# Weyl Character Rings¶

class sage.combinat.root_system.weyl_characters.FusionRing(ct, base_ring=Integer Ring, prefix=None, style='lattice', k=None)

Return the Fusion Ring (Verlinde Algebra) of level k.

INPUT:

• ct – the Cartan type of a simple (finite-dimensional) Lie algebra
• k – a nonnegative integer

This algebra has a basis indexed by the weights of level $$\leq k$$. It is implemented as a variant of the WeylCharacterRing.

EXAMPLES:

sage: A22 = FusionRing("A2",2)
sage: [f1, f2] = A22.fundamental_weights()
sage: M = [A22(x) for x in [0*f1, 2*f1, 2*f2, f1+f2, f2, f1]]
sage: [M * x for x in M]
[A22(1,1),
A22(0,1),
A22(1,0),
A22(0,0) + A22(1,1),
A22(0,1) + A22(2,0),
A22(1,0) + A22(0,2)]


You may assign your own labels to the basis elements. In the next example, we create the $$SO(5)$$ fusion ring of level $$2$$, check the weights of the basis elements, then assign new labels to them:

sage: B22 = FusionRing("B2", 2)
sage: basis = sorted(B22.basis(), key=str)
sage: basis
[B22(0,0), B22(0,1), B22(0,2), B22(1,0), B22(1,1), B22(2,0)]
sage: [x.highest_weight() for x in basis]
[(0, 0), (1/2, 1/2), (1, 1), (1, 0), (3/2, 1/2), (2, 0)]
sage: B22.fusion_labels(['1','X','Y2','Y1','Xp','Z'])
sage: relabeled_basis = sorted(B22.basis(), key=str)
sage: relabeled_basis
[1, X, Xp, Y1, Y2, Z]
sage: [(x, x.highest_weight()) for x in relabeled_basis]
[(1, (0, 0)),
(X, (1/2, 1/2)),
(Xp, (3/2, 1/2)),
(Y1, (1, 0)),
(Y2, (1, 1)),
(Z, (2, 0))]
sage: X*Y1
X + Xp
sage: Z*Z
1

sage: C22 = FusionRing("C2", 2)
sage: sorted(C22.basis(), key=str)
[C22(0,0), C22(0,1), C22(0,2), C22(1,0), C22(1,1), C22(2,0)]


REFERENCES:

fusion_labels(labels=None, key=<class 'str'>)

Set the labels of the basis.

INPUT:

• labels – (default: None) a list of strings
• key – (default: str) key to use to sort basis

The length of the list labels must equal the number of basis elements. These become the names of the basis elements. If labels is None, then this resets the labels to the default.

Note that the basis is stored as unsorted data, so to obtain consistent results, it should be sorted when applying labels. The argument key (default str) specifies how to sort the basis. If you call this with key=None, then no sorting is done. This may lead to random results, at least with Python 3.

EXAMPLES:

sage: A13 = FusionRing("A1", 3)
sage: A13.fusion_labels(['x0','x1','x2','x3'])
sage: fb = list(A13.basis()); fb
[x0, x1, x2, x3]
sage: Matrix([[x*y for y in A13.basis()] for x in A13.basis()])
[     x0      x1      x2      x3]
[     x1 x0 + x2 x1 + x3      x2]
[     x2 x1 + x3 x0 + x2      x1]
[     x3      x2      x1      x0]


We reset the labels to the default:

sage: A13.fusion_labels()
sage: fb
[A13(0), A13(1), A13(2), A13(3)]

some_elements()

Return some elements of self.

EXAMPLES:

sage: D41 = FusionRing('D4', 1)
sage: D41.some_elements()
[D41(1,0,0,0), D41(0,0,1,0), D41(0,0,0,1)]

class sage.combinat.root_system.weyl_characters.WeightRing(parent, prefix)

The weight ring, which is the group algebra over a weight lattice.

A Weyl character may be regarded as an element of the weight ring. In fact, an element of the weight ring is an element of the Weyl character ring if and only if it is invariant under the action of the Weyl group.

The advantage of the weight ring over the Weyl character ring is that one may conduct calculations in the weight ring that involve sums of weights that are not Weyl group invariant.

EXAMPLES:

sage: A2 = WeylCharacterRing(['A',2])
sage: a2 = WeightRing(A2)
sage: wd = prod(a2(x/2)-a2(-x/2) for x in a2.space().positive_roots()); wd
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: chi = A2([5,3,0]); chi
A2(5,3,0)
sage: a2(chi)
a2(1,2,5) + 2*a2(1,3,4) + 2*a2(1,4,3) + a2(1,5,2) + a2(2,1,5)
+ 2*a2(2,2,4) + 3*a2(2,3,3) + 2*a2(2,4,2) + a2(2,5,1) + 2*a2(3,1,4)
+ 3*a2(3,2,3) + 3*a2(3,3,2) + 2*a2(3,4,1) + a2(3,5,0) + a2(3,0,5)
+ 2*a2(4,1,3) + 2*a2(4,2,2) + 2*a2(4,3,1) + a2(4,4,0) + a2(4,0,4)
+ a2(5,1,2) + a2(5,2,1) + a2(5,3,0) + a2(5,0,3) + a2(0,3,5)
+ a2(0,4,4) + a2(0,5,3)
sage: a2(chi)*wd
-a2(-1,3,6) + a2(-1,6,3) + a2(3,-1,6) - a2(3,6,-1) - a2(6,-1,3) + a2(6,3,-1)
sage: sum((-1)^w.length()*a2([6,3,-1]).weyl_group_action(w) for w in a2.space().weyl_group())
-a2(-1,3,6) + a2(-1,6,3) + a2(3,-1,6) - a2(3,6,-1) - a2(6,-1,3) + a2(6,3,-1)
sage: a2(chi)*wd == sum((-1)^w.length()*a2([6,3,-1]).weyl_group_action(w) for w in a2.space().weyl_group())
True

class Element

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

A class for weight ring elements.

cartan_type()

Return the Cartan type.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: a2 = WeightRing(A2)
sage: a2([0,1,0]).cartan_type()
['A', 2]

character()

Assuming that self is invariant under the Weyl group, this will express it as a linear combination of characters. If self is not Weyl group invariant, this method will not terminate.

EXAMPLES:

sage: A2 = WeylCharacterRing(['A',2])
sage: a2 = WeightRing(A2)
sage: W = a2.space().weyl_group()
sage: mu = a2(2,1,0)
sage: nu = sum(mu.weyl_group_action(w) for w in W) ; nu
a2(1,2,0) + a2(1,0,2) + a2(2,1,0) + a2(2,0,1) + a2(0,1,2) + a2(0,2,1)
sage: nu.character()
-2*A2(1,1,1) + A2(2,1,0)

demazure(w, debug=False)

Return the result of applying the Demazure operator $$\partial_w$$ to self.

INPUT:

• w – a Weyl group element, or its reduced word

If $$w = s_i$$ is a simple reflection, the operation $$\partial_w$$ sends the weight $$\lambda$$ to

$\frac{\lambda - s_i \cdot \lambda + \alpha_i}{1 + \alpha_i}$

where the numerator is divisible the denominator in the weight ring. This is extended by multiplicativity to all $$w$$ in the Weyl group.

EXAMPLES:

sage: B2 = WeylCharacterRing("B2",style="coroots")
sage: b2 = WeightRing(B2)
sage: b2(1,0).demazure()
b2(1,0) + b2(-1,2)
sage: b2(1,0).demazure()
b2(1,0)
sage: r = b2(1,0).demazure([1,2]); r
b2(1,0) + b2(-1,2)
sage: r.demazure()
b2(1,0) + b2(-1,2)
sage: r.demazure()
b2(0,0) + b2(1,0) + b2(1,-2) + b2(-1,2)

demazure_lusztig(i, v)

Return the result of applying the Demazure-Lusztig operator $$T_i$$ to self.

INPUT:

• i – an element of the index set (or a reduced word or Weyl group element)
• v – an element of the base ring

If $$R$$ is the parent WeightRing, the Demazure-Lusztig operator $$T_i$$ is the linear map $$R \to R$$ that sends (for a weight $$\lambda$$) $$R(\lambda)$$ to

$(R(\alpha_i)-1)^{-1} \bigl(R(\lambda) - R(s_i\lambda) - v(R(\lambda) - R(\alpha_i + s_i \lambda)) \bigr)$

where the numerator is divisible by the denominator in $$R$$. The Demazure-Lusztig operators give a representation of the Iwahori–Hecke algebra associated to the Weyl group. See

• Lusztig, Equivariant $$K$$-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985), no. 2, 337-342.
• Cherednik, Nonsymmetric Macdonald polynomials. IMRN 10, 483-515 (1995).

In the examples, we confirm the braid and quadratic relations for type $$B_2$$.

EXAMPLES:

sage: P.<v> = PolynomialRing(QQ)
sage: B2 = WeylCharacterRing("B2",style="coroots",base_ring=P); b2 = B2.ambient()
sage: def T1(f): return f.demazure_lusztig(1,v)
sage: def T2(f): return f.demazure_lusztig(2,v)
sage: T1(T2(T1(T2(b2(1,-1)))))
(v^2-v)*b2(0,-1) + v^2*b2(-1,1)
sage: [T1(T1(f))==(v-1)*T1(f)+v*f for f in [b2(0,0), b2(1,0), b2(2,3)]]
[True, True, True]
sage: [T1(T2(T1(T2(b2(i,j))))) == T2(T1(T2(T1(b2(i,j))))) for i in [-2..2] for j in [-1,1]]
[True, True, True, True, True, True, True, True, True, True]


Instead of an index $$i$$ one may use a reduced word or Weyl group element:

sage: b2(1,0).demazure_lusztig([2,1],v)==T2(T1(b2(1,0)))
True
sage: W = B2.space().weyl_group(prefix="s")
sage: [s1,s2]=W.simple_reflections()
sage: b2(1,0).demazure_lusztig(s2*s1,v)==T2(T1(b2(1,0)))
True

scale(k)

Multiply a weight by $$k$$.

The operation is extended by linearity to the weight ring.

INPUT:

• k – a nonzero integer

EXAMPLES:

sage: g2 = WeylCharacterRing("G2",style="coroots").ambient()
sage: g2(2,3).scale(2)
g2(4,6)

shift(mu)

Add $$\mu$$ to any weight.

Extended by linearity to the weight ring.

INPUT:

• mu – a weight

EXAMPLES:

sage: g2 = WeylCharacterRing("G2",style="coroots").ambient()
sage: [g2(1,2).shift(fw) for fw in g2.fundamental_weights()]
[g2(2,2), g2(1,3)]

weyl_group_action(w)

Return the action of the Weyl group element w on self.

EXAMPLES:

sage: G2 = WeylCharacterRing(['G',2])
sage: g2 = WeightRing(G2)
sage: L = g2.space()
sage: [fw1, fw2] = L.fundamental_weights()
sage: sum(g2(fw2).weyl_group_action(w) for w in L.weyl_group())
2*g2(-2,1,1) + 2*g2(-1,-1,2) + 2*g2(-1,2,-1) + 2*g2(1,-2,1) + 2*g2(1,1,-2) + 2*g2(2,-1,-1)

cartan_type()

Return the Cartan type.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: WeightRing(A2).cartan_type()
['A', 2]

fundamental_weights()

Return the fundamental weights.

EXAMPLES:

sage: WeightRing(WeylCharacterRing("G2")).fundamental_weights()
Finite family {1: (1, 0, -1), 2: (2, -1, -1)}

one_basis()

Return the index of $$1$$.

EXAMPLES:

sage: A3 = WeylCharacterRing("A3")
sage: WeightRing(A3).one_basis()
(0, 0, 0, 0)
sage: WeightRing(A3).one()
a3(0,0,0,0)

parent()

Return the parent Weyl character ring.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: a2 = WeightRing(A2)
sage: a2.parent()
The Weyl Character Ring of Type A2 with Integer Ring coefficients
sage: a2.parent() == A2
True

positive_roots()

Return the positive roots.

EXAMPLES:

sage: WeightRing(WeylCharacterRing("G2")).positive_roots()
[(0, 1, -1), (1, -2, 1), (1, -1, 0), (1, 0, -1), (1, 1, -2), (2, -1, -1)]

product_on_basis(a, b)

Return the product of basis elements indexed by a and b.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: a2 = WeightRing(A2)
sage: a2(1,0,0) * a2(0,1,0) # indirect doctest
a2(1,1,0)

simple_roots()

Return the simple roots.

EXAMPLES:

sage: WeightRing(WeylCharacterRing("G2")).simple_roots()
Finite family {1: (0, 1, -1), 2: (1, -2, 1)}

some_elements()

Return some elements of self.

EXAMPLES:

sage: A3 = WeylCharacterRing("A3")
sage: a3 = WeightRing(A3)
sage: a3.some_elements()
[a3(1,0,0,0), a3(1,1,0,0), a3(1,1,1,0)]

space()

Return the weight space realization associated to self.

EXAMPLES:

sage: E8 = WeylCharacterRing(['E',8])
sage: e8 = WeightRing(E8)
sage: e8.space()
Ambient space of the Root system of type ['E', 8]

weyl_character_ring()

Return the parent Weyl Character Ring.

A synonym for self.parent().

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: a2 = WeightRing(A2)
sage: a2.weyl_character_ring()
The Weyl Character Ring of Type A2 with Integer Ring coefficients

wt_repr(wt)

Return a string representing the irreducible character with highest weight vector wt.

Uses coroot notation if the associated Weyl character ring is defined with style="coroots".

EXAMPLES:

sage: G2 = WeylCharacterRing("G2")
sage: [G2.ambient().wt_repr(x) for x in G2.fundamental_weights()]
['g2(1,0,-1)', 'g2(2,-1,-1)']
sage: G2 = WeylCharacterRing("G2",style="coroots")
sage: [G2.ambient().wt_repr(x) for x in G2.fundamental_weights()]
['g2(1,0)', 'g2(0,1)']

class sage.combinat.root_system.weyl_characters.WeylCharacterRing(ct, base_ring=Integer Ring, prefix=None, style='lattice', k=None)

A class for rings of Weyl characters.

Let $$K$$ be a compact Lie group, which we assume is semisimple and simply-connected. Its complexified Lie algebra $$L$$ is the Lie algebra of a complex analytic Lie group $$G$$. The following three categories are equivalent: finite-dimensional representations of $$K$$; finite-dimensional representations of $$L$$; and finite-dimensional analytic representations of $$G$$. In every case, there is a parametrization of the irreducible representations by their highest weight vectors. For this theory of Weyl, see (for example):

• Adams, Lectures on Lie groups
• Broecker and Tom Dieck, Representations of Compact Lie groups
• Bump, Lie Groups
• Fulton and Harris, Representation Theory
• Goodman and Wallach, Representations and Invariants of the Classical Groups
• Hall, Lie Groups, Lie Algebras and Representations
• Humphreys, Introduction to Lie Algebras and their representations
• Procesi, Lie Groups
• Samelson, Notes on Lie Algebras
• Varadarajan, Lie groups, Lie algebras, and their representations
• Zhelobenko, Compact Lie Groups and their Representations.

Computations that you can do with these include computing their weight multiplicities, products (thus decomposing the tensor product of a representation into irreducibles) and branching rules (restriction to a smaller group).

There is associated with $$K$$, $$L$$ or $$G$$ as above a lattice, the weight lattice, whose elements (called weights) are characters of a Cartan subgroup or subalgebra. There is an action of the Weyl group $$W$$ on the lattice, and elements of a fixed fundamental domain for $$W$$, the positive Weyl chamber, are called dominant. There is for each representation a unique highest dominant weight that occurs with nonzero multiplicity with respect to a certain partial order, and it is called the highest weight vector.

EXAMPLES:

sage: L = RootSystem("A2").ambient_space()
sage: [fw1,fw2] = L.fundamental_weights()
sage: R = WeylCharacterRing(['A',2], prefix="R")
sage: [R(1),R(fw1),R(fw2)]
[R(0,0,0), R(1,0,0), R(1,1,0)]


Here R(1), R(fw1), and R(fw2) are irreducible representations with highest weight vectors $$0$$, $$\Lambda_1$$, and $$\Lambda_2$$ respectively (the first two fundamental weights).

For type $$A$$ (also $$G_2$$, $$F_4$$, $$E_6$$ and $$E_7$$) we will take as the weight lattice not the weight lattice of the semisimple group, but for a larger one. For type $$A$$, this means we are concerned with the representation theory of $$K = U(n)$$ or $$G = GL(n, \CC)$$ rather than $$SU(n)$$ or $$SU(n, \CC)$$. This is useful since the representation theory of $$GL(n)$$ is ubiquitous, and also since we may then represent the fundamental weights (in sage.combinat.root_system.root_system) by vectors with integer entries. If you are only interested in $$SL(3)$$, say, use WeylCharacterRing(['A',2]) as above but be aware that R([a,b,c]) and R([a+1,b+1,c+1]) represent the same character of $$SL(3)$$ since R([1,1,1]) is the determinant.

For more information, see the thematic tutorial Lie Methods and Related Combinatorics in Sage, available at:

https://doc.sagemath.org/html/en/thematic_tutorials/lie.html

class Element

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

A class for Weyl characters.

adams_operation(r)

Return the $$r$$-th Adams operation of self.

INPUT:

• r – a positive integer

This is a virtual character, whose weights are the weights of self, each multiplied by $$r$$.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
A2(2,2,2) - A2(3,2,1) + A2(3,3,0)

branch(S, rule='default')

Return the restriction of the character to the subalgebra.

If no rule is specified, we will try to specify one.

INPUT:

• S – a Weyl character ring for a Lie subgroup or subalgebra
• rule – a branching rule

See branch_weyl_character() for more information about branching rules.

EXAMPLES:

sage: B3 = WeylCharacterRing(['B',3])
sage: A2 = WeylCharacterRing(['A',2])
sage: [B3(w).branch(A2,rule="levi") for w in B3.fundamental_weights()]
[A2(0,0,0) + A2(1,0,0) + A2(0,0,-1),
A2(0,0,0) + A2(1,0,0) + A2(1,1,0) + A2(1,0,-1) + A2(0,-1,-1) + A2(0,0,-1),
A2(-1/2,-1/2,-1/2) + A2(1/2,-1/2,-1/2) + A2(1/2,1/2,-1/2) + A2(1/2,1/2,1/2)]

cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: A2([1,0,0]).cartan_type()
['A', 2]

degree()

Return the degree of self.

This is the dimension of the associated module.

EXAMPLES:

sage: B3 = WeylCharacterRing(['B',3])
sage: [B3(x).degree() for x in B3.fundamental_weights()]
[7, 21, 8]

dual()

The involution that replaces a representation with its contragredient. (For Fusion rings, this is the conjugation map.)

EXAMPLES:

sage: A3 = WeylCharacterRing("A3", style="coroots")
sage: A3(1,0,0)^2
A3(0,1,0) + A3(2,0,0)
sage: (A3(1,0,0)^2).dual()
A3(0,1,0) + A3(0,0,2)

exterior_power(k)

Return the $$k$$-th exterior power of self.

INPUT:

• k – a nonnegative integer

The algorithm is based on the identity $$k e_k = \sum_{r=1}^k (-1)^{k-1} p_k e_{k-r}$$ relating the power-sum and elementary symmetric polynomials. Applying this to the eigenvalues of an element of the parent Lie group in the representation self, the $$e_k$$ become exterior powers and the $$p_k$$ become Adams operations, giving an efficient recursive implementation.

EXAMPLES:

sage: B3 = WeylCharacterRing("B3",style="coroots")
sage: spin = B3(0,0,1)
sage: spin.exterior_power(6)
B3(1,0,0) + B3(0,1,0)

exterior_square()

Return the exterior square of the character.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2",style="coroots")
sage: A2(1,0).exterior_square()
A2(0,1)

frobenius_schur_indicator()

Return:

• $$1$$ if the representation is real (orthogonal)
• $$-1$$ if the representation is quaternionic (symplectic)
• $$0$$ if the representation is complex (not self dual)

The Frobenius-Schur indicator of a character $$\chi$$ of a compact group $$G$$ is the Haar integral over the group of $$\chi(g^2)$$. Its value is 1, -1 or 0. This method computes it for irreducible characters of compact Lie groups by checking whether the symmetric and exterior square characters contain the trivial character.

Todo

Try to compute this directly without actually calculating the full symmetric and exterior squares.

EXAMPLES:

sage: B2 = WeylCharacterRing("B2",style="coroots")
sage: B2(1,0).frobenius_schur_indicator()
1
sage: B2(0,1).frobenius_schur_indicator()
-1

highest_weight()

Return the parametrizing dominant weight of an irreducible character or simple element of a FusionRing.

This method is only available for basis elements.

EXAMPLES:

sage: G2 = WeylCharacterRing("G2", style="coroots")
sage: [x.highest_weight() for x in [G2(1,0),G2(0,1)]]
[(1, 0, -1), (2, -1, -1)]
sage: A21 = FusionRing("A2",1)
sage: sorted([x.highest_weight() for x in A21.basis()])
[(0, 0, 0), (1/3, 1/3, -2/3), (2/3, -1/3, -1/3)]

inner_product(other)

Compute the inner product with another character.

The irreducible characters are an orthonormal basis with respect to the usual inner product of characters, interpreted as functions on a compact Lie group, by Schur orthogonality.

INPUT:

• other – another character

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: [f1,f2] = A2.fundamental_weights()
sage: r1 = A2(f1)*A2(f2); r1
A2(1,1,1) + A2(2,1,0)
sage: r2 = A2(f1)^3; r2
A2(1,1,1) + 2*A2(2,1,0) + A2(3,0,0)
sage: r1.inner_product(r2)
3

invariant_degree()

Return the multiplicity of the trivial representation in self.

Multiplicities of other irreducibles may be obtained using multiplicity().

EXAMPLES:

sage: A2 = WeylCharacterRing("A2",style="coroots")
sage: rep = A2(1,0)^2*A2(0,1)^2; rep
2*A2(0,0) + A2(0,3) + 4*A2(1,1) + A2(3,0) + A2(2,2)
sage: rep.invariant_degree()
2

is_irreducible()

Return whether self is an irreducible character.

EXAMPLES:

sage: B3 = WeylCharacterRing(['B',3])
sage: [B3(x).is_irreducible() for x in B3.fundamental_weights()]
[True, True, True]
sage: sum(B3(x) for x in B3.fundamental_weights()).is_irreducible()
False

multiplicity(other)

Return the multiplicity of the irreducible other in self.

INPUT:

• other – an irreducible character

EXAMPLES:

sage: B2 = WeylCharacterRing("B2",style="coroots")
sage: rep = B2(1,1)^2; rep
B2(0,0) + B2(1,0) + 2*B2(0,2) + B2(2,0) + 2*B2(1,2) + B2(0,4) + B2(3,0) + B2(2,2)
sage: rep.multiplicity(B2(0,2))
2

symmetric_power(k)

Return the $$k$$-th symmetric power of self.

INPUT:

• $$k$$ – a nonnegative integer

The algorithm is based on the identity $$k h_k = \sum_{r=1}^k p_k h_{k-r}$$ relating the power-sum and complete symmetric polynomials. Applying this to the eigenvalues of an element of the parent Lie group in the representation self, the $$h_k$$ become symmetric powers and the $$p_k$$ become Adams operations, giving an efficient recursive implementation.

EXAMPLES:

sage: B3 = WeylCharacterRing("B3",style="coroots")
sage: spin = B3(0,0,1)
sage: spin.symmetric_power(6)
B3(0,0,0) + B3(0,0,2) + B3(0,0,4) + B3(0,0,6)

symmetric_square()

Return the symmetric square of the character.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2",style="coroots")
sage: A2(1,0).symmetric_square()
A2(2,0)

weight_multiplicities()

Return the dictionary of weight multiplicities for the Weyl character self.

The character does not have to be irreducible.

EXAMPLES:

sage: B2 = WeylCharacterRing("B2",style="coroots")
sage: B2(0,1).weight_multiplicities()
{(-1/2, -1/2): 1, (-1/2, 1/2): 1, (1/2, -1/2): 1, (1/2, 1/2): 1}

adjoint_representation()

Return the adjoint representation as an element of the WeylCharacterRing.

EXAMPLES:

sage: G2 = WeylCharacterRing("G2",style="coroots")
G2(0,1)

affine_reflect(wt, k=0)

INPUT:

• wt – a weight
• k – (optional) a positive integer

Returns the reflection of wt in the hyperplane $$\theta$$. Optionally shifts by a multiple $$k$$.

EXAMPLES:

sage: B22 = FusionRing("B2",2)
sage: fw = B22.fundamental_weights(); fw
Finite family {1: (1, 0), 2: (1/2, 1/2)}
sage: [B22.affine_reflect(x,2) for x in fw]
[(2, 1), (3/2, 3/2)]

ambient()

Return the weight ring of self.

EXAMPLES:

sage: WeylCharacterRing("A2").ambient()
The Weight ring attached to The Weyl Character Ring of Type A2 with Integer Ring coefficients

base_ring()

Return the base ring of self.

EXAMPLES:

sage: R = WeylCharacterRing(['A',3], base_ring = CC); R.base_ring()
Complex Field with 53 bits of precision

cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: WeylCharacterRing("A2").cartan_type()
['A', 2]

char_from_weights(mdict)

Construct a Weyl character from an invariant linear combination of weights.

INPUT:

• mdict – a dictionary mapping weights to coefficients, and representing a linear combination of weights which shall be invariant under the action of the Weyl group

OUTPUT: the corresponding Weyl character

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: v = A2._space([3,1,0]); v
(3, 1, 0)
sage: d = dict([(x,1) for x in v.orbit()]); d
{(1, 3, 0): 1,
(1, 0, 3): 1,
(3, 1, 0): 1,
(3, 0, 1): 1,
(0, 1, 3): 1,
(0, 3, 1): 1}
sage: A2.char_from_weights(d)
-A2(2,1,1) - A2(2,2,0) + A2(3,1,0)

demazure_character(hwv, word, debug=False)

Compute the Demazure character.

INPUT:

• hwv – a (usually dominant) weight
• word – a Weyl group word

Produces the Demazure character with highest weight hwv and word as an element of the weight ring. Only available if style="coroots". The Demazure operators are also available as methods of WeightRing elements, and as methods of crystals. Given a CrystalOfTableaux with given highest weight vector, the Demazure method on the crystal will give the equivalent of this method, except that the Demazure character of the crystal is given as a sum of monomials instead of an element of the WeightRing.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2",style="coroots")
sage: h = sum(A2.fundamental_weights()); h
(2, 1, 0)
sage: A2.demazure_character(h,word=[1,2])
a2(0,0) + a2(-2,1) + a2(2,-1) + a2(1,1) + a2(-1,2)
sage: A2.demazure_character((1,1),word=[1,2])
a2(0,0) + a2(-2,1) + a2(2,-1) + a2(1,1) + a2(-1,2)

dot_reduce(a)

Auxiliary function for product_on_basis().

Return a pair $$[\epsilon, b]$$ where $$b$$ is a dominant weight and $$\epsilon$$ is 0, 1 or -1. To describe $$b$$, let $$w$$ be an element of the Weyl group such that $$w(a + \rho)$$ is dominant. If $$w(a + \rho) - \rho$$ is dominant, then $$\epsilon$$ is the sign of $$w$$ and $$b$$ is $$w(a + \rho) - \rho$$. Otherwise, $$\epsilon$$ is zero.

INPUT:

• a – a weight

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: weights = sorted(A2(2,1,0).weight_multiplicities().keys(), key=str); weights
[(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0)]
sage: [A2.dot_reduce(x) for x in weights]
[[0, (0, 0, 0)], [-1, (1, 1, 1)], [-1, (1, 1, 1)], [1, (1, 1, 1)], [0, (0, 0, 0)], [0, (0, 0, 0)], [1, (2, 1, 0)]]

dynkin_diagram()

Return the Dynkin diagram of self.

EXAMPLES:

sage: WeylCharacterRing("E7").dynkin_diagram()
O 2
|
|
O---O---O---O---O---O
1   3   4   5   6   7
E7

extended_dynkin_diagram()

Return the extended Dynkin diagram, which is the Dynkin diagram of the corresponding untwisted affine type.

EXAMPLES:

sage: WeylCharacterRing("E7").extended_dynkin_diagram()
O 2
|
|
O---O---O---O---O---O---O
0   1   3   4   5   6   7
E7~

fundamental_weights()

Return the fundamental weights.

EXAMPLES:

sage: WeylCharacterRing("G2").fundamental_weights()
Finite family {1: (1, 0, -1), 2: (2, -1, -1)}

highest_root()

Return the highest root.

EXAMPLES:

sage: WeylCharacterRing("G2").highest_root()
(2, -1, -1)

irr_repr(hwv)

Return a string representing the irreducible character with highest weight vector hwv.

EXAMPLES:

sage: B3 = WeylCharacterRing("B3")
sage: [B3.irr_repr(v) for v in B3.fundamental_weights()]
['B3(1,0,0)', 'B3(1,1,0)', 'B3(1/2,1/2,1/2)']
sage: B3 = WeylCharacterRing("B3", style="coroots")
sage: [B3.irr_repr(v) for v in B3.fundamental_weights()]
['B3(1,0,0)', 'B3(0,1,0)', 'B3(0,0,1)']

level(wt)

Return the level of the weight, defined to be the value of the weight on the coroot associated with the highest root.

EXAMPLES:

sage: R = FusionRing("F4",2); [R.level(x) for x in R.fundamental_weights()]
[2, 3, 2, 1]
sage: [CartanType("F4~").dual().a()[x] for x in [1..4]]
[2, 3, 2, 1]

lift()

The embedding of self into its weight ring.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: A2.lift
Generic morphism:
From: The Weyl Character Ring of Type A2 with Integer Ring coefficients
To:   The Weight ring attached to The Weyl Character Ring of Type A2 with Integer Ring coefficients

sage: x = -A2(2,1,1) - A2(2,2,0) + A2(3,1,0)
sage: A2.lift(x)
a2(1,3,0) + a2(1,0,3) + a2(3,1,0) + a2(3,0,1) + a2(0,1,3) + a2(0,3,1)


As a shortcut, you may also do:

sage: x.lift()
a2(1,3,0) + a2(1,0,3) + a2(3,1,0) + a2(3,0,1) + a2(0,1,3) + a2(0,3,1)


Or even:

sage: a2 = WeightRing(A2)
sage: a2(x)
a2(1,3,0) + a2(1,0,3) + a2(3,1,0) + a2(3,0,1) + a2(0,1,3) + a2(0,3,1)

lift_on_basis(irr)

Expand the basis element indexed by the weight irr into the weight ring of self.

INPUT:

• irr – a dominant weight

This is used to implement lift().

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: v = A2._space([2,1,0]); v
(2, 1, 0)
sage: A2.lift_on_basis(v)
2*a2(1,1,1) + a2(1,2,0) + a2(1,0,2) + a2(2,1,0) + a2(2,0,1) + a2(0,1,2) + a2(0,2,1)


This is consistent with the analogous calculation with symmetric Schur functions:

sage: s = SymmetricFunctions(QQ).s()
sage: s[2,1].expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2

maximal_subgroup(ct)

Return a branching rule or a list of branching rules.

INPUT:

• ct – the Cartan type of a maximal subgroup of self.

In rare cases where there is more than one maximal subgroup (up to outer automorphisms) with the given Cartan type, the function returns a list of branching rules.

EXAMPLES:

sage: WeylCharacterRing("E7").maximal_subgroup("A2")
miscellaneous branching rule E7 => A2
sage: WeylCharacterRing("E7").maximal_subgroup("A1")
[iii branching rule E7 => A1, iv branching rule E7 => A1]


For more information, see the related method maximal_subgroups().

maximal_subgroups()

This method is only available if the Cartan type of self is irreducible and of rank no greater than 8. This method produces a list of the maximal subgroups of self, up to (possibly outer) automorphisms. Each line in the output gives the Cartan type of a maximal subgroup followed by a command that creates the branching rule.

EXAMPLES:

sage: WeylCharacterRing("E6").maximal_subgroups()
D5:branching_rule("E6","D5","levi")
C4:branching_rule("E6","C4","symmetric")
F4:branching_rule("E6","F4","symmetric")
A2:branching_rule("E6","A2","miscellaneous")
G2:branching_rule("E6","G2","miscellaneous")
A2xG2:branching_rule("E6","A2xG2","miscellaneous")
A1xA5:branching_rule("E6","A1xA5","extended")
A2xA2xA2:branching_rule("E6","A2xA2xA2","extended")


Note that there are other embeddings of (for example $$A_2$$ into $$E_6$$ as nonmaximal subgroups. These embeddings may be constructed by composing branching rules through various subgroups.

Once you know which maximal subgroup you are interested in, to create the branching rule, you may either paste the command to the right of the colon from the above output onto the command line, or alternatively invoke the related method maximal_subgroup():

sage: branching_rule("E6","G2","miscellaneous")
miscellaneous branching rule E6 => G2
sage: WeylCharacterRing("E6").maximal_subgroup("G2")
miscellaneous branching rule E6 => G2


It is believed that the list of maximal subgroups is complete, except that some subgroups may be not be invariant under outer automorphisms. It is reasonable to want a list of maximal subgroups that is complete up to conjugation, but to obtain such a list you may have to apply outer automorphisms. The group of outer automorphisms modulo inner automorphisms is isomorphic to the group of symmetries of the Dynkin diagram, and these are available as branching rules. The following example shows that while a branching rule from $$D_4$$ to $$A_1\times C_2$$ is supplied, another different one may be obtained by composing it with the triality automorphism of $$D_4$$:

sage: [D4,A1xC2]=[WeylCharacterRing(x,style="coroots") for x in ["D4","A1xC2"]]
sage: fw = D4.fundamental_weights()
sage: b = D4.maximal_subgroup("A1xC2")
sage: [D4(fw).branch(A1xC2,rule=b) for fw in D4.fundamental_weights()]
[A1xC2(1,1,0),
A1xC2(2,0,0) + A1xC2(2,0,1) + A1xC2(0,2,0),
A1xC2(1,1,0),
A1xC2(2,0,0) + A1xC2(0,0,1)]
sage: b1 = branching_rule("D4","D4","triality")*b
sage: [D4(fw).branch(A1xC2,rule=b1) for fw in D4.fundamental_weights()]
[A1xC2(1,1,0),
A1xC2(2,0,0) + A1xC2(2,0,1) + A1xC2(0,2,0),
A1xC2(2,0,0) + A1xC2(0,0,1),
A1xC2(1,1,0)]

one_basis()

Return the index of 1 in self.

EXAMPLES:

sage: WeylCharacterRing("A3").one_basis()
(0, 0, 0, 0)
sage: WeylCharacterRing("A3").one()
A3(0,0,0,0)

positive_roots()

Return the positive roots.

EXAMPLES:

sage: WeylCharacterRing("G2").positive_roots()
[(0, 1, -1), (1, -2, 1), (1, -1, 0), (1, 0, -1), (1, 1, -2), (2, -1, -1)]

product_on_basis(a, b)

Compute the tensor product of two irreducible representations a and b.

EXAMPLES:

sage: D4 = WeylCharacterRing(['D',4])
sage: spin_plus = D4(1/2,1/2,1/2,1/2)
sage: spin_minus = D4(1/2,1/2,1/2,-1/2)
sage: spin_plus * spin_minus # indirect doctest
D4(1,0,0,0) + D4(1,1,1,0)
sage: spin_minus * spin_plus
D4(1,0,0,0) + D4(1,1,1,0)


Uses the Brauer-Klimyk method.

rank()

Return the rank.

EXAMPLES:

sage: WeylCharacterRing("G2").rank()
2

retract()

The partial inverse map from the weight ring into self.

EXAMPLES:

sage: A2 = WeylCharacterRing("A2")
sage: a2 = WeightRing(A2)
sage: A2.retract
Generic morphism:
From: The Weight ring attached to The Weyl Character Ring of Type A2 with Integer Ring coefficients
To:   The Weyl Character Ring of Type A2 with Integer Ring coefficients

sage: v = A2._space([3,1,0]); v
(3, 1, 0)
sage: chi = a2.sum_of_monomials(v.orbit()); chi
a2(1,3,0) + a2(1,0,3) + a2(3,1,0) + a2(3,0,1) + a2(0,1,3) + a2(0,3,1)
sage: A2.retract(chi)
-A2(2,1,1) - A2(2,2,0) + A2(3,1,0)


The input should be invariant:

sage: A2.retract(a2.monomial(v))
Traceback (most recent call last):
...
ValueError: multiplicity dictionary may not be Weyl group invariant


As a shortcut, you may use conversion:

sage: A2(chi)
-A2(2,1,1) - A2(2,2,0) + A2(3,1,0)
sage: A2(a2.monomial(v))
Traceback (most recent call last):
...
ValueError: multiplicity dictionary may not be Weyl group invariant

simple_coroots()

Return the simple coroots.

EXAMPLES:

sage: WeylCharacterRing("G2").simple_coroots()
Finite family {1: (0, 1, -1), 2: (1/3, -2/3, 1/3)}

simple_roots()

Return the simple roots.

EXAMPLES:

sage: WeylCharacterRing("G2").simple_roots()
Finite family {1: (0, 1, -1), 2: (1, -2, 1)}

some_elements()

Return some elements of self.

EXAMPLES:

sage: WeylCharacterRing("A3").some_elements()
[A3(1,0,0,0), A3(1,1,0,0), A3(1,1,1,0)]

space()

Return the weight space associated to self.

EXAMPLES:

sage: WeylCharacterRing(['E',8]).space()
Ambient space of the Root system of type ['E', 8]

sage.combinat.root_system.weyl_characters.irreducible_character_freudenthal(hwv, debug=False)

Return the dictionary of multiplicities for the irreducible character with highest weight $$\lambda$$.

The weight multiplicities are computed by the Freudenthal multiplicity formula. The algorithm is based on recursion relation that is stated, for example, in Humphrey’s book on Lie Algebras. The multiplicities are invariant under the Weyl group, so to compute them it would be sufficient to compute them for the weights in the positive Weyl chamber. However after some testing it was found to be faster to compute every weight using the recursion, since the use of the Weyl group is expensive in its current implementation.

INPUT:

• hwv – a dominant weight in a weight lattice.
• L – the ambient space

EXAMPLES:

sage: WeylCharacterRing("A2")(2,1,0).weight_multiplicities() # indirect doctest
{(1, 1, 1): 2, (1, 2, 0): 1, (1, 0, 2): 1, (2, 1, 0): 1,
(2, 0, 1): 1, (0, 1, 2): 1, (0, 2, 1): 1}