Nonvanishing of Kronecker coefficients for rectangular shapes
aa r X i v : . [ m a t h . G R ] J un Nonvanishing of Kronecker coefficientsfor rectangular shapes
Peter B¨urgisser a,1 , Matthias Christandl b,2 , Christian Ikenmeyer a,1 a Institute of Mathematics, University of Paderborn, D-33098 Paderborn, Germany b Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerland
Abstract
We prove that for any partition ( λ , . . . , λ d ) of size ℓd there exists k ≥ S kℓd with respect to the rectangular partition ( kℓ, . . . , kℓ ) con-tains the irreducible representation corresponding to the stretched partition( kλ , . . . , kλ d ). We also prove a related approximate version of this state-ment in which the stretching factor k is effectively bounded in terms of d .We further discuss the consequences for geometric complexity theory whichprovided the motivation for this work. Keywords:
Kronecker coefficients, quantum marginal problem, geometriccomplexity theory, quantum information theory
1. Introduction
Kronecker coefficients are the multiplicities occurring in tensor productdecompositions of irreducible representations of the symmetric groups. Thesecoefficients play a crucial role in geometric complexity theory [15, 16], whichis an approach to arithmetic versions of the famous P versus NP problem and
Email addresses: [email protected] (Peter B¨urgisser), [email protected] (Matthias Christandl), [email protected] (Christian Ikenmeyer) Supported by the German Science Foundation (grant BU 1371/3-1 of the SPP 1388on Representation Theory) Supported by the Swiss National Science Foundation (grant PP00P2-128455) and theGerman Science Foundation (grant CH 843/1-1 of the SPP 1388 on Representation Theoryand grant CH 843/2-1)
Preprint submitted to Elsevier August 27, 2018 elated questions in computational complexity via geometric representationtheory. As pointed out in [3] (see Section 4), for implementing this approach,one needs to identify certain partitions λ ⊢ d ℓd with the property that asymmetric version of the Kronecker coefficient associated with λ, (cid:3) , (cid:3) van-ishes, where (cid:3) := ( ℓ, . . . , ℓ ) stands for the rectangle partition of length d .Computer experiments show that such λ occur rarely. Our main result con-firms this experimental finding. We prove that for any λ ⊢ d ℓd there existsa stretching factor k such that the Kronecker coefficient of kλ, k (cid:3) , k (cid:3) isnonzero (Theorem 1). Here, kλ stands for the partition arising by multiply-ing all components of λ by k . We also prove a related approximate versionof this statement (Theorem 2) that suggests that the stretching factor k maybe chosen not too large. Similar results are shown to hold for the symmetricversion of the Kronecker coefficient and thus have a bearing on geometriccomplexity theory (see Lemma 3 and Section 4).Our proof relies on a recently discovered connection between Kroneckercoefficients and the spectra of composite quantum states [11, 5]. Let ρ AB bethe density operator of a bipartite quantum system and let ρ A , ρ B denotethe density operators corresponding to the systems A and B , respectively. Itturns out that the set of possible triples of spectra (spec ρ AB , spec ρ A , spec ρ B )is obtained as the closure of the set of triples ( λ, µ, ν ) of normalized parti-tions λ, µ, ν with nonvanishing Kronecker coefficient, where we set λ := | λ | λ .For proving the main theorem it is therefore sufficient to construct, for anyprescribed spectrum λ , a density matrix ρ AB having this spectrum and suchthat the spectra of ρ A and ρ B are uniform distributions.The set of possible triples of spectra (spec ρ AB , spec ρ A , spec ρ B ) is inter-preted in [11] as the moment polytope of a complex algebraic group variety,thus linking the problem to geometric invariant theory. We do not not usethis connection in our paper. Instead we argue as in [5] using the estimationtheorem of [9]. The exponential decrease rate in this estimation allows us toderive the bound on the stretching factor in Theorem 2.
2. Preliminaries A partition λ of n ∈ N is a monotonically decreasing sequence λ =( λ , λ , . . . ) of natural numbers such that λ i = 0 for all but finitely many i .The length ℓ ( λ ) of λ is defined as the number of its nonzero parts and itssize as | λ | := P i λ i . One writes λ ⊢ d n to express that λ is a partition of n ℓ ( λ ) ≤ d . Note that ¯ λ := λ/n = ( λ /n, λ /n, . . . ) defines a probabilitydistribution on N .It is well known [8] that the complex irreducible representations of thesymmetric group S n can be labeled by partitions λ ⊢ n of n . We shall denoteby S λ the irreducible representation of S n associated with λ . The Kroneckercoefficient g λ,µ,ν associated with three partitions λ, µ, ν of n is defined as thedimension of the space of S n -invariants in the tensor product S λ ⊗ S µ ⊗ S ν .Note that g λ,µ,ν is invariant with respect to a permutation of the partitions.It is known that g λ,µ,ν = 0 vanishes if ℓ ( λ ) > ℓ ( µ ) ℓ ( ν ). Equivalently, g λ,µ,ν may also be defined as the multiplicity of S λ in the tensor product S µ ⊗ S ν .If µ = ν we define the symmetric Kronecker coefficient sg λµ as the multiplicityof S λ in the symmetric square Sym ( S µ ). We note that sg λµ ≤ g λ,µ,µ .The Kronecker coefficients also appear when studying representations ofthe general linear groups GL d over C . We recall that rational irreducible GL d -modules are labeled by their highest weight, a monotonically decreasinglist of d integers, cf. Fulton and Harris [8]. We will only be concerned withhighest weights consisting of nonnegative numbers, which are therefore of theform λ ⊢ d k for modules of degree k . We shall denote by V λ the irreducible GL d -module with highest weight λ .Suppose now that λ ⊢ d d k . When restricting with respect to the mor-phism GL d × GL d → GL d d , ( α, β ) α ⊗ β, then the module V λ splits asfollows: V λ = M µ ⊢ d k,ν ⊢ d k g λ,µ,ν V µ ⊗ V ν . (1)Even though being studied for more than fifty years, Kronecker coef-ficients are only understood in some special cases. For instance, giving acombinatorial interpretation of the numbers g λ,µ,ν is a major open problem,cf. Stanley [17, 18] for more information.We are mainly interested in whether g λ,µ,ν vanishes or not. For studyingthis in an asymptotic way one may consider, for fixed d = ( d , d , d ) ∈ N with d ≤ d ≤ d ≤ d d , the set Kron ( d ) := n n ( λ, µ, ν ) | n ∈ N , λ ⊢ d n, µ ⊢ d n, ν ⊢ d n g λ,µ,ν = 0 o . It turns out that
Kron ( d ) is a rational polytope in Q d + d + d . This fol-lows from general principles from geometric invariant theory, namely Kron ( d )equals the moment polytope of the projective variety P ( C d ⊗ C d ⊗ C d ) with3espect to the standard action of the group GL d × GL d × GL d , cf. [14, 7, 11].For an elementary proof that Kron ( d ) is a polytope see [4]. Let H be a d -dimensional complex Hilbert space and denote by L ( H )the space of linear operators mapping H into itself. For ρ ∈ L ( H ) we write ρ ≥ ρ is positive semidefinite. By the spectrum spec ρ of ρ we will understand the vector ( r , . . . , r d ) of eigenvalues of ρ in decreasingorder, that is, r ≥ · · · ≥ r d . The set of density operators on H is defined as S ( H ) := { ρ ∈ L ( H ) | ρ ≥ , tr ρ = 1 } . Density operators are the mathematical formalism to describe the statesof quantum objects. The spectrum of a density operator is a probabilitydistribution on [ d ] := { , . . . , d } .The state of a system composed of particles A and B is described by adensity operator on a tensor product of two Hilbert spaces, ρ AB ∈ L ( H A ⊗H B ). The partial trace ρ A = tr B ( ρ AB ) ∈ L ( H A ) of ρ AB obtained by tracingover B then defines the state of particle A . We recall that the partial trace tr B is the linear map tr B : L ( H A ⊗ H B ) → L ( H A ) uniquely characterized bythe property tr( R tr B ( ρ AB )) = tr( ρ AB R ⊗ id) for all ρ AB ∈ L ( H A ⊗ H B ) and R ∈ L ( H A ). The quantum marginal problem asks for a description of the set of possibletriples of spectra (spec ρ AB , spec ρ A , spec ρ B ) for fixed d A = dim H A and d B =dim H B . In [5, 11, 4] it was shown that this set equals the closure of themoment polytope for Kronecker coefficients, so Kron ( d A , d B , d A d B ) = n (spec ρ AB , spec ρ A , spec ρ B ) | ρ AB ∈ L ( H A ⊗ H B ) o . We remark that this result is related to
Horn’s problem that asks for the com-patibility conditions of the spectra of Hermitian operators A , B , and A + B on finite dimensional Hilbert spaces. Klyachko [10] gave a similar charac-terization of these triples of spectra in terms of the Littlewood-Richardsoncoefficients. The latter are the multiplicities occurring in tensor productsof irreducible representations of the general linear groups. For Littlewood-Richardson coefficients one can actually avoid the asymptotic descriptionsince the so called saturation conjecture is true [12].4 .4. Estimation theorem We will need a consequence of the estimation theorem of [9]. The group S k × GL d naturally acts on the tensor power ( C d ) ⊗ k . Schur-Weyl dualitydescribes the isotypical decomposition of this module as( C d ) ⊗ k = M λ ⊢ d k S λ ⊗ V λ . (2)We note that this is an orthogonal decomposition with respect to the stan-dard inner product on ( C d ) ⊗ k . Let P λ denote the orthogonal projection of( C d ) ⊗ k onto S λ ⊗ V λ . The estimation theorem of Keyl and Werner [9] statesthat for any density operator ρ ∈ L ( C d ) with spectrum r we havetr( P λ ρ ⊗ k ) ≤ ( k + 1) d ( d − / exp (cid:0) − k k λ − r k (cid:1) (3)(see [5] for a simple proof). This shows that the probability distribution λ tr( P λ ρ ⊗ k ) is concentrated around r with exponential decay in the distance k λ − r k .
3. Main results
By a decreasing probability distribution r on [ d ] we understand r ∈ R d such that r ≥ · · · ≥ r d ≥ P i r i = 1. We denote by u d = ( d , . . . , d )the uniform probability distribution on [ d ]. Theorem 1.
The following statements are true: (1)
For all decreasing probability distributions r on [ d ] , the triple ( r, u d , u d ) is contained in Kron ( d , d, d ) . (2) Let λ ⊢ ℓd be a partition into at most d parts for ℓ, d ≥ and let (cid:3) := ( ℓ, . . . , ℓ ) denote the rectangular partition of ℓd into d parts. Thenthere exists a stretching factor k ≥ such that g kλ,k (cid:3) ,k (cid:3) = 0 . The next result indicates that the stretching factor k may be chosen nottoo large. Theorem 2.
Let λ ⊢ d ℓd and ǫ > . Then there exists a stretching factor k = O ( d ǫ log dǫ ) and there exist partitions Λ ⊢ d kℓd and R , R ⊢ d kℓd of kℓd such that g kλ,R ,R = 0 and k Λ − kλ k ≤ ǫ | Λ | , k R i − k (cid:3) k ≤ ǫ | R i | for i = 1 , . g λ,µ,µ = 0. By stretching the partitions λ, µ with two, wecan guarantee that the corresponding symmetric Kronecker coefficients doesnot vanish either. Lemma 3.
Let λ, µ ⊢ n . If S λ occurs in S µ ⊗ S µ , then S λ occurs in Sym ( S µ ) . In other words, g λ,µ,µ = 0 implies sg λ µ = 0 . This lemma, when combined with Theorems 1 and 2, shows that find-ing partitions λ with sg λ (cid:3) = 0, as required for the purposes of geometriccomplexity theory (see below), requires a careful search.
4. Connection to geometric complexity theory
The most important problem of algebraic complexity theory is Valiant’sHypothesis [19, 20], which is an arithmetic analogue of the famous P versusNP conjecture (see [2] for background information). Valiant’s Hypothesiscan be easily stated in precise mathematical terms.Consider the determinant det d = det[ x ij ] ≤ i,j ≤ d of a d by d matrix ofvariables x ij , and for m < d , the permanent of its m by m submatrix definedas per m := X σ ∈ S m x ,σ (1) · · · x m,σ ( m ) . We choose z := x dd as a homogenizing variable and view det d and z d − m per m as homogeneous functions C d → C of degree d . How large has d to be inrelation to m such that there is a linear map A : C d → C d with the propertythat z d − m per m = det d ◦ A ? (*)It is known that such A exists for d = O ( m m ). Valiant’s Hypothesis statesthat (*) is impossible for d polynomially bounded in m .Mulmuley and Sohoni [15] suggested to study an orbit closure problemrelated to (*). Note that the group GL d = GL d ( C ) acts on the spaceSym d ( C d × d ) ∗ of homogeneous polynomials of degree d in the variables x ij by substitution. Instead of (*), we ask now whether z d − m per m ∈ GL d · det d . (**)Mulmuley and Sohoni [15] conjectured that (**) is impossible for d polyno-mially bounded in m , which would imply Valiant’s Hypothesis.6oreover, in [15, 16] it was proposed to show that (**) is impossible forspecific values m, d by exhibiting an irreducible representation of SL d in thecoordinate ring of the orbit closure of z d − m per m , that does not occur in thecoordinate ring C [ GL d · det d ] of GL d · det d . We call such a representation of SL d an obstruction for (**) for the values m, d .We can label the irreducible SL d -representations by partitions λ into atmost d − λ ∈ N d such that λ ≥ . . . ≥ λ d − ≥ λ d = 0 we shalldenote by V λ ( SL d ) the irreducible SL d -representation obtained from theirreducible GL d -representation V λ with the highest weight λ by restriction.If V λ ( SL d ) is an obstruction for m, d , then we must have | λ | = P i λ i = ℓd for some ℓ , see [3, Prop. 5.6.2]. We call the representation V λ ( SL d )a candidate for an obstruction iff V λ ( SL d ) does not occur in C [ GL d · det d ].The following proposition relates the search for obstructions to the symmetricKronecker coefficient. Proposition 1.
Suppose that | λ | = ℓd and write (cid:3) = ( ℓ, . . . , ℓ ) with ℓ occurring d times. Then V λ ( SL d ) is a candidate for an obstruction iff thesymmetric Kronecker coefficient sg λ (cid:3) vanishes.Proof. This is an immediate consequence of Prop. 4.4.1 and Prop. 5.2.1in [3].We may thus interpret this paper’s main results by saying that candidatesfor obstructions are rare.
5. Proofs
We know that
Kron ( d, d, d ) is a rational polytope, i.e., defined by finitelymany affine linear inequalities with rational coefficients. This easily impliesthat a rational point in Kron ( d, d, d ) actually lies in Kron ( d, d, d ). Hencethe second part of Theorem 1 follows from the first part.The first part of Theorem 1 follows from the spectral characterization of Kron ( d, d, d ) described in Section 2.3 and the following result. Proposition 2.
For any decreasing probability distribution r on [ d ] thereexists a density operator ρ AB ∈ S ( H A ⊗ H B ) with spectrum r such that tr A ( ρ AB ) = tr B ( ρ AB ) = u d , where H A ≃ H B ≃ C d . H A and H B are d -dimensional Hilbert spaces. We recall first the Schmidt decomposition : for any | ψ i ∈ H A ⊗ H B , there exist orthonormalbases {| u i i} of H A and {| v i i} of H B as well as nonnegative real numbers α i ,called Schmidt coefficients , such that | ψ i = P i α i | u i i ⊗ | v i i . Indeed, the α i are just the singular values of | ψ i when we interpret it as a linear operatorin L ( H ∗ A , H B ) ≃ H A ⊗ H B . Lemma 4.
Suppose that | ψ i ∈ H A ⊗ H B has the Schmidt coefficients α i andconsider ρ := | ψ ih ψ | ∈ L ( H A ⊗ H B ) . Then tr B ( ρ ) ∈ L ( H A ) , obtained bytracing over the B -spaces, has eigenvalues α i .Proof. We have | ψ i = P i α i | u i i ⊗ | v i i for some orthonormal bases {| u i i} and {| v i i} of H A and H B , respectively. This implies ρ = | ψ ih ψ | = X i,j α i α j | u i ih u j | ⊗ | v i ih v j | and tracing over the B -spaces yields tr B ( | ψ ih ψ | ) = P i α i | u i ih u i | .Let | i , . . . , | d − i denote the standard orthonormal basis of C d . Weconsider the discrete Weyl operators X, Z ∈ L ( C d ) defined by X | i i = | i + 1 i , Z | i i = ω i | i i , where ω denotes a primitive d th root of unity and the addition is modulo d (see for instance [6]). We note that X and Z are unitary matrices and X − ZX = ωZ .We consider now two copies H A and H B of C d and define the “maximalentangled state” | ψ i := √ d P ℓ | ℓ i| ℓ i of H A ⊗ H B . By definition, | ψ i hasthe Schmidt coefficients √ d . Hence the vectors | ψ ij i := (id ⊗ X i Z j ) | ψ i , obtained from | ψ i by applying a tensor product of unitary matrices, havethe Schmidt coefficients √ d as well. Lemma 5.
The vectors | ψ ij i , for ≤ i, j < d , form an orthonormal basesof H A ⊗ H B . roof. We have, for some d th root of unity θ , h ψ ij | ψ kℓ i = h ψ | (id ⊗ Z − j X − i )(id ⊗ X k Z ℓ ) | ψ i = θ h ψ | id ⊗ X k − i Z ℓ − j | ψ i = θd X m,m ′ h mm | id ⊗ X k − i Z ℓ − j | m ′ m ′ i = θd X m h m | X k − i Z ℓ − j | m i = θd tr (cid:0) X k − i Z ℓ − j (cid:1) . It is easy to check that θd tr (cid:0) X k − i Z ℓ − j (cid:1) = 0 if ℓ = j or k = i . Proof of Proposition 2.
Let r ij be the given probability distribution assum-ing some bijection [ d ] ≃ [ d ] . According to Lemma 5, the density opera-tor ρ AB := P ij r ij | ψ ij ih ψ ij | has the eigenvalues r ij . Lemma 4 tells us thattr B ( | ψ ij ih ψ ij | ) has the eigenvalues 1 /d , hence tr B ( | ψ ij ih ψ ij | ) = u d . It followsthat tr B ( ρ AB ) = u d . Analogously, we get tr A ( ρ AB ) = u d . The proof is essentially the one of Theorem 2 in [5] carried out in thespecial case at hand. Suppose that λ ⊢ d ℓd . By Proposition 2 there isa density operator ρ AB having the spectrum λ such that tr A ( ρ AB ) = u d ,tr B ( ρ AB ) = u d . Let P X denote the orthogonal projection of ( H A ) ⊗ k onto thesum of its isotypical components S µ ⊗ V µ satisfying k µ − u d k ≤ ǫ . Then P X := id − P X is the orthogonal projection of ( H A ) ⊗ k onto the sum of itsisotypical components S µ ⊗ V µ satisfying k µ − u d k > ǫ . The estimationtheorem (3) implies thattr( P X ( ρ A ) ⊗ k ) ≤ ( k + 1) d ( k + 1) d ( d − / e − k ǫ ≤ ( k + 1) d ( d +1) / e − k ǫ , since there are at most ( k + 1) d partitions of k of length at most d .Let P Y denote the orthogonal projection of ( H B ) ⊗ k onto the sum of itsisotypical components S ν ⊗ V ν satisfying k ν − u k ≤ ǫ , and let P Z denotethe orthogonal projection of ( H A ⊗ H B ) ⊗ k onto the sum of its isotypicalcomponents S Λ ⊗ V Λ satisfying k Λ − λ k ≤ ǫ . We set P Y := id − P Y and P Z := id − P Z . Then we have, similarly as for P X ,tr( P Y ( ρ B ) ⊗ k ) ≤ ( k + 1) d ( d +1) / e − k ǫ , tr( P Z ( ρ AB ) ⊗ k ) ≤ ( k + 1) d ( d +1) / e − k ǫ .
9y choosing k = O ( d ǫ log dǫ ) we can achieve thattr( P X ( ρ A ) ⊗ k ) < , tr( P Y ( ρ B ) ⊗ k ) < , tr( P Z ( ρ AB ) ⊗ k ) < . We put σ := ( ρ AB ) ⊗ k in order to simplify notation and claim thattr(( P X ⊗ P Y ) σP Z ) > . (4)In order to see this, we decompose id = P X ⊗ P Y + P X ⊗ id + P X ⊗ P Y . Fromthe definition of the partial trace we havetr (cid:0) ( P X ⊗ id) σ (cid:1) = tr (cid:0) P X ( ρ A ) ⊗ k (cid:1) < . Similarly, tr (cid:0) ( P X ⊗ P Y ) σ (cid:1) ≤ tr (cid:0) (id ⊗ P Y ) σ (cid:1) = tr (cid:0) P Y ( ρ B ) ⊗ k (cid:1) < . Hence tr (cid:0) ( P X ⊗ P Y ) σ (cid:1) > . Using tr (cid:0) ( P X ⊗ P Y ) σP Z (cid:1) ≤ tr( σP Z ) < , we gettr (cid:0) ( P X ⊗ P Y ) σP Z (cid:1) = tr (cid:0) ( P X ⊗ P Y ) σ (cid:1) − tr (cid:0) ( P X ⊗ P Y ) σP Z (cid:1) > −
13 = 0 , which proves Claim (4).Claim (4) implies that there exist partitions µ, ν, Λ with normalizations ǫ -close to u d , u d , r , respectively, such that ( P µ ⊗ P ν ) P Λ = 0. Recalling theisotypical decomposition (2), we infer that( S Λ ⊗ V Λ ) ∩ ( S µ ⊗ V µ ) ⊗ ( S ν ⊗ V ν ) = 0 . Statement (1) implies that g µ,ν, Λ = 0 and hence the assertion follows for R = µ, R = ν . (cid:3) We assume that λ, µ ⊢ d n . The group GL d × GL d × GL d operates on C d ⊗ C d ⊗ C d by tensor product, which induces an action on the polynomialring A on C d ⊗ C d ⊗ C d . Schur-Weyl duality implies that the submodule A n of homogeneous polynomials of degree n splits as follows (cf. [13]): A n = M λ,µ,ν ⊢ d n (cid:0) S λ ⊗ S µ ⊗ S ν (cid:1) S n ⊗ V ∗ λ ⊗ V ∗ µ ⊗ V ∗ ν .
10e assume now that g λ,µ,µ = dim (cid:0) S λ ⊗ S µ ⊗ S µ (cid:1) S n = 0 for some λ, µ ⊢ d n .Hence there exists a highest weight vector F ∈ A n of weight ( λ, µ, µ ). Wemay assume that the coefficients of F are real (cf. [1]).Consider the linear automorphism that exchanges the last two factors of C d ⊗ C d ⊗ C d . This induces an automorphism σ of the algebra A . It iseasy to see that F ′ := σ ( F ) is a highest weight vector of weight ( λ, µ, µ ).Therefore, both squares F and ( F ′ ) are highest weight vectors of weight(2 λ, µ, µ ). Since F + ( F ′ ) is nonzero and invariant under σ , we see that (cid:0) S λ ⊗ S µ ⊗ S µ (cid:1) S n has a nonzero invariant with respect to σ . Hence (cid:0) S λ ⊗ Sym ( S µ ) (cid:1) S n = 0 , which means that sg λ µ = 0. (cid:3) References [1] B¨urgisser, P., Christandl, M., Ikenmeyer, C., 2011. Even partitions inplethysms. Journal of Algebra 328, 322–329.[2] B¨urgisser, P., Clausen, M., Shokrollahi, M.A., 1997. Algebraic Com-plexity Theory. Volume 315 of