Multiplicities of compact groups representations : Qualitative properties and computations

Multiplicities of compact groups representations : Qualitative properties and computations

WORK IN COMMON WITH WELLEDA BALDONI

Multiplicities of compact group representations.

K compact connected Lie group.

Kˆ set of irreducible finite dimensional complex representations ofK.

LetV be a complex vector space with a representation ofK, which is a direct sum (possibly infinite) of irreducible finite dimensional complex representationsπofK, each of them occurring with finite multiplicitym(π,V).

We write

V =⊕_π∈Kˆm(π,V)π.

This means ;k ∈K acts onV : Then Tr_V(k) =X

π

m(π,V)Tr_π(π(k)).

We call the functionπ →m(π,V)onKˆ the multiplicity function ofV.

Example1 : Littlewood-Richardson coefficients

HereV =V_λ⊗V_µis the tensor product of two irreducible representations ofK, andK acts diagonally. Then

V_λ⊗V_µ=⊕c_λ,µ^ν V_ν.

Question : Computations and properties ofc_λ,µ^ν ?.

Advertisement for computation ofc_λ,µ^ν for classical root systems by Cochet ( a Maple program, available on my webpage). In particular, it computes the function

(λ, µ, ν)→c_kλ,kµ^kν

when(λ, µ, ν)are given andk ∈N: a periodic polynomial ink with periods among 1,2 (for classical root systems)

Example 2 : Kronecker coefficients

We consider here

V =Sym(C^p1⊗C^p2⊗C^p3).

(Space of polynomials functions inp₁p₂p₃variables).

ThenV is a representation space forU(p₁)×U(p₂)×U(p₃).

V =⊕g(λ₁, λ₂, λ₃)V_λ1⊗V_λ2⊗V_λ3. Theg(λ₁, λ₂, λ₃)are the Kronecker coefficients.

Question : computation and properties ofg(λ₁, λ₂, λ₃)?

What about Quantum marginals ? ?

ρ_A, ρ_B, ρ_C densities matrices with rational coefficients (assume common denominator`).

Thusρ_A, ρ_B, ρ_C are the quantum marginals of a pure statev in (C^A⊗C^B⊗C^C), if and only if for some largek :

Sym<sup>k`</sup>(C<sup>A</sup>⊗C<sup>B</sup>⊗C<sup>C</sup>)contains the irreducible representation V<sub>k`ρA</sub>⊗V_{k`ρB</sub>⊗V<sub>k`ρC} of the groupU(A)×U(B)×U(C).

FIRST PART : QUALITATIVE GENERAL RESULTS

V =Q(M)

quantization of a general symplectic manifoldM(classical exampleM =R^2N =C^N,Q(M)the Fock space). We will see that, in this general context, the functionπ→m(π,V)is a piecewise periodic function, with some ”continuity properties”.

SECOND PART : TOOLS FOR COMPUTATIONS In the second part,M =H whereH=C^N with a unitary representation ofK =U(n).

Then

V =Q(M) =Sym(H),

and related computation of volumes or number of integral points in rational polytopes.

Periodic polynomial functions

Consider a real vector spaceE with a latticeL(that isE =R^d andL=Z^d for somed). LetC(L)be the space of functions on the latticeL.

The restriction toLof a polynomial function onE will be called a polynomial function onL.

Letqbe an integer, a function onL/qLwill be called a periodic function onΛof periodq. Ifq=1, just a constant function.

A periodic polynomial function onLis a function onLwhich is a linear combination of products of polynomials with periodic functions (of various periods).

For example :

m(k) = 1

2k²+k +3 4 +1

4(−1)^k

is a periodic polynomial function (with periods 1 and 2 ) of k ∈Zand of degree 2. (Furthermore the functionm(k)takes integral values on eachk ∈Z).

Quantization of Hamiltonian manifolds.

I will start with recalling some general qualitative results on properties of the multiplicity functionπ7→m(π,V)when

V =Q(M)

is constructed as the geometric quantization of a

K-Hamiltonian manifoldM. Then I will give some specific examples, whenK =U(n)is the unitary group, andM =C^N is a Hermitian vector space with a unitary representation ofU(n) (that is we have an homomorphismU(n)→U(N)).

Notations

•K compact connected Lie group with maximal torusT.

•Lie algebrastandk.

•R₊a choice of positive roots.

•t^∗_≥0the positive Weyl chamber. Thust^∗_≥0⊂k^∗

•Weight latticeΛofT identified to a lattice int^∗.

•Λ≥0the setΛ∩t^∗_≥0”cone” of dominant weights.

•V_λ^K the corresponding irreducible representation ofK with highest weightλ.

EXAMPLEK =U(n). ThenΛ≥0= [λ₁, λ₂, . . . , λn]withλ_i ∈Z and

λ1≥λ2≥ · · · ≥λn.

Notations

•K compact connected Lie group with maximal torusT.

•Lie algebrastandk.

•R₊a choice of positive roots.

•t^∗_≥0the positive Weyl chamber. Thust^∗_≥0⊂k^∗

•Weight latticeΛofT identified to a lattice int^∗.

•Λ≥0the setΛ∩t^∗_≥0”cone” of dominant weights.

•V_λ^K the corresponding irreducible representation ofK with highest weightλ.

EXAMPLEK =U(n). ThenΛ≥0= [λ₁, λ₂, . . . , λn]withλ_i ∈Z and

λ₁≥λ₂≥ · · · ≥λn.

Kirwan polyhedron

(M,Ω): a Hamiltonian compact connectedK manifold.

Moment mapΦK :M →k^∗ commuting with the action ofK. DEFINITION :The Kirwan polytope is∆_K(M) = Φ_K(M)∩t^∗_≥0. THIS IS A CONVEX POLYTOPE ( forM projective, see Nicole Berline lectures on my webpage following Mumford argument.) IfM is non compact, (and some mild restrictions), thenΦ_K(M) is aCONVEX POLYHEDRON, ifΦ_K is proper.

A typical example forMis the sphere.

We considerT the group of rotations around thez axis. Here is the drawing of∆_T(M). The moment mapΦ_T is the height. Thus

∆_T(M)is the interval[−r,r].

Kirwan polyhedron

(M,Ω): a Hamiltonian compact connectedK manifold.

Moment mapΦK :M →k^∗ commuting with the action ofK. DEFINITION :The Kirwan polytope is∆_K(M) = Φ_K(M)∩t^∗_≥0. THIS IS A CONVEX POLYTOPE ( forM projective, see Nicole Berline lectures on my webpage following Mumford argument.) IfM is non compact, (and some mild restrictions), thenΦ_K(M) is aCONVEX POLYHEDRON, ifΦ_K is proper.

A typical example forMis the sphere.

We considerT the group of rotations around thez axis. Here is the drawing of∆_T(M). The moment mapΦ_T is the height. Thus

∆_T(M)is the interval[−r,r].

”the Example”

M=Hom(C^A,C^B), with action ofU(A)andU(B). Then the moment map is

Φ(X) = [XX^∗,X^∗X]

Dilated Kirwan polytope

We consider the Kirwan polytope∆_K(M), and all its dilated.

Thus we introduce

C_K(M) ={[t,tξ],t ≥0, ξ ∈∆_K(M)}

the cone generated by∆_K(M)in the extended vector space E =R⊕t^∗ with extended latticeL=Z⊕Λ.

EXAMPLE : The dilated Kirwan polyhedron for the sphere

We thus considerMthe sphere withr =1, andT the group of rotations around thezaxis. Here is the drawing of the cone C_T(M): the cone over∆_T(M).

Multiplicities

IfM is provided with a Kostant line bundle (the curvature of this line bundle is−iΩ), then, for eachk ≥0, we can define a representation of the compact groupK in a quantized space Q(M,k)and :

Q(M,k) =⊕_µ∈Λ≥0m(k, µ)V_µ^K.

The support of the functionµ→m(k, µ)is contained in the dilated Kirwan polytopek∆_K(M)∩Λ. (We have dilated the symplectic formΩinkΩand the moment map inkΦ_K) We are interested in this multiplicity functionm(k, µ)as a function of(k, µ)for(k, µ)∈C_K(M)(that isµ∈k∆_K(M))

Qualitative behavior

There exists :

•a decomposition of the coneC_K(M) =∪_aCa, inclosed polyhedral conesCa, withCa∩C_bcontained in the boundary of C_a,C_b.

•for eacha, a periodic polynomial functionp_aon the lattice Z⊕Λsuch that

m(k, µ) =p_a(k, µ) if(k, µ)∈C_a∩(Z⊕Λ).

In particulark →m(k,kµ)is a periodic polynomial function on N={0,1,2, . . .}for anyµ∈∆_K(M)∩Λ.

Duistermaat-Heckman measure

There is a homogeneous polynomial functiond_a(k, µ)(the Duistermaat-Heckman function) such that for

(k, µ)∈C_a∩(Z⊕Λ),

m(k, µ) =da(k, µ) modulo lower degree terms (in(k, µ).

We thus call theCachambers for polynomiality .

All the above results follows from the[Q,R] =0 conjecture, proved by Meinrenken-Sjamaar. See Poster ( Saeid

Molladavoudi) for the geometric meaning ofDHK(y)and reduced spaces).

” Continuity properties”

•The Duistermaat-Heckman functionDHK(y)is a multispline on the coneC_K(M); that is, it is given by polynomials on each convex coneC_a, these polynomials match onC_a∩C_bwith some order of differentiability.

•The multiplicity functionm(k, µ)is a ”discrete multispline” on the coneC_K(M)∩L; that is : it is given by periodic polynomials p_aon each convex coneC_a∩L, these periodic polynomials match onCa∩C_b∩L, and in fact they match on a

”neighborhood” ofCa∩C_b∩L. Paradan jump formula to comparep_a,p_b.

EXAMPLE : Three qubits M = Proj ( C

⊗ C

⊗ C

) acted by K = U(2) × U(2) × U (2).

We can draw the Kirwan polytope inR⁺×R⁺×R⁺, as in fact this is onlySU(2)timesSU(2)timesSU(2)which is acting.

Here is the Kirwan polytope is inR⁺×R⁺×R⁺, Its

decomposition in6 chambersfor polynomiality (see Christandl, Doran, Kousidis, Walter).

The multiplicities onC_K(M)(cone over this polytope) have been described by Briand- Orellana-Rosas.On each chamber

m(λ, µ, ν)is a periodic polynomial of degree 1 and period 2.

Duistermaat-Heckman measure for three qubits

Drawing along the line[[1/2,1/2],[1/2,1/2],[1/2,1/2]]to [[1,0],[1,0],[1,0]]between the bottom vertex of the Kirwan polytope.

EXAMPLE : Four qubits : M = Proj ( C

⊗ C

⊗ C

⊗ C

) acted by K = U(2) × U(2) × U(2) × U (2).

The Kirwan polytope has been described by

Higuchi-Sudbery-Szulc. I have no idea of the number of polynomiality chambers. Nevertheless, given highest weights α, β, γ, δ, we can computeg(kα,kβ,kγ,kδ)as a periodic polynomial ink (a polynomial of degree at most 7 and period 6).

EXAMPLE :m(k ∗[2,1],k∗[2,1],k ∗[2,1],k∗[2,1]) =

23/241920∗k⁷+13/5760∗k⁶+155/6912∗k⁵+139/1152∗k⁴ +(81601/207360+1/1536∗(−1)^k)∗k³

+(9799/11520+ (−1)^k ∗5/256)k² +(38545/32256+ (−1)^k∗179/1536)∗k+P(k) whereP(k)is of period 6 and such that such that its values on 0,1,2,3,4,5 are

[1,5725/10368,76/81,77/128,77/81,5597/10368]

Duistermaat-Heckman measure for four qubits

Drawing along the line

[[1/2,1/2],[1/2,1/2],[1/2,1/2],[1/2,1/2]]to

[[1,0],[1,0],[1,0],[1,0]]between the bottom vertex of the Kirwan polytope and the top vertex. It is a spline of degree 7 on [0,¹₅,¹₃,¹₂,1]. .

Second part : Hermitian vector spaces

We thus consider a representation ofK in a Hermitian space H=C^N. That is, we have an homomorphismh:K →U(N).

We will discuss computational tools for Duistermaat-Heckman functions and multiplicities

WhenK =T is abelian, computation of multiplicities is the same problem than computing the number of integral points in a (rational) polytope, while computation of

Duistermaat-Heckman measure is the same problem than computing the volume of a polytope. Thus we will apply our (Baldoni-Beck-Cochet-V.) technics for computing volumes or number of integral points in polytopes (different from Latte, Barvinok, and more efficient for specific polytopes, but...). They are based on iterated residue formulae (Szenes-Vergne) for number of integral points in rational polytopes.

Quantization of a Hermitian vector space

LetHbe a Hermitian vector space with an action ofK. We assume thatK contains the homothetyv →e^iφv.

Then the quantization ofHisSym(H) =⊕^∞_k=0Sym^k(H).

The simplest example

H=C²with action ofe^iφ.Sym(H) =⊕^kS^k(C²).

Dimension ofS^k(C²) = (k +1). Verify the trace equation

(

∞

X

i=0

zⁱ)(

∞

X

j=0

z^j) =

∞

X

k=0

(k+1)z^k = 1 (1−z)² Indeed Cardinal

{(i,j);i+j=k}=k +1

Multiplicities

Find the Fourier coefficient ofTr_S(H)(t): 1

2iπ Z

|z|=

z⁻ⁿ (1−z)²

dz z We compute the right hand side by residues :

= 1 2iπ

Z

|u|=

(1+u)⁻⁽¹⁺ⁿ⁾ u² du

Multiplicities

Action ofK inH=C^N.

LetT be the maximal torus ofK and letr =dim(T)be the rank ofK. We will use both groupsT andK, and the moment maps for the abelian groupT, and the compact groupK. Thus we obtain two conesC_K(H)andC_T(H).

LetΦ⊂t^∗ be the list of weights[φ1, φ2, . . . , φ_N]ofT inH =C^N. We assume (to simplify) thatΦgeneratingt^∗.

Example

U(2)action on

H=Ce₁²⊕Ce₁e₂⊕Ce²₂ :

ThenΦ ={[2,0],[1,1],[0,2]}and Tr_S(H)= 1

(1−z₁²) 1 (1−z₁z₂)

1 (1−z₂²)

Polytopes

It is very easy to determineC_T(H)the Kirwan polyhedron for the action of the abelian groupT : just the cone generated byΦ.

Lety ∈t^∗. If we consider the polytope

P_Φ(y) ={x_i ∈R^N,x_i ≥0,

N

X

i=1

x_iφ_i =y},

thenDHT(y)is the volume of this polytope.

DefineC_T^reg(H)the open subset of the coneC_T(H), where we removed fromC_T(H)the union of the boundaries of the cones generated by all subsets ofΦ.

Then a connected componentc_aof this set is a ”chamber for polynomiality” ofDHT(y).

Duistermaat-Heckman measures for K

Assume (to simplify) thatK acts onM with generic finite stabilizers. ThenC_K(H)is a cone int^∗_≥0with non empty interior and is the union of some of the setsc_a∩t^∗_≥0. Then the

Duistermaat-Heckman measureDHK(y)is equal to DHK(y) = Y

α∈R+

(∂(α)DHT)(y)

(this is a spline on the coneC_K(H)).

In some cases, it is also possible to describeDHK(y)as the volume of a (smaller dimensional) polytope. But no general results.

Iterated residues

A list−→σ = [α₁, α₂, . . . , αr]of elements ofΦwill be called a ordered basisif the elementsα_k form a basis oft^∗. Denote by d_σ the volume of the parallelepiped generated byσwith respect to the measure determined byΛ. Thend_σ is an integer.

Let−→

B(Φ)be the set of ordered basis.

For−→σ = [α₁, α₂, . . . , α_r]∈−→

B(Φ), consider the associated iterated residue functional :

Forz ∈t_C, letz_j =hz, α_ji.

We then express a meromorphic functionf(z)ont_Cwith poles on the union of the hyperplanesφ_k(z) =0 as a function f(z) =f(z₁,z₂, . . . ,zr).(in particularf may have poles on z_i =0) DEFINE

Res^−→σ(f(z)) :=Res_z1=0(Res_z2=0· · ·(Res_zr=0f(z₁,z₂, . . . ,zr))· · ·).

Jeffrey Kirwan theorem for DHT (y )

Lety ∈t^∗. Define the following function ofz ∈t_C. s_T(y,z) =e<y,z> 1

Q

φ∈Φ < φ,z >.

The functionz →s_T(y,z)has poles on the union of the hyperplanes< φ,z >=0.

THEOREMLetcbe a chamber for polynomiality. There exists a subset−→

B(c)of−→

B(Φ), depending oncsuch that for anyy ∈c, DHT(y) = X

σ∈−→ B(c)

Res^−→σs_T(y,z).

Furthermore, there is a nice algorithm to determine−→

B(c)using the family of admissible hyperplanes. We can compute our iterated residues with a symbolic variabley and we obtain as a result the polynomial function ofy which coincides withDHT(y) on the chamberc.

Example

Φ = [2,0],[1,1],[0,2].

Ify₁>y₂,

res_z2=0(res_z1=0(exp(y₁z₁+y₂z₂) z₁z₂(z₁+z₂) =y₂. Ify₂>y₁,

res_z1=0(res_z2=0(exp(y₁z₁+y₂z₂) z₁z₂(z₁+z₂) =y₁.

Jeffrey Kirwan theorem for DHK (y )

Lety ∈t_C. Define the following function ofz ∈t_C: s_K(y,z) =e<y,z>

Q

α∈R⁺ < α,z >

Q

φ∈Φ < φ,z > .

Thus we obtain

DHK(y) = X

σ∈−→ B(c)

Res^−→σs_K(y,z).

This residue formula for the Duistermaat-Heckman measure DHK forK is easier to compute that the one forDHT(y)as the orders of poles inzof the functions_K(y,z)are smaller that those ofs_T(y,z). However, to know on whichc_athe resulting formula gives 0 is difficult. So some sort of paradox, easier to compute this functionDHK(y)thanDHT(y), but harder to know its support...

Multiplicities

Letµ∈Λ, and define the following functions ofz ∈t_C: S_T(µ,z) =e<µ,z> 1

Q

φ∈Φ(1−e−<φ,z>). S_K(µ,z) =e<µ,z>

Q

α∈R⁺(1−e−<α,z>) Q

φ∈Φ(1−e−<φ,z>) .

Residue formulae for Multiplicities for T , K (Szenes-Vergne)

LetΓbe the dual lattice toΛ. Letqbe the lcm of the integersd_σ for all basis oft^∗ made up of elements ofΦ

For anyµ∈c_a∩Λ, then m_T(µ) = X

σ∈B(c)

X

γ∈Γ/qΓ

Res_σS_T(µ,z+2iπ q γ).

Letc_asuch thatc_a∩t^∗_≥0is contained inC_K(M). Then for any µ∈c_a∩Λ≥0

m_K(µ) = X

σ∈B(c)

X

γ∈Γ/qΓ

ResσS_K(µ,z+2iπ q γ).

Similarly, once we knowB(c)the calculation is not more difficult to do with symbolic variableµ, and we obtain the periodic polynomial which coincides withm_K(µ)onc_a∩Λ_≥0.

Applications

The method works

•H =C³⊗C³⊗C³: Three quthrits.

Rather quick time for the actual numerical valuesg(α, β, γ).

We can also compute the quasi polynomialg(kα,kβ,kγ). The periods are among 1,2,3,4. The degree is at most 11.

•Four qubitsH =C²⊗C²⊗C²⊗C²

We can compute the quasi polynomialg(kα,kβ,kγ,kδ). The periods are among 1,2,3,4.

No idea of the periods in higher cases, nor the possible set of admissible hyperplanes (among them, the facets of the Kirwan polytope, talk of Ressayre). Very challenging problems...

Three quthrits ; C

⊗ C

⊗ C

The Kirwan polytope has been determined (independently by Bravyi, and Matthias Franz).

The multiplicityg(kα,kβ,kγ)is a periodic polynomial of degree (at most) 11 and of period (at most) 12 ;

We can compute the full periodic polynomial with a Maple program. Here is the periodic term for the coefficient of degree 0 forg(k[4,3,2],k[4,3,2],k[4,3,2]) :

[1,1166651/5308416,13403/20736, 29899/65536,59/81,1166651/5308416,

235/256,980027/5308416,59/81, 32203/65536,13403/20736,980027/5308416]