The Weyl Character Ring#

Weyl character rings#

The Weyl character ring is the representation ring of a compact Lie group. It has a basis consisting of the irreducible representations of \(G\), or equivalently, their characters. The addition and multiplication in the Weyl character ring correspond to direct sum and tensor product of representations.

Methods of the ambient space#

In Sage, many useful features of the Lie group are available as methods of the ambient space:

sage: S = RootSystem("B2").ambient_space(); S
Ambient space of the Root system of type ['B', 2]
sage: S.roots()
[(1, -1), (1, 1), (1, 0), (0, 1), (-1, 1), (-1, -1), (-1, 0), (0, -1)]
sage: S.fundamental_weights()
Finite family {1: (1, 0), 2: (1/2, 1/2)}
sage: S.positive_roots()
[(1, -1), (1, 1), (1, 0), (0, 1)]
sage: S.weyl_group()
Weyl Group of type ['B', 2] (as a matrix group acting on the ambient space)

Methods of the Weyl character ring#

If you are going to work with representations, you may want to create a Weyl Character ring. Many methods of the ambient space are available as methods of the Weyl character ring:

sage: B3 = WeylCharacterRing("B3")
sage: B3.fundamental_weights()
Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1/2, 1/2, 1/2)}
sage: B3.simple_roots()
Finite family {1: (1, -1, 0), 2: (0, 1, -1), 3: (0, 0, 1)}
sage: B3.dynkin_diagram()
O---O=>=O
1   2   3
B3

Other useful methods of the Weyl character ring include:

cartan_type
highest_root
positive_root
extended_dynkin_diagram
rank

Some methods of the ambient space are not implemented as methods of the Weyl character ring. However, the ambient space itself is a method, and so you have access to its methods from the Weyl character ring:

sage: B3 = WeylCharacterRing("B3")
sage: B3.space().weyl_group()
Weyl Group of type ['B', 3] (as a matrix group acting on the ambient space)
sage: B3.space()
Ambient space of the Root system of type ['B', 3]
sage: B3.space().rho()
(5/2, 3/2, 1/2)
sage: B3.cartan_type()
['B', 3]

Coroot notation#

It is useful to give the Weyl character ring a name that corresponds to its Cartan type. This has the effect that the ring can parse its own output:

sage: G2 = WeylCharacterRing("G2")
sage: [G2(fw) for fw in G2.fundamental_weights()]
[G2(1,0,-1), G2(2,-1,-1)]
sage: G2(1,0,-1)
G2(1,0,-1)

Actually the prefix for the ring is configurable, so you don’t really have to call this ring G2. Type WeylCharacterRing? at the sage: prompt for details.

There is one important option that you may want to know about. This is coroot notation. You select this by specifying the option style="coroots" when you create the ring. With the coroot style, the fundamental weights are represented (1,0,0, ...), (0,1,0,...) instead of as vectors in the ambient space:

sage: B3 = WeylCharacterRing("B3", style="coroots")
sage: [B3(fw) for fw in B3.fundamental_weights()]
[B3(1,0,0), B3(0,1,0), B3(0,0,1)]
sage: B3(0,0,1)
B3(0,0,1)
sage: B3(0,0,1).degree()
8

The last representation is the eight dimensional spin representation of \(G = spin(7)\), the double cover of the orthogonal group \(SO(7)\). In the default notation it would be represented B3(1/2,1/2,1/2).

With the coroot notation every irreducible representation is represented B3(a,b,c) where a, b and c are nonnegative integers. This is often convenient. For many purposes the coroot style is preferable.

One disadvantage: in the coroot style the Lie group or Lie algebra is treated as semisimple, so you lose the distinction between \(GL(n)\) and \(SL(n)\); you also some information about representations of E6 and E7 for the same reason.

Tensor products of representations#

The multiplication in the Weyl character ring corresponds to tensor product. This gives us a convenient way of decomposing a tensor product into irreducibles:

sage: B3 = WeylCharacterRing("B3")
sage: fw = B3.fundamental_weights()
sage: spinweight = fw[3]; spinweight
(1/2, 1/2, 1/2)
sage: spin = B3(spinweight); spin
B3(1/2,1/2,1/2)
sage: spin.degree()
8

The element \(spin\) of the WeylCharacterRing is the representation corresponding to the third highest weight representation, the eight-dimensional spin representation of \(spin(7)\). We could just as easily construct it with the command:

sage: spin = B3(1/2,1/2,1/2)

We may compute its tensor product with itself, using the multiplicative structure of the Weyl character ring:

sage: chi = spin*spin; chi
B3(0,0,0) + B3(1,0,0) + B3(1,1,0) + B3(1,1,1)

We have taken the eight-dimensional spin representation and tensored with itself. We see that the tensor square splits into four irreducibles, each with multiplicity one.

The highest weights that appear here are available (with their coefficients) through the usual free module accessors:

sage: from pprint import pprint
sage: list(chi)                           # random
[((1, 1, 1), 1), ((0, 0, 0), 1), ((1, 0, 0), 1), ((1, 1, 0), 1)]
sage: sorted(chi, key=str)
[((0, 0, 0), 1), ((1, 0, 0), 1), ((1, 1, 0), 1), ((1, 1, 1), 1)]
sage: pprint(dict(chi))
{(0, 0, 0): 1, (1, 0, 0): 1, (1, 1, 0): 1, (1, 1, 1): 1}
sage: M = sorted(chi.monomials(), key=lambda x: tuple(x.support())); M
[B3(0,0,0), B3(1,0,0), B3(1,1,0), B3(1,1,1)]
sage: sorted(chi.support())
[(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1)]
sage: chi.coefficients()
[1, 1, 1, 1]
sage: [r.degree() for r in M]
[1, 7, 21, 35]
sage: sum(r.degree() for r in chi.monomials())
64

Here we have extracted the individual representations, computed their degrees and checked that they sum up to \(64\).

Weight multiplicities#

The weights of the character are available (with their coefficients) through the method weight_multiplicities. Continuing from the example in the last section:

sage: pprint(chi.weight_multiplicities())
{(0, 0, 0): 8, (-1, 0, 0): 4, (-1, -1, 0): 2, (-1, -1, -1): 1,
 (-1, -1, 1): 1, (-1, 1, 0): 2, (-1, 1, -1): 1, (-1, 1, 1): 1,
 (-1, 0, -1): 2, (-1, 0, 1): 2, (1, 0, 0): 4, (1, -1, 0): 2,
 (1, -1, -1): 1, (1, -1, 1): 1, (1, 1, 0): 2, (1, 1, -1): 1,
 (1, 1, 1): 1, (1, 0, -1): 2, (1, 0, 1): 2, (0, -1, 0): 4,
 (0, -1, -1): 2, (0, -1, 1): 2, (0, 1, 0): 4, (0, 1, -1): 2,
 (0, 1, 1): 2, (0, 0, -1): 4, (0, 0, 1): 4}

Each key of this dictionary is a weight, and its value is the multiplicity of that weight in the character.

Example#

Suppose that we wish to compute the integral

\[\int_{U(n)} |tr(g)|^{2k}\,dg\]

for various \(n\). Here \(U(n)\) is the unitary group, which is the maximal compact subgroup of \(GL(n,\mathbb{C})\). The irreducible unitary representations of \(U(n)\) may be regarded as the basis elements of the WeylCharacterRing of type \(A_r\), where \(r=n-1\) so we might work in that ring. The trace \(tr(g)\) is then just the character of the standard representation. We may realize it in the WeylCharacterRing by taking the first fundamental weight and coercing it into the ring. For example, if \(k=5\) and \(n=3\) so \(r=2\):

sage: A2 = WeylCharacterRing("A2")
sage: fw = A2.fundamental_weights(); fw
Finite family {1: (1, 0, 0), 2: (1, 1, 0)}
sage: tr = A2(fw[1]); tr
A2(1,0,0)

We may compute the norm square the character tr^5 by decomposing it into irreducibles, and taking the sum of the squares of their multiplicities. By Schur orthogonality, this gives the inner product of the \(tr(g)^5\) with itself, that is, the integral of \(|tr(g)|^{10}\):

sage: sum(d^2 for d in (tr^5).coefficients())
103

So far we have been working with \(n=3\). For general \(n\):

sage: def f(n,k):
....:     R = WeylCharacterRing(['A',n-1])
....:     tr = R(R.fundamental_weights()[1])
....:     return sum(d^2 for d in (tr^k).coefficients())
sage: [f(n,5) for n in [2..7]]
[42, 103, 119, 120, 120, 120]

We see that the 10-th moment of \(tr(g)\) is just \(5!\) when \(n\) is sufficiently large. What if we fix \(n\) and vary \(k\)?

sage: [f(2,k) for k in [1..10]]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]
sage: [catalan_number(k) for k in [1..10]]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]

Frobenius-Schur indicator#

The Frobeinus-Schur indicator of an irreducible representation of a compact Lie group \(G\) with character \(\chi\) is:

\[\int_G\chi(g^2) \, dg\]

The Haar measure is normalized so that \(vol(G) = 1\). The Frobenius-Schur indicator equals \(1\) if the representation is real (orthogonal), \(-1\) if it is quaternionic (symplectic) and \(0\) if it is complex (not self-contragredient). This is a method of weight ring elements corresponding to irreducible representations. Let us compute the Frobenius-Schur indicators of the spin representations of some odd spin groups:

sage: def spinrepn(r):
....:     R = WeylCharacterRing(['B',r])
....:     return R(R.fundamental_weights()[r])
....:
sage: spinrepn(3)
B3(1/2,1/2,1/2)
sage: for r in [1..4]:
....:     print("{} {}".format(r, spinrepn(r).frobenius_schur_indicator()))
1 -1
2 -1
3 1
4 1

Here we have defined a function that returns the spin representation of the group \(spin(2r+1)\) with Cartan type \(['B',r]\), then computed the Frobenius-Schur indicators for a few values. From this experiment we see that the spin representations of \(spin(3)\) and \(spin(5)\) are symplectic, while those of \(spin(7)\) and \(spin(9)\) are orthogonal.

Symmetric and exterior powers#

Sage can compute symmetric and exterior powers of a representation:

sage: B3 = WeylCharacterRing("B3",style="coroots")
sage: spin = B3(0,0,1); spin.degree()
8
sage: spin.exterior_power(2)
B3(1,0,0) + B3(0,1,0)
sage: spin.exterior_square()
B3(1,0,0) + B3(0,1,0)
sage: spin.exterior_power(5)
B3(0,0,1) + B3(1,0,1)
sage: spin.symmetric_power(5)
B3(0,0,1) + B3(0,0,3) + B3(0,0,5)

The \(k\)-th exterior square of a representation is zero if \(k\) is greater than the degree of the representation. However the \(k\)-th symmetric power is nonzero for all \(k\).

The tensor square of any representation decomposes as the direct sum of the symmetric and exterior squares:

sage: C4 = WeylCharacterRing("C4",style="coroots")
sage: chi = C4(1,0,0,0); chi.degree()
8
sage: chi.symmetric_square()
C4(2,0,0,0)
sage: chi.exterior_square()
C4(0,0,0,0) + C4(0,1,0,0)
sage: chi^2 == chi.symmetric_square() + chi.exterior_square()
True

Since in this example the exterior square contains the trivial representation we expect the Frobenius-Schur indicator to be \(-1\), and indeed it is:

sage: chi = C4(1,0,0,0)
sage: chi.frobenius_schur_indicator()
-1

This is not surprising since this is the standard representation of a symplectic group, which is symplectic by definition!

Weyl dimension formula#

If the representation is truly large you will not be able to construct it in the Weyl character ring, since internally it is represented by a dictionary of its weights. If you want to know its degree, you can still compute that since Sage implements the Weyl dimension formula. The degree of the representation is implemented as a method of its highest weight vector:

sage: L = RootSystem("E8").ambient_space()
sage: [L.weyl_dimension(f) for f in L.fundamental_weights()]
[3875, 147250, 6696000, 6899079264, 146325270, 2450240, 30380, 248]

It is a fact that for any compact Lie group if \(\rho\) is the Weyl vector (half the sum of the positive roots) then the degree of the irreducible representation with highest weight \(\rho\) equals \(2^N\) where \(N\) is the number of positive roots. Let us check this for \(E_8\). In this case \(N = 120\):

sage: L = RootSystem("E8").ambient_space()
sage: len(L.positive_roots())
120
sage: 2^120
1329227995784915872903807060280344576
sage: L.weyl_dimension(L.rho())
1329227995784915872903807060280344576

SL versus GL#

Sage takes the weight space for type ['A',r] to be \(r+1\) dimensional. As a by-product, if you create the Weyl character ring with the command:

sage: A2 = WeylCharacterRing("A2")

Then you are effectively working with \(GL(3)\) instead of \(SL(3)\). For example, the determinant is the character A2(1,1,1). However, as we will explain later, you can work with \(SL(3)\) if you like, so long as you are willing to work with fractional weights. On the other hand if you create the Weyl character ring with the command:

sage: A2 = WeylCharacterRing("A2", style="coroots")

Then you are working with \(SL(3)\).

There are some advantages to this arrangement:

The group \(GL(r+1)\) arises frequently in practice. For example, even if you care mainly about semisimple groups, the group \(GL(r+1)\) may arise as a Levi subgroup.
It avoids fractional weights. If you want to work with \(SL(3)\) the fundamental weights are (2/3,-1/3,-1/3) and (1/3,1/3,-2/3). If you work instead with \(GL(3)\), they are (1,0,0) and (1,1,0). For many mathematical purposes it doesn’t make any difference which you use. This is because the difference between (2/3,-1/3,-1/3) and (1,0,0) is a vector that is orthogonal to all the simple roots. Thus these vectors are interchangeable. But for convenience avoiding fractional weights is advantageous.

However if you want to be an \(SL\) purist, Sage will support you. The weight space for \(SL(3)\) can be taken to be the hyperplane in \(\mathbf{Q}^3\) consisting of vectors \((a,b,c)\) with \(a+b+c = 0\). The fundamental weights for SL(3) are then (2/3,-1/3,-1/3) and (1/3,1/3,-2/3), though Sage will tell you they are (1,0,0) and (1,1,0). The work-around is to filter them through the method coerce_to_sl as follows:

sage: A2 = WeylCharacterRing("A2")
sage: [fw1,fw2] = [w.coerce_to_sl() for w in A2.fundamental_weights()]
sage: [standard, contragredient] = [A2(fw1), A2(fw2)]
sage: standard, contragredient
(A2(2/3,-1/3,-1/3), A2(1/3,1/3,-2/3))
sage: standard*contragredient
A2(0,0,0) + A2(1,0,-1)

Sage is not confused by the fractional weights. Note that if you use coroot notation, you are working with \(SL\) automatically:

sage: A2 = WeylCharacterRing("A2", style="coroots")
sage: A2(1,0).weight_multiplicities()
{(-1/3, -1/3, 2/3): 1, (-1/3, 2/3, -1/3): 1, (2/3, -1/3, -1/3): 1}

There is no convenient way to create the determinant in the Weyl character ring if you adopt the coroot style.

Just as we coerced the fundamental weights into the \(SL\) weight lattice, you may need to coerce the Weyl vector \(\rho\) if you are working with \(SL\). The default value for \(\rho\) in type \(A_r\) is \((r,r-1,\dots,0)\), but if you are an \(SL\) purist you want

\[\left(\frac{r}{2}, \frac{r}{2}-1,\dots,-\frac{r}{2}\right).\]

Therefore take the value of \(\rho\) that you get from the method of the ambient space and coerce it into \(SL\):

sage: A2 = WeylCharacterRing("A2", style="coroots")
sage: rho = A2.space().rho().coerce_to_sl(); rho
(1, 0, -1)
sage: rho == (1/2)*sum(A2.space().positive_roots())
True

You do not need to do this for other Cartan types. If you are working with (say) \(F4\) then a \(\rho\) is a \(\rho\):

sage: F4 = WeylCharacterRing("F4")
sage: L = F4.space()
sage: rho = L.rho()
sage: rho == (1/2)*sum(L.positive_roots())
True

Integration#

Suppose that we wish to compute the integral

\[\int_{U(n)} |tr(g)|^{2k}\,dg\]

for various \(n\). Here \(U(n)\) is the unitary group, which is the maximal compact subgroup of \(GL(n,\mathbf{C})\), and \(dg\) is the Haar measure on \(U(n)\), normalized so that the volume of the group is 1.

The irreducible unitary representations of \(U(n)\) may be regarded as the basis elements of the WeylCharacterRing of type \(A_r\), where \(r=n-1\) so we might work in that ring. The trace \(tr(g)\) is then just the character of the standard representation. We may realize it in the WeylCharacterRing by taking the first fundamental weight and coercing it into the ring. For example, if \(k=5\) and \(n=3\) so \(r=2\):

sage: A2 = WeylCharacterRing("A2")
sage: fw = A2.fundamental_weights(); fw
Finite family {1: (1, 0, 0), 2: (1, 1, 0)}
sage: tr = A2(fw[1]); tr
A2(1,0,0)

sage: tr^5
5*A2(2,2,1) + 6*A2(3,1,1) + 5*A2(3,2,0) + 4*A2(4,1,0) + A2(5,0,0)
sage: sorted((tr^5).monomials(), key=lambda x: tuple(x.support()))
[A2(2,2,1), A2(3,1,1), A2(3,2,0), A2(4,1,0), A2(5,0,0)]
sage: sorted((tr^5).coefficients())
[1, 4, 5, 5, 6]
sage: sum(x^2 for x in (tr^5).coefficients())
103

So far we have been working with \(n=3\). For general \(n\):

sage: def f(n,k):
....:    R = WeylCharacterRing(['A',n-1])
....:    tr = R(R.fundamental_weights()[1])
....:    return sum(x^2 for x in (tr^k).coefficients())
....:
sage: [f(n,5) for n in [2..7]]  # long time (31s on sage.math, 2012)
[42, 103, 119, 120, 120, 120]

We see that the 10-th moment of \(tr(g)\) is just \(5!\) when \(n\) is sufficiently large. What if we fix \(n\) and vary \(k\)?

sage: [f(2,k) for k in [1..10]]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]
sage: [catalan_number(k) for k in [1..10]]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]

Invariants and multiplicities#

Sometimes we are only interested in the multiplicity of the trivial representation in some character. This may be found by the method invariant_degree. Continuing from the preceding example,

sage: A2 = WeylCharacterRing("A2",style="coroots")
sage: ad = A2(1,1)
sage: [ad.symmetric_power(k).invariant_degree() for k in [0..6]]
[1, 0, 1, 1, 1, 1, 2]
sage: [ad.exterior_power(k).invariant_degree() for k in [0..6]]
[1, 0, 0, 1, 0, 1, 0]

If we want the multiplicity of some other representation, we may obtain that using the method multiplicity:

sage: (ad^3).multiplicity(ad)
8

Weight Rings#

You may wish to work directly with the weights of a representation.

Weyl character ring elements are represented internally by a dictionary of their weights with multiplicities. However these are subject to a constraint: the coefficients must be invariant under the action of the Weyl group.

The WeightRing is also a ring whose elements are represented internally by a dictionary of their weights with multiplicities, but it is not subject to this constraint of Weyl group invariance. The weights are allowed to be fractional, that is, elements of the ambient space. In other words, the weight ring is the group algebra over the ambient space of the weight lattice.

To create a WeightRing first construct the WeylCharacterRing, then create the WeightRing as follows:

sage: A2 = WeylCharacterRing(['A',2])
sage: a2 = WeightRing(A2)

You may coerce elements of the WeylCharacterRing into the weight ring. For example, if you want to see the weights of the adjoint representation of \(GL(3)\), you may use the method mlist, but another way is to coerce it into the weight ring:

sage: from pprint import pprint
sage: A2 = WeylCharacterRing(['A',2])
sage: ad = A2(1,0,-1)
sage: pprint(ad.weight_multiplicities())
{(0, 0, 0): 2, (-1, 1, 0): 1, (-1, 0, 1): 1, (1, -1, 0): 1,
 (1, 0, -1): 1, (0, -1, 1): 1, (0, 1, -1): 1}

This command produces a dictionary of the weights that appear in the representation, together with their multiplicities. But another way of getting the same information, with an aim of working with it, is to coerce it into the weight ring:

sage: a2 = WeightRing(A2)
sage: a2(ad)
2*a2(0,0,0) + a2(-1,1,0) + a2(-1,0,1) + a2(1,-1,0) + a2(1,0,-1) + a2(0,-1,1) + a2(0,1,-1)

For example, the Weyl denominator formula is usually written this way:

\[\prod_{\alpha\in\Phi^+}\left(e^{\alpha/2}-e^{-\alpha/2}\right) = \sum_{w\in W} (-1)^{l(w)}e^{w(\rho)}.\]

The notation is as follows. Here if \(\lambda\) is a weight, or more generally, an element of the ambient space, then \(e^\lambda\) means the image of \(\lambda\) in the group algebra of the ambient space of the weight lattice \(\lambda\). Since this group algebra is just the weight ring, we can interpret \(e^\lambda\) as its image in the weight ring.

Let us confirm the Weyl denominator formula for A2:

sage: A2 = WeylCharacterRing("A2")
sage: a2 = WeightRing(A2)
sage: L = A2.space()
sage: W = L.weyl_group()
sage: rho = L.rho().coerce_to_sl()
sage: lhs = prod(a2(alpha/2)-a2(-alpha/2) for alpha in L.positive_roots()); lhs
a2(-1,1,0) - a2(-1,0,1) - a2(1,-1,0) + a2(1,0,-1) + a2(0,-1,1) - a2(0,1,-1)
sage: rhs = sum((-1)^(w.length())*a2(w.action(rho)) for w in W); rhs
a2(-1,1,0) - a2(-1,0,1) - a2(1,-1,0) + a2(1,0,-1) + a2(0,-1,1) - a2(0,1,-1)
sage: lhs == rhs
True

Note that we have to be careful to use the right value of \(\rho\). The reason for this is explained in SL versus GL.

We have seen that elements of the WeylCharacterRing can be coerced into the WeightRing. Elements of the WeightRing can be coerced into the WeylCharacterRing provided they are invariant under the Weyl group.