Lecture 14: the Weyl Character Formula (II)

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The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Lecture 14: The Weyl Character Formula (II)

Daniel Bump

May 21, 2020 The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The weight lattice and the root lattice

Let G be a compact, connected Lie group. For simplicity let us assume that G is semisimple. Thus we assume that the roots span the weight lattice. For example, SU(n) is semisimple, but U(n) is not.

The sublattice Λroot spanned by the root lattice is of ﬁnite codimension in the root lattice Λ. For example, for SU(n) we have [Λ : Λroot] = n.

We haven’t talked about Dynkin diagrams and extended Dynkin diagrams yet, but we mention that the ﬁnite quotient group Λ/Λroot acts by automorphisms on the extended Dynkin diagram and is related to both the center of G and its fundamental group (Chapter 23). The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The root lattice and the Weyl group

Proposition

If λ ∈ Λ and w ∈ W then λ − w(λ) ∈ Λroot.

∨ We recall that we proved last time that α (λ) ∈ Z if α ∈ Φ and λ ∈ Λ.

First we check the Proposition if w = sα is a reﬂection. Then it follows from formula ∨ sα(λ) = λ − α (λ)α. We may now prove the proposition by induction on the length of w. If `(w) = 0, then w = 1 and this is trivial. Otherwise write 0 0 0 w = sαw where `(w ) = `(w) − 1 and we have λ − w (λ) ∈ Λroot. 0 0 w (λ) − w(λ) = µ − sα(µ) where µ = w (λ) and the statement follows from the special case already proved. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

A note about the Weyl vector

The Weyl vector ρ may or may not be in Λ.

If G is semisimple and simply connected, then ρ ∈ Λ. This follows from facts that are proved in Chapter 23.

In any case ρ − w(ρ) ∈ Λroot. For example

ρ − sα(ρ) = α

if α is a simple root. More generally X ρ − w(ρ) = α, α∈Φ+∩w−1Φ−

and the statement follows. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Review: the Weyl character formula

If λ is a dominant weight we deﬁned

P (−1)`(w)ew(λ+ρ) χ = w∈W , λ ∆ where the Weyl denominator Y ∆ = eρ (1 − e−α). α∈Φ+

We proved that χλ is the character of an irreducible representation.

We have a partial order on weights in which µ 4 λ if λ − µ is a linear combination with nonnegative integer coefﬁcients of the simple roots Σ. If µ is a weight of χλ then λ < µ, so we say λ is the highest weight of this representation. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The support of χλ

We proved last time that if µ is a weight of χλ, then µ 4 λ. Now the weights are w invariant, so we have other inequalities w(µ) 4 λ for w ∈ W. The effect of these inequalities is that µ must lie in the convex hull of the Weyl orbit Wλ.

Furthermore, all weights lie in the same coset of the root lattice Λroot. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The support of χλ: an example

λ = (3, 1, 0)

λ − α2 λ − 2α1

The dotted line delineates the region µ 4 λ. Other conditions come from w(µ) 4 λ placing the support inside the convex hull of Wλ. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The weights lie in a single coset of Λroot

λ = (3, 1, 0)

λ − α2 λ − 2α1 Lighter dots: Λ

Darker dots: Λroot + λ

The inequality µ 4 λ not only imposes a constraint that µ lie within a region, but it also requires that λ − µ lie in a particular coset of Λ. This differs from the convention in the book, but is consistent with Kac, Inﬁnite-dimensional Lie algebras. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The rings E and E2

In discussing the Weyl character formula we worked in a formal λ ring isomorphic to Z[Λ] with a basis e with λ ∈ Λ.

1 Sometimes we use a larger ring E2 corresponding to Z 2 Λ .

We can interpret elements of E as functions on T. Elements of E2 could be interpreted as functions on a ﬁnite cover of T.

We do not need to interpret E2 this way. But we do need a home for the Weyl vector

1 X ρ = α 2 α∈Φ+

which may or may not be in Λ. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Flexibility about ρ

We note that in the Weyl character formula

P(−1)`(w)ew(λ+ρ) χλ = ρ Q −α e α∈Φ+ (1 − e ) we could replace ρ by ρ + κ where κ is any vector such that w(κ) = κ for all w. Then both the numerator and the denominator are multiplied by eκ and the result is unchanged.

The condition that w(κ) = κ for all w is equivalent to the condition that κ is orthogonal to all roots, due to the formula 2hκ, αi r (κ) = κ − α. α hα, αi This is only useful if G is not semisimple, for if Φ spans Λ, the only such κ is zero. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

ρ for U(n)

However if G = U(n), and if we identify the weight lattice with n Z , then n − 1 n − 3 1 − n ρ = , , ··· , . 2 2 2 n−1 Instead, we may add κ = 2 (1, ··· , 1) and work with ρ0 = (n − 1, n − 2, ··· , 0). The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Partitions as dominant weights

Now for U(n) or GL(n, C), let λ be a dominant weight. Thus λ = (λ1, ··· , λn) where λ1 > ··· > λn. If λn > 0, then λ is a partition.

Thus a partition (of length 6 n) may be regarded as a dominant weight for U(n) or GL(n, C). Partitions parametrize both irreducible representations of the symmetric groups.

If λn < 0 then the dominant weight λ is not a partition, but it may be translated by a multiple of (1n) (representing a power of the determinant) to obtain a partition. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The Vandermonde determinant

Assuming this, the Weyl denominator (using ρ0 instead of ρ) can be written

ρ0 Y Y ∆(t) = e (1 − tj/ti) = (ti − tj). i

The Weyl denominator formula

X 0 ∆(t) = (−1)`(w)ew(λ+ρ ) w∈W

can then be identiﬁed as the Vandermonde determinant identity: Y n−j (ti − tj) = det(ti ) i

Schur polynomials

Thus λi+n−j det(ti ) χλ(t) = n−j . det(ti ) If λ is a partition (so λn > 0) this is a polynomial, called the Schur polynomial sλ(t). For more general g ∈ GL(n, C), conjugating g into TC shows that the character χλ(g) is obtained by applying the Schur polynomial to the eigenvalues ti of g.

If λ is not a partition, that is, if λn < 0, then χλ is a Schur polynomial divided by a power of det(g). The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Fundamental weights

Let us assume that G is semisimple, so Φ spans V = R ⊗ Λ as a vector space. Let r = dim(V) be the rank. Let us deﬁne vectors $i (i = 1, ··· , r) in V called the fundamental weights by the condition that ∨ αi ($j) = δij. The walls of the positive Weyl chamber are determined by the equations si(x) = x and since

∨ si(x) = x − αi (x)αi,

boundary of the positive Weyl chamber. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The fundamental weights and ρ

Assuming that they are weights (true for simply-connected groups) the fundamental weights are dominant weights and span the cone of dominant weights over N. ∨ Moreover since αi (ρ) = 1, we have

r X ρ = $i. i=1

The fundamental weights may or may not be elements of Λ. It may be shown that if G is simply-connected, then the fundamental weights are indeed in Λ. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Fundamental weights for Type A

As we know, the weight lattice of U(n) or GL(n, C) is isomorphic n to Z . The determinant is represented by the vector

(1n) = (1, ··· , 1).

n So the weight lattice of SU(n) or SL(n, C) consists of Z modulo n Z · (1 ). The fundamental weights are:

$i = (1, ··· , 1, 0, ··· , 0), i leading 1’s. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Isobaric fundamental weights

n n n n n n Embedding Z in R , we may embed Λ = Z /Z(1 ) in R /R(1 ). n Thus we may optionally translate by an element of R(1 ) to make the weights isobaric. This means that for SU(3) the fundamental weights

(1, 0, 0), (1, 1, 0)

become 2 1 1 1 1 2 , − , − , , , − . 3 3 3 3 3 3 The group SU(n) is simply connected, so the fundamental weights, and ρ, are in Λ. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The orthogonal group

Now let us consider the orthogonal group SO(7). The simple positive roots are

α1 = (1, −1, 0), α2 = (0, 1, −1), α3 = (0, 0, 1).

∨ The coroots αi can then be identiﬁed with the vectors 2αi/hαi, αii, thus:

∨ ∨ ∨ α1 = (1, −1, 0), α2 = (0, 1, −1), α3 = (0, 0, 2).

From this, we see that the fundamental weights are:

1 1 1 $ = (1, 0, 0), $ = (1, 1, 0), $ = , , . 1 2 3 2 2 2

The ﬁrst two fundamental weights are in Λ. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Spin

As for the third weight, this is not a weight of SO(7). However the group SO(7) has a simply-connected double cover spin(7), 1 whose weight lattice Λspin consists of (λ1, λ2, λ3) with λi ∈ 2 Z subject to λi ≡ λj modulo Z. For the spin group, all the fundamental weights are indeed weights and. The fundamental weights correspond to irreducible representations of spin(7) of degrees 7, 21 and 8, respectively. They are the standard 7-dimensional representation, its exterior square, and the “spin” representation which, for spin(2n + 1) has degree 2n. We have

X 5 3 1 ρ = $ = , , . i 2 2 2 The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Computing weight multiplicities

The Weyl character formula is not very good for computing weight multiplicites since it expresses the character as a ratio of two other polynomials. Various methods exist: The Demazure character formula, The Freudenthal dimension formula, The Kostant multiplicity formula. The Demazure character formula and the Kostant multiplicity formula also have great theoretical significance. The Freudenthal dimension formula is another efficient method that can be extended to infinite-dimensional Lie algebras. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The Kostant partition function

The Kostant partition function P(λ) is deﬁned for weights λ but it is zero unless λ < 0. With our convention this means that λ is in the root lattice. The generating function is: X Y P(λ)e−λ = (1 + e−α + e−2α + ...). λ α∈Φ

This is the character of a Verma module, a certain

inﬁnite-dimensional representation of gC in the Bernstein-Gelfand-Gelfand category O. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Starting with the Weyl character formula

As in the previous lecture we may write the Weyl character formula   −ρ Y −α −1 X `(w) w(λ+ρ) χλ = e  (1 − e )  (−1) e α∈Φ+ w∈W   Y −α −2α X `(w) w(λ+ρ)−ρ  (1 + e + e + ...) (−1) e , α∈Φ+ w∈W valid in completion denoted Eˆ in Tuesday’s lecture. Note that there are inﬁnitely many terms on the right-hand side, so there is considerable cancellation to produce just a ﬁnite sum. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The Kostant multiplicity formula

This equals X X (−1)`(w)ew(λ+ρ)−ρ P(µ)e−µ. w∈W µ The coefﬁcient of eν is the sum of P(µ) over solutions to ν = w(λ + ρ) − ρ − µ, µ = w(λ + ρ) − ρ − ν. Therefore " # X X χ = (−1)wP(w(λ + ρ) − ρ − µ) eν. ν w∈W The formula X (−1)wP(w(λ + ρ) − ρ − µ) w∈W

for the multiplicity of ν in χλ is called the Kostant multiplicity formula. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Remarks about application

If ν is near λ, there may be only one term w = 1, in which case it is a very efﬁcient way to know the multiplicity. For ν farther from λ, this is a less efﬁcient way to compute the character. In practice, one only needs to know the weight for ν in a fundamental domain. The fact that the weight is W-invariant is not manifest in this formula.

In practice, the Freudenthal multiplicity formula and the Demazure character formula are faster ways to compute the character.

Apart from the problem of computing the weight multiplicities, the Kostant formula has considerable theoretical importance. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The Brauer-Klimyk or Racah-Speiser algorithm

Let χλ be the character of πλ. We expand it in terms of weight µ, each with multiplicity K(λ, µ):

X µ χλ(t) = K(λ, µ)t . µ

for t ∈ T. Then we can try to decompose χλχν, which is the character of πλ ⊗ πν.

The simplest case is where ν + µ is dominant for all weights µ of λ. Then X χλχν = K(λ, µ)χµ+ν, µ M πλ ⊗ πν = K(λ, µ)πµ+ν, µ The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof

We substitute the weight expansion for χλ and the Weyl character formula for χν:

−1 X µ X `(w) w(ν+ρ) χλχν = ∆ K(λ, µ) t (−1) t . µ w

Interchange the order of summation, so that the sum over ν is the inner sum, and make the variable change ν −→ w(ν). Since K(λ, µ) = K(λ, wµ), we get X X ∆−1 K(λ, µ)(−1)`(w) tw(ν+µ+ρ). w ν

Now we may interchange the order of summation again and P apply the Weyl character formula to obtain K(λ, µ)χν+µ. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Example

λ + ν

As long as µ + ν is dominant for every weight µ of λ, we get the decomposition of πλ ⊗ πµ as X K(λ, µ)πν+µ. µ The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

What if µ + ν is not dominant?

The proof of the Brauer-Klimyk-Steinberg-Racah-Speiser formula goes through but we have to reinterpret X ∆−1 (−1)`(w)tµ+ν+ρ w∈W

Proposition Let λ be given, not assumed dominant. Write λ = wξ where w ∈ W and ξ is dominant. Let η = ξ + w−1ρ − ρ. Then

X (−1)`(w)χ if η is dominant ∆−1 (−1)wtw(λ+ρ) = η 0 otherwise. w

This is easily proved by making a change of variables in the Weyl character formula. It may be seen that η is dominant unless ξ lies on a wall of the positive Weyl chamber. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The algorithm, continued

We may describe the algorithm as follows. There is a modiﬁed action of the Weyl group known as the “dot” action in which w · λ = w(λ + ρ) − ρ. The point ﬁxed by this action is −ρ. Let H be the set of hyperplanes perpendicular to the roots. It includes the walls of C. If µ + ν lies on one of these hyperplanes, it contributes zero to the sum: X X K(λ, µ)∆−1 (−1)`(w)tµ+ν+ρ µ w∈W But if µ + ν does not lie on one of these hyperplanes, then η = w · (µ + ν) is dominant for some w ∈ W and we get a term

±K(λ, µ)χη. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Example

−ρ

The green weights lie on a hyperplane through −ρ so they are discarded. Three other red weights to the left of the hyperplane are reﬂected and subtracted. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

After cancellation

−ρ

χ(3,1,0)χ(3,3,0) = χ(4,3,3)+χ(4,4,2)+χ(5,3,2)+χ(5,4,1)+χ(6,3,1)+χ(6,4,0) The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

The dimension of an irreducible representation

Weyl gave a formula for the dimension of the irreducible representation with character χλ. Theorem (Weyl) The dimension of π(λ) is Q α∈Φ+ hλ + ρ, αi Q . α∈Φ+ hρ, αi

This is the value χλ at the identity element of G. We cannot evaluate the Weyl character formula directly since the numerator and denominator both vanish at t = 1:

P `(w) w(λ+ρ) w∈W (−1) t ρ Q −α e α∈Φ+ (1 − t ) The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof

Let Ω: E2 −→ Z be the map ! X λ X Ω nλ · e = nλ. λ λ

The dimension we wish to compute is Ω(χλ). The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof

If α ∈ Φ, let ∂α : E2 −→ E2 be the map ! X λ X λ ∂α nλ · e = nλ hλ, αi · e . λ λ

It is straightforward to check that ! X λ X λ ∂α nλ · e = nλ hλ, αi · e . λ λ

is a derivation and that the operators ∂α commute with each other. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof (continued)

We have ∂w(α) ◦ w = w ◦ ∂α since applying the operator on the left-hand side to eλ gives hw(λ), w(α)i ew(λ), while the second gives hλ, αi ewλ, and these are equal. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof (continued)

Q Let ∂ = α∈Φ+ ∂α. We show that if w ∈ W and f ∈ E2, we have

w∂(f ) = (−1)l(w)∂w(f ).

We may assume that w = sβ is a simple reﬂection. Then we have   Y w ◦  ∂w(α) = ∂ ◦ w. α∈Φ+ But w(α) consists of Φ+ with just one element, namely β, replaced by its negative. So sβ∂(f ) = −∂sβ(f ). The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof (continued)

We consider now what happens when we apply Ω ◦ ∂ to both sides of the identity

X l(w) w(λ+ρ) Y α/2 −α/2 (−1) e = χλ · e − e . w∈W α∈Φ+

On the left-hand side, applying ∂ gives   X λ+ρ X Y λ+ρ w ∂e = w hλ + ρ, αi e  . w∈W w∈W α∈Φ+ Q Now applying Ω gives |W| α∈Φ+ hλ + ρ, αi. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof (continued)

Q On the other hand, apply ∂ = ∂β one derivation at a time to the right-hand side

Y α/2 −α/2 χλ · e − e . α∈Φ+

Expanding by the Leibnitz product rule to obtain a sum of terms, each of which is a product of χλ and the terms α/2 −α/2 e − e , with each ∂β applied to some factor. When we apply Ω, any term in which a eα/2 − e−α/2 is not hit by at least one ∂β will be killed. Since the number of operators ∂β and the number of factors eα/2 − e−α/2 are equal, only the terms in α/2 −α/2 which each e − e is hit by exactly one ∂β survive. So χλ is not hit by a ∂β in any such term. The support of the character Fundamental weights The Kostant character formula The Brauer-Klimyk formula The Weyl dimension formula

Proof (continued)

Therefore   Y α/2 −α/2 Ω ◦ ∂ χλ · e − e  = θ · Ω(χλ), α∈Φ+ where   Y α/2 −α/2 θ = Ω ◦ ∂  e − e  α∈Φ+ is independent of λ. We have proved that Y |W| hλ + ρ, αi = θ · Ω(χλ). α∈Φ+

To evaluate θ, we take λ = 0. Since Ω(χ0) = Ω(1) = 1 Y θ = |W| hρ, αi . α∈Φ+