The Weyl character ring is the representation ring of a compact Lie group. It
has a basis consisting of the irreducible representations of , or
equivalently, their characters. The addition and multiplication in the Weyl
character ring correspond to direct sum and tensor product of representations.
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)
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:
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]
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 , the double cover of the orthogonal group
. 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 and
; you also some information about representations of E6 and E7
for the same reason.
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 of the WeylCharacterRing is the representation
corresponding to the third highest weight representation, the
eight-dimensional spin representation of
. We could
just as easily construct it with the commmand:
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: list(chi)
[((1, 1, 1), 1), ((1, 0, 0), 1), ((1, 1, 0), 1), ((0, 0, 0), 1)]
sage: dict(chi)
{(1, 1, 1): 1, (1, 0, 0): 1, (1, 1, 0): 1, (0, 0, 0): 1}
sage: chi.monomials()
[B3(0,0,0), B3(1,0,0), B3(1,1,0), B3(1,1,1)]
sage: 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 chi.monomials()]
[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 .
The weights of the character are available (with their coefficients) through the method weight_multiplicities. Continuing from the example in the last section:
sage: chi.weight_multiplicities()
{(0, 1, 0): 4, (1, -1, 1): 1, (-1, -1, 1): 1, (0, 1, 1): 2,
(0, -1, -1): 2, (0, -1, 0): 4, (1, -1, -1): 1, (0, 1, -1): 2,
(-1, 0, 1): 2, (1, 1, 0): 2, (-1, 0, 0): 4, (-1, 1, 0): 2,
(1, 0, 0): 4, (-1, -1, 0): 2, (1, 0, -1): 2, (0, 0, -1): 4,
(0, -1, 1): 2, (1, 1, -1): 1, (0, 0, 1): 4, (-1, 0, -1): 2,
(-1, 1, 1): 1, (-1, 1, -1): 1, (0, 0, 0): 8, (-1, -1, -1): 1,
(1, 0, 1): 2, (1, 1, 1): 1, (1, -1, 0): 2}
Each key of this dictionary is a weight, and its value is the multiplicity of that weight in the character.
Suppose that we wish to compute the integral
for various . Here
is the unitary group, which is the maximal
compact subroup of
. The irreducible unitary representations
of
may be regarded as the basis elements of the WeylCharacterRing of
type
, where
so we might work in that ring. The trace
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
and
so
:
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 with
itself, that is, the integral of
:
sage: sum(d^2 for d in (tr^5).coefficients())
103
So far we have been working with . For general
:
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 is just
when
is sufficiently
large. What if we fix
and vary
?
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]
The Frobeinus-Schur indicator of an irreducible representation of a
compact Lie group with character
is:
The Haar measure is normalized so that . The
Frobenius-Schur indicator equals
if the representation is real
(orthogonal),
if it is quaternionic (symplectic) and
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 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 with Cartan type
, then computed
the Frobenius-Schur indicators for a few values. From this experiment
we see that the spin representations of
and
are
symplectic, while those of
and
are orthogonal.
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 -th exterior square of a representation is zero if
is greater than the degree of the representation. However the
-th symmetric power is nonzero for all
.
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 , 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!
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 is the Weyl vector
(half the sum of the positive roots) then the degree of the irreducible
representation with highest weight
equals
where
is the number
of positive roots. Let us check this for
. In this case
:
sage: L = RootSystem("E8").ambient_space()
sage: len(L.positive_roots())
120
sage: 2^120
1329227995784915872903807060280344576
sage: L.weyl_dimension(L.rho())
1329227995784915872903807060280344576
Sage takes the weight space for type ['A',r] to be
dimensional. As a biproduct, if you create the Weyl character ring
with the command:
sage: A2 = WeylCharacterRing("A2")
Then you are effectively working with instead of
. For
example, the determinant is the character A2(1,1,1). However, as
we will explain later, you can work with
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 .
There are some advantages to this arrangement:
However if you want to be an purist, Sage will support you. The
weight space for
can be taken to be the hyperplane in
consisting of vectors
with
. 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 automatically:
sage: A2 = WeylCharacterRing("A2", style="coroots")
sage: A2(1,0).weight_multiplicities()
{(-1/3, -1/3, 2/3): 1, (2/3, -1/3, -1/3): 1, (-1/3, 2/3, -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 weight
lattice, you may need to coerce the Weyl vector
if you are
working with
. The default value for
in type
is
, but if you are an
purist you want
Therefore take the value of that you get from the method of the
ambient space and coerce it into
:
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) then a
is a
:
sage: F4 = WeylCharacterRing("F4")
sage: L = F4.space()
sage: rho = L.rho()
sage: rho == (1/2)*sum(L.positive_roots())
True
Suppose that we wish to compute the integral
for various . Here
is the unitary group, which is the maximal
compact subroup of
, and
is the Haar measure on
, normalized so that the volume of the group is 1.
The irreducible unitary representations of may be regarded as the basis
elements of the WeylCharacterRing of type
, where
so we might
work in that ring. The trace
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
and
so
:
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 with
itself, that is, the integral of
:
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: (tr^5).monomials()
[A2(2,2,1), A2(3,1,1), A2(3,2,0), A2(4,1,0), A2(5,0,0)]
sage: (tr^5).coefficients()
[5, 6, 5, 4, 1]
sage: sum(x^2 for x in (tr^5).coefficients())
103
So far we have been working with . For general
:
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 is just
when
is sufficiently
large. What if we fix
and vary
?
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]
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