Lecture 14: The Weyl Character Formula (II)sporadic.stanford.edu/Math210C/lecture14.pdf · Lecture 14: The Weyl Character Formula (II) Daniel Bump May 21, 2020. The support of the

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 weight lattice and the root lattice

Let G be a compact, connected Lie group. For simplicity let usassume that G is semisimple. Thus we assume that the rootsspan the weight lattice. For example, SU(n) is semisimple, butU(n) is not.

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

We haven’t talked about Dynkin diagrams and extended Dynkindiagrams yet, but we mention that the finite quotient groupΛ/Λroot acts by automorphisms on the extended Dynkindiagram and is related to both the center of G and itsfundamental group (Chapter 23).


The root lattice and the Weyl group

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

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

First we check the Proposition if w = sα is a reflection. Then itfollows from formula

sα(λ) = λ− α∨(λ)α.

We may now prove the proposition by induction on the length ofw. If `(w) = 0, then w = 1 and this is trivial. Otherwise writew = sαw′ where `(w′) = `(w)− 1 and we have λ− w′(λ) ∈ Λroot.w′(λ)− w(λ) = µ− sα(µ) where µ = w′(λ) and the statementfollows from the special case already proved.


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. Thisfollows 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

ρ− w(ρ) =∑

α∈Φ+∩w−1Φ−

α,

and the statement follows.


Review: the Weyl character formula

If λ is a dominant weight we defined

χλ =

∑w∈W(−1)`(w)ew(λ+ρ)

∆,

where the Weyl denominator

∆ = eρ∏α∈Φ+

(1− e−α).

We proved that χλ is the character of an irreduciblerepresentation.

We have a partial order on weights in which µ 4 λ if λ− µ is alinear combination with nonnegative integer coefficients of thesimple roots Σ. If µ is a weight of χλ then λ < µ, so we say λ isthe highest weight of this representation.


The support of χλ

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

Furthermore, all weights lie in the same coset of the root latticeΛroot.


The support of χλ: an example

λ = (3, 1, 0)

λ− α2λ− 2α1

The dotted line delineates the region µ 4 λ. Other conditionscome from w(µ) 4 λ placing the support inside the convex hullof Wλ.


The weights lie in a single coset of Λroot

λ = (3, 1, 0)

λ− α2λ− 2α1

Lighter dots: Λ

Darker dots: Λroot + λ

The inequality µ 4 λ not only imposes a constraint that µ liewithin a region, but it also requires that λ− µ lie in a particularcoset of Λ. This differs from the convention in the book, but isconsistent with Kac, Infinite-dimensional Lie algebras.


The rings E and E2

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

Sometimes we use a larger ring E2 corresponding to Z[1

2Λ].

We can interpret elements of E as functions on T. Elements ofE2 could be interpreted as functions on a finite cover of T.

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

ρ =12

∑α∈Φ+

α

which may or may not be in Λ.


Flexibility about ρ

We note that in the Weyl character formula

χλ =

∑(−1)`(w)ew(λ+ρ)

eρ∏α∈Φ+(1− e−α)

we could replace ρ by ρ+ κ where κ is any vector such thatw(κ) = κ for all w. Then both the numerator and thedenominator are multiplied by eκ and the result is unchanged.

The condition that w(κ) = κ for all w is equivalent to thecondition that κ is orthogonal to all roots, due to the formula

rα(κ) = κ− 2〈κ, α〉〈α, α〉

α.

This is only useful if G is not semisimple, for if Φ spans Λ, theonly such κ is zero.


ρ for U(n)

However if G = U(n), and if we identify the weight lattice withZn, then

ρ =

(n− 1

2,

n− 32

, · · · , 1− n2

).

Instead, we may add κ = n−12 (1, · · · , 1) and work with

ρ′ = (n− 1, n− 2, · · · , 0).


Partitions as dominant weights

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

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

If λn < 0 then the dominant weight λ is not a partition, but it maybe translated by a multiple of (1n) (representing a power of thedeterminant) to obtain a partition.


The Vandermonde determinant

Assuming this, the Weyl denominator (using ρ′ instead of ρ) canbe written

∆(t) = eρ′∏

i<j

(1− tj/ti) =∏i<j

(ti − tj).

The Weyl denominator formula

∆(t) =∑w∈W

(−1)`(w)ew(λ+ρ′)

can then be identified as the Vandermonde determinantidentity: ∏

i<j

(ti − tj) = det(tn−ji )


Schur polynomials

Thus

χλ(t) =det(tλi+n−j

i )

det(tn−ji )

.

If λ is a partition (so λn > 0) this is a polynomial, called theSchur polynomial sλ(t). For more general g ∈ GL(n,C),conjugating g into TC shows that the character χλ(g) is obtainedby 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 Schurpolynomial divided by a power of det(g).


Fundamental weights

Let us assume that G is semisimple, so Φ spans V = R⊗ Λ asa vector space. Let r = dim(V) be the rank. Let us definevectors $i (i = 1, · · · , r) in V called the fundamental weights bythe condition that

α∨i ($j) = δij.

The walls of the positive Weyl chamber are determined by theequations si(x) = x and since

si(x) = x− α∨i (x)αi,

boundary of the positive Weyl chamber.


The fundamental weights and ρ

Assuming that they are weights (true for simply-connectedgroups) the fundamental weights are dominant weights andspan the cone of dominant weights over N.Moreover since α∨i (ρ) = 1, we have

ρ =

r∑i=1

$i.

The fundamental weights may or may not be elements of Λ. Itmay be shown that if G is simply-connected, then thefundamental weights are indeed in Λ.


Fundamental weights for Type A

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

(1n) = (1, · · · , 1).

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

$i = (1, · · · , 1, 0, · · · , 0), i leading 1’s.


Isobaric fundamental weights

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

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

become (23,−1

3,−1

3

),

(13,

13,−2

3

).

The group SU(n) is simply connected, so the fundamentalweights, and ρ, are in Λ.


The orthogonal group

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

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

The coroots α∨i can then be identified with the vectors2αi/〈αi, αi〉, thus:

α∨1 = (1,−1, 0), α∨2 = (0, 1,−1), α∨3 = (0, 0, 2).

From this, we see that the fundamental weights are:

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

(12,

12,

12

).

The first two fundamental weights are in Λ.


Spin

As for the third weight, this is not a weight of SO(7). Howeverthe group SO(7) has a simply-connected double cover spin(7),whose weight lattice Λspin consists of (λ1, λ2, λ3) with λi ∈ 1

2Zsubject to λi ≡ λj modulo Z. For the spin group, all thefundamental weights are indeed weights and. The fundamentalweights correspond to irreducible representations of spin(7) ofdegrees 7, 21 and 8, respectively. They are the standard7-dimensional representation, its exterior square, and the “spin”representation which, for spin(2n + 1) has degree 2n. We have

ρ =∑

$i =

(52,

32,

12

).


Computing weight multiplicities

The Weyl character formula is not very good for computingweight multiplicites since it expresses the character as a ratio oftwo other polynomials. Various methods exist:

The Demazure character formula,The Freudenthal dimension formula,The Kostant multiplicity formula.

The Demazure character formula and the Kostant multiplicityformula also have great theoretical significance. TheFreudenthal dimension formula is another efficient method thatcan be extended to infinite-dimensional Lie algebras.


The Kostant partition function

The Kostant partition function P(λ) is defined for weights λ but itis zero unless λ < 0. With our convention this means that λ is inthe root lattice. The generating function is:∑

λ

P(λ)e−λ =∏α∈Φ

(1 + e−α + e−2α + . . .).

This is the character of a Verma module, a certaininfinite-dimensional representation of gC in theBernstein-Gelfand-Gelfand category O.


Starting with the Weyl character formula

As in the previous lecture we may write the Weyl characterformula

χλ = e−ρ

∏α∈Φ+

(1− e−α)−1

∑w∈W

(−1)`(w)ew(λ+ρ)

∏α∈Φ+

(1 + e−α + e−2α + . . .)

∑w∈W

(−1)`(w)ew(λ+ρ)−ρ,

valid in completion denoted E in Tuesday’s lecture. Note thatthere are infinitely many terms on the right-hand side, so thereis considerable cancellation to produce just a finite sum.


The Kostant multiplicity formula

This equals ∑w∈W

(−1)`(w)ew(λ+ρ)−ρ∑µ

P(µ)e−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

χ =∑ν

[∑w∈W

(−1)wP(w(λ+ ρ)− ρ− µ)

]eν .

The formula ∑w∈W

(−1)wP(w(λ+ ρ)− ρ− µ)

for the multiplicity of ν in χλ is called the Kostant multiplicityformula.


Remarks about application

If ν is near λ, there may be only one term w = 1, in which caseit is a very efficient way to know the multiplicity. For ν fartherfrom λ, this is a less efficient way to compute the character. Inpractice, one only needs to know the weight for ν in afundamental domain. The fact that the weight is W-invariant isnot manifest in this formula.

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

Apart from the problem of computing the weight multiplicities,the Kostant formula has considerable theoretical importance.


The Brauer-Klimyk or Racah-Speiser algorithm

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

χλ(t) =∑µ

K(λ, µ)tµ.

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

The simplest case is where ν + µ is dominant for all weights µof λ. Then

χλχν =∑µ

K(λ, µ)χµ+ν ,

πλ ⊗ πν =⊕µ

K(λ, µ)πµ+ν ,


Proof

We substitute the weight expansion for χλ and the Weylcharacter formula for χν :

χλχν = ∆−1∑µ

K(λ, µ) tµ∑

w

(−1)`(w) tw(ν+ρ).

Interchange the order of summation, so that the sum over ν isthe inner sum, and make the variable change ν −→ w(ν). SinceK(λ, µ) = K(λ,wµ), we get

∆−1∑

w

∑ν

K(λ, µ) (−1)`(w) tw(ν+µ+ρ).

Now we may interchange the order of summation again andapply the Weyl character formula to obtain

∑K(λ, µ)χν+µ.


Example

λ+ ν

As long as µ+ ν is dominant for every weight µ of λ, we get thedecomposition of πλ ⊗ πµ as∑

µ

K(λ, µ)πν+µ.


What if µ+ ν is not dominant?

The proof of the Brauer-Klimyk-Steinberg-Racah-Speiserformula goes through but we have to reinterpret

∆−1∑w∈W

(−1)`(w)tµ+ν+ρ

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

∆−1∑

w

(−1)wtw(λ+ρ) =

(−1)`(w)χη if η is dominant0 otherwise.

This is easily proved by making a change of variables in theWeyl character formula. It may be seen that η is dominantunless ξ lies on a wall of the positive Weyl chamber.


The algorithm, continued

We may describe the algorithm as follows. There is a modifiedaction 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 ofhyperplanes perpendicular to the roots. It includes the walls ofC. If µ+ ν lies on one of these hyperplanes, it contributes zeroto the sum: ∑

µ

K(λ, µ)∆−1∑w∈W

(−1)`(w)tµ+ν+ρ

But if µ+ ν does not lie on one of these hyperplanes, thenη = w · (µ+ ν) is dominant for some w ∈ W and we get a term

±K(λ, µ)χη.


Example

−ρ

The green weights lie on a hyperplane through −ρ so they arediscarded. Three other red weights to the left of the hyperplaneare reflected and subtracted.


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 dimension of an irreducible representation

Weyl gave a formula for the dimension of the irreduciblerepresentation with character χλ.

Theorem (Weyl)The dimension of π(λ) is∏

α∈Φ+ 〈λ+ ρ, α〉∏α∈Φ+ 〈ρ, α〉

.

This is the value χλ at the identity element of G. We cannotevaluate the Weyl character formula directly since thenumerator and denominator both vanish at t = 1:∑

w∈W(−1)`(w)tw(λ+ρ)

eρ∏α∈Φ+(1− t−α)


Proof

Let Ω : E2 −→ Z be the map

Ω

(∑λ

nλ · eλ)

=∑λ

nλ.

The dimension we wish to compute is Ω(χλ).


Proof

If α ∈ Φ, let ∂α : E2 −→ E2 be the map

∂α

(∑λ

nλ · eλ)

=∑λ

nλ 〈λ, α〉 · eλ.

It is straightforward to check that

∂α

(∑λ

nλ · eλ)

=∑λ

nλ 〈λ, α〉 · eλ.

is a derivation and that the operators ∂α commute with eachother.


Proof (continued)

We have∂w(α) w = w ∂α

since applying the operator on the left-hand side to eλ gives〈w(λ),w(α)〉 ew(λ), while the second gives 〈λ, α〉 ewλ, and theseare equal.


Proof (continued)

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 wehave

w

∏α∈Φ+

∂w(α)

= ∂ w.

But w(α) consists of Φ+ with just one element, namely β,replaced by its negative. So sβ∂(f ) = −∂sβ(f ).


Proof (continued)

We consider now what happens when we apply Ω ∂ to bothsides of the identity∑

w∈W

(−1)l(w)ew(λ+ρ) = χλ ·∏α∈Φ+

(eα/2 − e−α/2

).

On the left-hand side, applying ∂ gives

∑w∈W

w(∂eλ+ρ

)=∑w∈W

w

∏α∈Φ+

〈λ+ ρ, α〉 eλ+ρ

.

Now applying Ω gives |W|∏α∈Φ+ 〈λ+ ρ, α〉.


Proof (continued)

On the other hand, apply ∂ =∏∂β one derivation at a time to

the right-hand side

χλ ·∏α∈Φ+

(eα/2 − e−α/2

).

Expanding by the Leibnitz product rule to obtain a sum ofterms, each of which is a product of χλ and the termseα/2 − e−α/2, with each ∂β applied to some factor. When weapply Ω, any term in which a eα/2 − e−α/2 is not hit by at leastone ∂β will be killed. Since the number of operators ∂β and thenumber of factors eα/2 − e−α/2 are equal, only the terms inwhich each eα/2 − e−α/2 is hit by exactly one ∂β survive. So χλis not hit by a ∂β in any such term.


Proof (continued)

Therefore

Ω ∂

χλ · ∏α∈Φ+

(eα/2 − e−α/2

) = θ · Ω(χλ),

where

θ = Ω ∂

∏α∈Φ+

(eα/2 − e−α/2

)is independent of λ. We have proved that

|W|∏α∈Φ+

〈λ+ ρ, α〉 = θ · Ω(χλ).

To evaluate θ, we take λ = 0. Since Ω(χ0) = Ω(1) = 1

θ = |W|∏α∈Φ+

〈ρ, α〉 .

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Lecture 14: The Weyl Character Formula (II)sporadic.stanford.edu/Math210C/lecture14.pdf · Lecture 14: The Weyl Character Formula (II) Daniel Bump May 21, 2020. The support of the