Since we must be brief here, this is not really a place to learn about Lie groups. Rather, the point of this section is to outline what you need to know to use Sage effectively for Lie computations, and to fix ideas and notations.
If \(g \in GL(n,\mathbf{C})\), then \(g\) may be uniquely factored as \(g_1 g_2\) where \(g_1\) and \(g_2\) commute, with \(g_1\) semisimple (diagonalizable) and \(g_2\) unipotent (all its eigenvalues equal to 1). This follows from the Jordan canonical form. If \(g = g_1\) then \(g\) is called semisimple and if \(g = g_2\) then \(g\) is called unipotent.
We consider a Lie group \(G\) and a class of representations such that if an element \(g \in G\) is unipotent (resp. semisimple) in one faithful representation from the class, then it is unipotent (resp. semisimple) in every faithful representation of the class. Thus the notion of being semisimple or unipotent is intrinsic. Examples:
A subgroup of \(G\) is called unipotent if it is connected and all its elements are unipotent. It is called a torus if it is connected, abelian, and all its elements are semisimple. The group \(G\) is called reductive if it has no nontrivial normal unipotent subgroup. For example, \(GL(2,\mathbf{C})\) is reductive, but its subgroup:
is not since it has a normal unipotent subgroup
A group has a unique largest normal unipotent subgroup, called the unipotent radical, so it is reductive if and only if the unipotent radical is trivial.
A Lie group is called semisimple it is reductive and furthermore has no nontrivial normal tori. For example \(GL(2,\mathbf{C})\) is reductive but not semisimple because it has a normal torus:
The group \(SL(2,\mathbf{C})\) is semisimple.
If \(G\) is a semisimple Lie group then its center and fundamental group are finite abelian groups. The universal covering group \(\tilde G\) is therefore a finite extension with the same Lie algebra. Any representation of \(G\) may be reinterpreted as a representation of the simply connected \(\tilde G\). Therefore we may as well consider representations of \(\tilde G\), and restrict ourselves to the simply connected group.
Let \(G\) be a reductive complex analytic group. A maximal solvable subgroup of \(G\) is called a Borel subgroup. All Borel subgroups are conjugate. Any subgroup \(P\) containing a Borel subgroup is called a parabolic subgroup. We may write \(P\) as a semidirect product of its maximal normal unipotent subgroup or unipotent radical \(P\) and a reductive subgroup \(M\), which is determined up to conjugacy. The subgroup \(M\) is called a Levi subgroup.
Example: Let \(G = GL_n(\mathbf{C})\) and let \(r_1, \dots, r_k\) be integers whose sum is \(n\). Then we may consider matrices of the form:
where \(g_i \in GL(r_i,\mathbf{C}\). The unipotent radical consists of the subgroup in which all \(g_i = I_{r_i}\). The Levi subgroup (determined up to conjugacy) is:
and is isomorphic to \(M = GL(r_1,\mathbf{C}) \times \cdots \times GL(r_k,\mathbf{C})\). Therefore \(M\) is a Levi subgroup.
The notion of a Levi subgroup can be extended to compact Lie groups. Thus \(U(r_1) \times \cdots \times U(r_k)\) is a Levi subgroup of \(U(n)\). However parabolic subgroups do not exist for compact Lie groups.
Semisimple Lie groups are classified by their Cartan types. There are both reducible and irreducible Cartan types in Sage. Let us start with the irreducible types. Such a type is implemented in Sage as a pair ['X',r] where ‘X’ is one of A, B, C, D, E, F or G and \(r\) is a positive integer. If ‘X’ is ‘D’ then we must have \(r > 1\) and if ‘X’ is one of the exceptional types ‘E’, ‘F’ or ‘G’ then \(r\) is limited to only a few possibilities. The exceptional types are:
['G',2], ['F',4], ['E',6], ['E',7] or ['E',8].
A simply-connected semisimple group is a direct product of simple Lie groups, which are given by the following table. The integer \(r\) is called the rank, and is the dimension of the maximal torus.
Here are the Lie groups corresponding to the classical types:
| compact group | complex analytic group | Cartan type |
|---|---|---|
| \(SU(r+1)\) | \(SL(r+1,\mathbf{C})\) | \(A_r\) |
| \(spin(2r+1)\) | \(spin(2r+1,\mathbf{C})\) | \(B_r\) |
| \(Sp(2r)\) | \(Sp(2r,\mathbf{C})\) | \(C_r\) |
| \(spin(2r)\) | \(spin(2r,\mathbf{C})\) | \(D_r\) |
You may create these Cartan types and their Dynkin diagrams as follows:
sage: ct = CartanType("D5"); ct
['D', 5]
Here "D5" is an abbreviation for ['D',5]. The group \(spin(n)\) is the simply-connected double cover of the orthogonal group \(SO(n)\).
Every Cartan type has a dual, which you can get from within Sage:
sage: CartanType("B4").dual()
['C', 4]
Types other than B and C are self-dual in the sense that the dual is isomorphic to the original type; however the isomorphism of a Cartan type with its dual might relabel the vertices. We can see this as follows:
sage: CartanType("F4").dynkin_diagram()
O---O=>=O---O
1 2 3 4
F4
sage: CartanType("F4").dual()
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1}
sage: CartanType("F4").dual().dynkin_diagram()
O---O=>=O---O
4 3 2 1
F4 relabelled by {1: 4, 2: 3, 3: 2, 4: 1}
If \(G\) is a Lie group of finite index in \(G_1 \times G_2\), where \(G_1\) and \(G_2\) are Lie groups of dimension \(>0\), then \(G\) is called reducible. In this case, the root system of \(G\) is the disjoint union of the root systems of \(G_1\) and \(G_2\), which lie in orthogonal subspaces of the ambient space of the weight space of \(G\). The Cartan type of \(G\) is thus reducible.
Reducible Cartan types are supported in Sage as follows:
sage: RootSystem("A1xA1")
Root system of type A1xA1
sage: WeylCharacterRing("A1xA1")
The Weyl Character Ring of Type A1xA1 with Integer Ring coefficients
There are some isomorphisms that occur in low degree.
| Cartan Type | Group | Equivalent Type | Isomorphic Group |
|---|---|---|---|
| B2 | \(spin(5)\) | C2 | \(Sp(4)\) |
| D3 | \(spin(6)\) | A3 | \(SL(4)\) |
| D2 | \(spin(4)\) | A1xA1 | \(SL(2)\times SL(2)\) |
| B1 | \(spin(3)\) | A1 | \(SL(2)\) |
| C1 | \(Sp(2)\) | A1 | \(SL(2)\) |
Sometimes the redundant Cartan types such as D3 and D2 are excluded from the list of Cartan types. However Sage allows them since excluding them leads to exceptions having to be made in algorithms. A better approach, which is followed by Sage, is to allow the redundant Cartan types, but to implement the isomorphisms explicitly as special cases of branching rules. The utility of this approach may be seen by considering that the rank one group \(SL(2)\) has different natural weight lattices realizations depending on whether we consider it to be \(SL(2)\), \(spin(2)\) or \(Sp(2)\):
sage: RootSystem("A1").ambient_space().simple_roots()
Finite family {1: (1, -1)}
sage: RootSystem("B1").ambient_space().simple_roots()
Finite family {1: (1)}
sage: RootSystem("C1").ambient_space().simple_roots()
Finite family {1: (2)}
There are also affine Cartan types, which correspond to (infinite) affine Lie algebras. There is an affine Cartan type of the of the form [`X`,r,1] if X=A,B,C,D,E,F,G and [`X`,r] is an ordinary Cartan type. There are also twisted affine types of the form [X,r,k] where \(k = 2\) or 3 if the Dynkin diagram of the ordinary Cartan type [X,r] has an automorphism of degree \(k\).
Illustrating some of the methods available for the untwisted affine Cartan type ['A',4,1]:
sage: ct = CartanType(['A',4,1]); ct
['A', 4, 1]
sage: ct.dual()
['A', 4, 1]
sage: ct.classical()
['A', 4]
sage: ct.dynkin_diagram()
0
O-----------+
| |
| |
O---O---O---O
1 2 3 4
A4~
The twisted affine Cartan types are relabeling of the duals of certain untwisted Cartan types:
sage: CartanType(['A',3,2])
['B', 2, 1]^*
sage: CartanType(['D',4,3])
['G', 2, 1]^* relabelled by {0: 0, 1: 2, 2: 1}
By default Sage uses the labeling of the Dynkin Diagram from Bourbaki, Lie Groups and Lie Algebras Chapters 4,5,6. There is another labeling of the vertices due to Dynkin. Most of the literature follows Bourbaki, though Kac’s book Infinite Dimensional Lie algebras follows Dynkin.
If you need to use Dynkin’s labeling you should be aware that Sage does support relabeled Cartan types. See the documentation in sage.combinat.root_system.type_relabel for further information.
These realizations follow the Appendix in Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6. See the Root system plot tutorial for how to visualize them.
For type \(A_r\) we use an \(r+1\) dimensional ambient space. This means that we are modeling the Lie group \(U(r+1)\) or \(GL(r+1,\mathbf{C})\) rather than \(SU(r+1)\) or \(SL(r+1,\mathbf{C})\). The ambient space is identified with \(\mathbf{Q}^{r+1}\):
sage: RootSystem("A3").ambient_space().simple_roots()
Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}
sage: RootSystem("A3").ambient_space().fundamental_weights()
Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1, 1, 1, 0)}
sage: RootSystem("A3").ambient_space().rho()
(3, 2, 1, 0)
The dominant weights consist of integer \(r+1\)-tuples \(\lambda = (\lambda_1,\dots,\lambda_{r+1})\) such that \(\lambda_1 \ge \dots \ge \lambda_{r+1}\).
See SL versus GL for further remarks about Type A.
For the remaining classical Cartan types \(B_r\), \(C_r\) and \(D_r\) we use an \(r\)-dimensional ambient space:
sage: RootSystem("B3").ambient_space().simple_roots()
Finite family {1: (1, -1, 0), 2: (0, 1, -1), 3: (0, 0, 1)}
sage: RootSystem("B3").ambient_space().fundamental_weights()
Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1/2, 1/2, 1/2)}
sage: RootSystem("B3").ambient_space().rho()
(5/2, 3/2, 1/2)
This is the Cartan type of \(spin(2r+1)\). The last fundamental weight (1/2, 1/2, ..., 1/2) is the highest weight of the \(2^r\) dimensional spin representation. All the other fundamental representations factor through the homomorphism \(spin(2r+1) \to SO(2r+1)\) and are representations of the orthogonal group.
The dominant weights consist of \(r\)-tuples of integers or half-integers \((\lambda_1,\dots,\lambda_r)\) such that \(\lambda_1 \ge \lambda_2 \dots \ge \lambda_r \ge 0\), and such that the differences \(\lambda_i - \lambda_j \in \mathbf{Z}\).
sage: RootSystem("C3").ambient_space().simple_roots()
Finite family {1: (1, -1, 0), 2: (0, 1, -1), 3: (0, 0, 2)}
sage: RootSystem("C3").ambient_space().fundamental_weights()
Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1, 1, 1)}
sage: RootSystem("C3").ambient_space().rho()
(3, 2, 1)
This is the Cartan type of the symplectic group \(Sp(2r)\).
The dominant weights consist of \(r\)-tuples of integers \(\lambda = (\lambda_1,\dots,\lambda_{r+1})\) such that \(\lambda_1 \ge \cdots \ge \lambda_r \ge 0\).
sage: RootSystem("D4").ambient_space().simple_roots()
Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1), 4: (0, 0, 1, 1)}
sage: RootSystem("D4").ambient_space().fundamental_weights()
Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1/2, 1/2, 1/2, -1/2), 4: (1/2, 1/2, 1/2, 1/2)}
sage: RootSystem("D4").ambient_space().rho()
(3, 2, 1, 0)
This is the Cartan type of \(spin(2r)\). The last two fundamental weights are the highest weights of the two \(2^{r-1}\)-dimensional spin representations.
The dominant weights consist of \(r\)-tuples of integers \(\lambda = (\lambda_1,\dots,\lambda_{r+1})\) such that \(\lambda_1 \ge \cdots \ge \lambda_{r-1} \ge |\lambda_r|\).
We leave the reader to examine the exceptional types. You can use Sage to list the fundamental dominant weights and simple roots.
Let \(G\) be a reductive complex analytic group. Let \(T\) be a maximal torus, \(\Lambda = X^{\ast} (T)\) be its group of analytic characters. Then \(T \cong (\mathbf{C}^{\times})^r\) for some \(r\) and \(\Lambda \cong \mathbf{Z}^r\).
Example 1: Let \(G = \hbox{GL}_{r+1} (\mathbf{C})\). Then \(T\) is the diagonal subgroup and \(X^{\ast} (T) \cong \mathbf{Z}^{r+1}\). If \(\lambda = (\lambda_1, \dots, \lambda_n)\) then \(\lambda\) is identified with the rational character
Example 2: Let \(G = \hbox{SL}_{r+1} (\mathbf{C})\). Again \(T\) is the diagonal subgroup but now if \(\lambda \in \mathbf{Z}^{\Delta} = \{(d, \cdots, d) | d \in \mathbf{Z}\} \subseteq \mathbf{Z}^{r+1}\) then \(\prod t_i^{\lambda_i} = \det ({\bf t})^d = 1\), so \(X^{\ast} (T) \cong \mathbf{Z}^{r+1} /\mathbf{Z}^{\Delta} \cong \mathbf{Z}^r\).
As we have mentioned, \(G\) acts on its complexified Lie algebra \(\mathfrak{g}_{\mathbf{C}}\) by the adjoint representation. The zero weight space \(\mathfrak{g}_{\mathbf{C}}(0)\) is just the Lie algebra of \(T\) itself. The other nonzero weights each appear with multiplicity one and form an interesting configuration of vectors called the root system \(\Phi\).
It is convenient to partition \(\Phi\) into two sets \(\Phi^+\) and \(\Phi^-\) such that \(\Phi^+\) consists of all roots lying on one side of a hyperplane. Often we arrange things so that \(G\) is embedded in \(GL(n,\mathbf{C})\) in such a way that the positive weights correspond to upper triangular matrices. Thus if \(\alpha\) is a positive root, its weight space \(\mathfrak{g}_{\mathbf{C}}(\alpha)\) is spanned by a vector \(X_\alpha\), and the exponential of this eigenspace in \(G\) is a one-parameter subgroup of unipotent matrices. It is always possible to arrange that this one-parameter subgroup consists of upper triangular matrices.
If \(\alpha\) is a positive root that cannot be decomposed as a sum of other positive roots, then \(\alpha\) is called a simple root. If \(G\) is semisimple of rank \(r\), then \(r\) is the number of positive roots. Let \(\alpha_1,\dots,\alpha_r\) be these.
Let \(G\) be a complex analytic group. Let \(T\) be a maximal torus, and let \(N(T)\) be its normalizer. Let \(W = N(T)/T\) be the Weyl group. It acts on \(T\) by conjugation; therefore it acts on the weight lattice \(\Lambda\) and its ambient space. The ambient space admits an inner product that is invariant under this action. Let \(\left<v,w\right>\) denote this inner product. If \(\alpha\) is a root let \(r_\alpha\) denote the reflection in the hyperplane of the ambient space that is perpendicular to \(\alpha\). If \(\alpha = \alpha_i\) is a simple root, then we use the notation \(s_i\) to denote \(r_\alpha\).
Then \(s_1,\dots,s_r\) generate \(W\), which is a Coxeter group. This means that it is generated by elements \(s_i\) of order two and that if \(m(i,j)\) is the order of \(s_i s_j\), then
is a presentation. An important function \(l: W \to \mathbf{Z}\) is the length function, where \(l(w)\) is the length of the shortest decomposition of \(w\) into a product of simple reflections.
The coroots are certain linear functionals on the ambient space that also form a root system. Since the ambient space admits a \(W\)-invariant inner product, they may be identified with elements of the ambient space itself. Then they are proportional to the roots, though if the roots have different lengths, long roots correspond to short coroots and conversely. The coroot corresponding to the root \(\alpha\) is
The Dynkin diagram is a graph whose vertices are in bijection with the set simple roots. We connect the vertices corresponding to roots that are not orthogonal. Usually two such vertices make an angle of \(2\pi/3\), in which case we connect them with a single bond. Occasionally they may make an angle of \(3\pi/4\) in which case we connect them with a double bond, or \(5\pi/6\) in which case we connect them with a triple bond. If the bond is single, the roots have the same length with respect to the inner product on the ambient space. In the case of a double or triple bond, the two simple roots in questions have different length, and the bond is drawn as an arrow from the long root to the short root. Only the exceptional group \(G_2\) has a triple bond.
There are various ways to get the Dynkin diagram:
sage: DynkinDiagram("D5")
O 5
|
|
O---O---O---O
1 2 3 4
D5
sage: ct = CartanType("E6"); ct
['E', 6]
sage: ct.dynkin_diagram()
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6
sage: B4 = WeylCharacterRing("B4"); B4
The Weyl Character Ring of Type B4 with Integer Ring coefficients
sage: B4.dynkin_diagram()
O---O---O=>=O
1 2 3 4
B4
sage: RootSystem("G2").dynkin_diagram()
3
O=<=O
1 2
G2
For the extended Dynkin diagram, we add one negative root \(\alpha_0\). This is the root whose negative is the highest weight in the adjoint representation. Sometimes this is called the affine root. We make the Dynkin diagram as before by measuring the angles between the roots. This extended Dynkin diagram is useful for many purposes, such as finding maximal subgroups and for describing the affine Weyl group.
The extended Dynkin diagram may be obtained as the Dynkin diagram of the corresponding untwisted affine type:
sage: ct = CartanType("E6"); ct
['E', 6]
sage: ct.affine()
['E', 6, 1]
sage: ct.affine() == CartanType(['E',6,1])
True
sage: ct.affine().dynkin_diagram()
O 0
|
|
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6~
The extended Dynkin diagram is also a method of the WeylCharacterRing:
sage: WeylCharacterRing("E7").extended_dynkin_diagram()
O 2
|
|
O---O---O---O---O---O---O
0 1 3 4 5 6 7
E7~
There are certain weights \(\omega_1,\dots,\omega_r\) that:
If \(G\) is semisimple then these are uniquely determined, whereas if \(G\) is reductive but not semisimple we may choose them conveniently.
Let \(\rho\) be the sum of the fundamental dominant weights. If \(G\) is semisimple, then \(\rho\) is half the sum of the positive roots. In case \(G\) is not semisimple, we have noted, the fundamental weights are not completely determined by the inner product condition given above. If we make a different choice, then \(\rho\) is altered by a vector that is orthogonal to all roots. This is a harmless change for many purposes such as the Weyl character formula.
In Sage, this issue arises only for Cartan type A. See SL versus GL.
Let \(\Lambda = X^{\ast} (T)\) be the group of rational characters. Then \(\Lambda \cong \mathbf{Z}^r\).
Assuming that \(G\) is simply-connected (or more generally, reductive with a simply-connected derived group) every dominant weight \(\lambda\) is the highest weight of a unique irreducible representation \(\pi_\lambda\), and \(\lambda \mapsto \pi_\lambda\) gives a parametrization of the isomorphism classes of irreducible representations of \(G\) by the dominant weights.
The character of \(\pi_\lambda\) is the function \(\chi_\lambda(g) = tr(\pi_\lambda(g))\). It is determined by its values on \(T\). If \(\mathbf(z) \in T\) and \(\mu \in \Lambda\), let us write \(\mathbf{z}^\mu\) for the value of \(\mu\) on \(\mathbf{z}\). Then the character:
Sometimes this is written
The meaning of \(e^\lambda\) is subject to interpretation, but we may regard it as the image of the additive group \(\Lambda\) in its group algebra. The character is then regarded as an element of this ring, the group algebra of \(\Lambda\).
In this example, \(G = \hbox{SL}(3,\mathbf{C})\). We have drawn the weights of an irreducible representation with highest weight \(\lambda\). The shaded region is \(\mathcal{C}^+\). \(\lambda\) is a dominant weight, and the labeled vertices are the weights with positive multiplicity in \(V(\lambda)\). The weights weights on the outside have \(m(\mu) = 1\), while the six interior weights (with double circles) have \(m(\mu) = 2\).
The considerations of this section are particular to type A. We review the relationship between characters of \(GL(n,\mathbf{C})\) and symmetric function theory.
A partition \(\lambda\) is a sequence of descending nonnegative integers:
We do not distinguish between two partitions if they differ only by some trailing zeros, so \((3, 2) = (3, 2, 0)\). If \(l\) is the last integer such that \(\lambda_l > 0\) then we say that \(l\) is the length of \(\lambda\). If \(k = \sum \lambda_i\) then we say that \(\lambda\) is a partition of \(k\) and write \(\lambda \vdash k\).
A partition of length \(\le n=r+1\) is therefore a dominant weight of type ['A',r]. Not every dominant weight is a partition, since the coefficients in a dominant weight could be negative. Let us say that an element \(\mu = (\mu_1, \mu_2, \cdots, \mu_n)\) of the ['A',r] root lattice is effective if the \(\mu_i \ge 0\). Thus an effective dominant weight of ['A',r] is a partition of length \(\le n\), where \(n = r+1\).
Let \(\lambda\) be a dominant weight, and let \(\chi_\lambda\) be the character of \(GL(n,\mathbf{C})\) with highest weight \(\lambda\). If \(k\) is any integer we may consider the weight \(\mu = (\lambda_1+k,\dots,\lambda_n+k)\) obtained by adding \(k\) to each entry. Then \(\chi_{\mu} = \det^k \otimes \chi_\lambda\). Clearly by choosing \(k\) large enough, we may make \(\mu\) effective.
So the characters of irreducible representations of \(GL(n,\mathbf{C})\) do not all correspond to partitions, but the characters indexed by partitions (effective dominant weights) are enough that we can write any character \(\det^{-k}\chi_{\mu}\) where \(\mu\) is a partition. If we take \(k = -\lambda_n\) we could also arrange that the last entry in \(\lambda\) is zero.
If \(\lambda\) is an effective dominant weight, then every weight that appears in \(\chi_\lambda\) is effective. (Indeed, it lies in the convex hull of \(w(\lambda)\) where \(w\) runs through the Weyl group \(W = S_n\).) This means that if
then \(\chi_\lambda(g)\) is a polynomial in the eigenvalues of \(g\). This is the Schur polynomial \(s_\lambda(z_1,\dots,z_n)\).