the same proportionality factor for all the roots); this weighted graph also uniquely determines the original root system.
The following is a complete list of the pairwise non-isomorphic, irreducible, reduced root systems, as defined by their simple root diagrams: vertices vertices vertices vertices
Figure: r082590a
Construction of the irreducible root systems.
Let be the canonical basis in
,
the standard scalar product in
for which
, and
the lattice in
generated by the vectors
.
1) Let be a hyperplane in
orthogonal to the vector
. Then
![]() |
![]() |
is a root system of type . For
, this root system has the form
Figure: r082590b
2) The set of vectors
![]() |
![]() |
in is a root system of type
. For
it has the form
Figure: r082590c
3) A root system of type is dual to a system of type
and consists of the vectors
![]() |
4) The set of vectors
![]() |
in is a root system of type
.
5) A root system of type has the form
Figure: r082590d
and it can be described as the set of algebraic integers of the cyclotomic field generated by a cubic root of unity with norm or
.
6) The set of vectors
![]() |
in is a root system of type
.
7) The set of vectors
![]() |
where
![]() |
in is a root system of type
.
8) A root system of type can be obtained as the intersection of a root system of type
with the subspace of
spanned by
.
9) A root system of type can be obtained as the intersection of a root system of type
with the subspace of
spanned by
.
10) For each dimension there exists (up to an isomorphism) exactly one non-reduced irreducible root system
, namely, the union of
and
(see above). For
the system has the form
Figure: r082590e
Concerning affine root systems, see [6].
References
[1] | N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Hermann (1968) pp. Chapts. 4–6 |
[2] | J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) |
[3] | R.G. Steinberg, "Lectures on Chevalley groups" , Yale Univ. Press (1967) |
[4] | Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian) |
[5] | J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) |
[6] | I.G. Macdonald, "Affine root systems and Dedekind's eta function" Invent. Math. , 15 (1972) pp. 91–143 |
[7] | V.I. Arnol'd, "Critical points of smooth functions and their normal forms" Russian Math. Surveys , 30 : 5 (1975) pp. 1–75 Uspekhi Mat. Nauk , 30 : 5 (1975) pp. 3–65 |
Comments
In the Western literature one usually says Dynkin graph instead of Coxeter graph, especially in connection with Lie theory. The directed Coxeter graph or simple root diagram is commonly called Dynkin diagram.
See also [a4] for an introduction to root systems. Note that the notion of Coxeter graph used in the article above differs slightly from the notion used by N. Bourbaki [1].
In a sense, root systems are the combinatorial remnants of semi-simple Lie groups and one would like to express results on semi-simple Lie groups (e.g. classification of representations) in terms of the combinatorics of root systems. An example of this principle are the Kazhdan–Lusztig polynomials, defined in purely combinatorial terms of Coxeter groups, which describe the multiplicities of the composition factors for Verma modules [a7].
Root systems admit a refined special function theory, partly motivated by and related to the theory of semi-simple Lie groups, such as Macdonald's identity (the affine version of the Weyl denominator formula [6]), the invariant theory for -functions [a8] and a hypergeometric-type function theory . In this latter framework also fit the constant-term conjectures of I.G. Macdonald (see below).
Most of these were first formulated and proved for or
without any reference to root systems. Dyson's conjecture can be stated as:
![]() |
where is a non-negative integer and
denotes the constant term coefficient in the Laurent series expansion in
. Macdonald's conjecture [a5] generalizes this from root system
to an arbitrary not necessarily reduced root system
as follows. For each
, let
be a non-negative integer depending only on the length of
. Let
be a choice of the set of positive roots. Put
and
if
. Then
![]() |
Selberg's integral is:
![]() |
![]() |
For taking values in a discrete set, it is equivalent to Macdonald's
conjecture. Similarly, the left-hand side in Macdonald's general conjecture can be rewritten as
![]() |
or
![]() |
where is the torus
and
is normalized Lebesgue measure on
. Macdonald's conjecture was proved for special root systems by various authors in various different ways. See the survey [a1] for references and for a description of
-analogues and of Moris' conjecture related to affine root systems. Finally, Macdonald's conjecture was proved in [a6] in full generality and without use of the classification of root systems. The proof used shift operators and orthogonal polynomials (Jacobi polynomials) related to the root system.
Let be the weight lattice for
and let
consist of all weights
for which
for all
in
. Let
be the partial order on
such that
if
is a linear combination of positive roots with non-negative integer coefficients. On the space of exponential polynomials on
(spanned by
,
) which are moreover invariant under the Weyl group
, define the Hermitian inner product
![]() |
where the are non-negative real numbers depending only on
. For
the Jacobi polynomial
is defined on
by:
1) (here
is the convex hull of
, intersected with
) with
and
,
.
2) for all
with
.
It was shown in [a2], and subsequently much easier in [a3], that whenever
.
References
[a1] | L. Habsieger, "Macdonald conjectures and the Selberg integral" , q-Series and Partitions , IMA Vol. Math. Appl. , 18 , Springer (1989) pp. 99–108 |
[a2] | G.J. Heckman, "Root systems and hypergeometric functions II" Compositio Math. , 64 (1987) pp. 353–373 |
[a3] | G.J. Heckman, "An elementary approch to the hypergeometric shift operators of Opdam" Invent. Math. , 103 (1991) pp. 341–350 |
[a4] | J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1980) |
[a5] | I.G. Macdonald, "Some conjectures for root systems" SIAM J. Math. Anal. , 13 (1982) pp. 988–1007 |
[a6] | E.M. Opdam, "Some applications of hypergeometric shift operators" Invent. Math. , 98 (1989) pp. 1–18 |
[a7] | D. Kahzdan, G. Lusztig, "Representations of Coxeter groups and Hecke algebras" Invent. Math. , 53 (1979) pp. 165–184 |
[a8] | E. Looyenga, "Root systems and elliptic curves" Invent. Math. , 38 (1976) pp. 17–32 |
[a9a] | G.J. Heckman, E.M. Opdam, "Root systems and hypergeometric functions, I, II" Comp. Math. , 64 (1987) pp. 329–352; 353–373 |
[a9b] | G.J. Heckman, E.M. Opdam, "Root systems and hypergeometric functions, III, IV" Comp. Math. , 67 (1988) pp. 21–49; 191–209 |
[a10] | H. Freudenthal, H. de Vries, "Linear Lie groups" , Acad. Press (1969) |
[a11] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) |
Root system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Root_system&oldid=23968