aa r X i v : . [ m a t h . R T ] D ec On the asymptotics of Kronecker coefficients, 2
Laurent
Manivel
UMI 3457 CNRS/Centre de Recherches Math´ematiques,Universit´e de Montr´eal, Canada [email protected]
September 17, 2018
Abstract
Kronecker coefficients encode the tensor products of complex irreducible representationsof symmetric groups. Their stability properties have been considered recently by several au-thors (Vallejo, Pak and Panova, Stembridge). In [3] we described a geometric method, basedon Schur-Weyl duality, that allows to produce huge series of instances of this phenomenon.In this note we show how to go beyond these so-called additive triples. We show that theset of stable triples defines a union of faces of the moment polytope. Moreover these facesmay have different dimensions, and many of them have codimension one.
Keywords . Symmetric group, Kronecker coefficient, stability, Schur-Weyl duality, Borel-Weiltheorem, face, facet, simplicial
The complex representation theory of symmetric groups is well understood: the irreducible rep-resentations, usually called Specht modules, are indexed by partitions and their dimensions aregiven by the famous hook length formula. But the multiplicative structure of the representationring has always remained elusive. The multiplicities in tensor products of Specht modules arecalled Kronecker coefficients. They are poorly undestood and notoriously hard to compute.We refer to the introduction to [3] for a discussion of some of the most basic questions aboutKronecker coefficients whose answers remain out of reach.That Kronecker coefficients enjoy certain stability properties has been observed by Mur-naghan in 1938 [4, 5]. Such properties are extremely surprising in that they involve repre-sentations of differents groups, but they become less mysterious once translated in terms ofrepresentations of general linear groups, thanks to Schur-Weyl duality. More stability phenom-ena have been discovered during the last twenty years, and the wealth of examples we are nowaware of makes more urgent the need to understand and organize them better. That is one ofthe goals of this note.We use the following terminology, taken from [6] and [3]. We denote by [ λ ] the Specht moduleassociated with the partition λ . This is an irreducible representation of the symmetric group S n , if λ is a partition of n . Kronecker coefficients are defined by the identity[ λ ] ⊗ [ µ ] = ⊕ ν g ( λ, µ, ν )[ ν ] . They are symmetic in λ, µ, ν and of course, non negative.
Definition.
A triple of partitions ( λ, µ, ν ) is weakly stable if the Kronecker coefficients g ( kλ, kµ, kν ) = 1 ∀ k ≥ . t is stable if g ( λ, µ, ν ) = 0 and for any triple ( α, β, γ ) , the sequence of Kronecker coefficients g ( α + kλ, β + kµ, γ + kν ) is bounded, or equivalently, eventually constant. We call the asymptoticvalue of this coefficient a stable Kronecker coefficient. Stability implies weak stabilty. The converse implication is not known. Conjecturally, thetwo notions should be equivalent.In order to get nice finiteness properties we restrict to partitions whose length, rather thatsize, are bounded; the length ℓ ( λ ) of a partition λ is the number of non zero parts. We wouldthen like to understand stability phenomena in relation with the Kronecker semigroup and theKronecker polytohedron. The former is Kron a,b,c := { ( λ, µ, ν ) , ℓ ( λ ) ≤ a, ℓ ( µ ) ≤ b, ℓ ( ν ) ≤ c, g ( λ, µ, ν ) = 0 } . This is a finitely generated semigroup. A more precise version of the semigroup property is theelementary, but useful monotonicity property : if g ( λ, µ, ν ) = 0, then for any triple ( α, β, γ ), g ( α + λ, β + µ, γ + ν ) ≥ g ( α, β, γ ) . The semigroup
Kron a,b,c lives inside a codimension two sublattice of Z a + b + c , because ofthe obvious condition | λ | = | µ | = | ν | for a Kronecker coefficient g ( λ, µ, ν ) to be non zero. Wecall this lattice the weight lattice . The cone generated by Kron a,b,c is a rational polyhedralcone
P Kron a,b,c , that we call the Kronecker polyhedron. it is defined by some finite list oflinear inequalities, giving the equations of its facets (the maximal faces, of codimension one).The number of facets is huge already for small values of the parameters, and certainly growsexponentially with a, b, c (see [1] and [8]).In [2, 3], we showed that certain minimal faces of the Kronecker polyhedron are made ofstable triples. These minimal faces were defined in terms of certain standard tableaux withthe additivity property . Let us suppose for simplicity that c = ab (this is not a restriction,since it is well known that for a Kronecker coefficient g ( λ, µ, ν ) to be non zero, the condition ℓ ( λ ) ≤ ℓ ( µ ) ℓ ( ν ) on the lengths is required). Consider a standard tableau T of rectangularshape a × b . Such a tableau is additive if there exist increasing sequences x < · · · < x a and y < · · · < y b of real (or rational) numbers such that T ( i, j ) < T ( k, l ) ⇐⇒ x i + y j < x k + y l . The main stability result in [3] was the following:
Proposition 1
Let T be any additive standard tableau of rectangular shape a × b . For anypartition λ = ( λ , . . . , λ ab ) , define two partitions a T ( λ ) , b T ( λ ) by a T ( λ ) i = b X j =1 λ T ( i,j ) , b T ( λ ) j = a X i =1 λ T ( i,j ) . Then ( λ, a T ( λ ) , b T ( λ )) is a stable triple. Moreover the set of these additive triples, for a fixed T , is exactly the set of lattice pointsinside a minimal face f T of P Kron a,b,ab defined by this standard tableau.We want to stress here that the fact that stable triples can be related to faces of the Kroneckerpolyhedron is by no means a surprise. A general statement is the following.
Proposition 2
The set
SKron a,b,c of weakly stable triples in
Kron a,b,c is the intersection of
Kron a,b,c with a union of faces of
P Kron a,b,c . P Kron a,b,c that are maximal in
SKron a,b,c . We will call the faces of
P Kron a,b,c whose intersection withthe weight lattice are contained in
SKron a,b,c the stable faces , and those that are maximal in
SKron a,b,c , the maximal stable faces (which we don’t expect a priori to be maximal faces in
P Kron a,b,c , or facets). Among many other questions, we can ask: what is the maximal dimen-sion of such a face? what can be their dimensions? could they all be of the same dimension?can the additive stable faces be maximal in
SKron a,b,c ? more generally, what are the stablefaces containing a given additive stable face?The main goal of this note is to answer some of these questions, in particular the last one,and draw some unexpected consequences. In [3] we explained how to describe the local structureof the Kronecker polyhedron around an additive face. Among the faces that contain such anadditive face, we will distinguish those that have a property that we will call strong simpliciality .We will prove:
Theorem 1
Among the faces of
P Kron a,b,c that contain an additive face, the stable ones areexactly those that are strongly simplicial.
A priori, we would have expected the stable faces to be very special, in particular to havehigh codimension. Surprisingly, our Theorem has the following consequence:
Corollary 1
The polyhedral cone
P Kron a,b,c always contains stable facets.
This means that there exist families of stable triples of the largest possible dimension. Itwould be extremely interesting to have a full classification. We can give many explicit examplesof strongly simplicial facets and show that there always exist many of them (Proposition 4). Wecan also describe their structure, which is that of a cone over hypercube (Proposition 3). Thevertices of this hypercube are in bijection with the additive faces contained in the facet.Another striking phenomenon is the following. Consider an additive face, and the maximalstable faces that contain it. It may very well happen that these maximal faces have differentdimensions! In fact it seems quite plausible that the maximal stable faces can have all thepossible dimensions between the smallest and maximal possible dimensions. In particular theset of stable triples seems to have a very intricate structure in general.
Let us briefly recall the main features of the geometric method used in [2, 3] in order to approachKronecker coefficients. Let
A, B be complex vector spaces of finite dimensions a, b . By Schur-Weyl duality, Kronecker coefficients are the multiplicities of the Schur powers S λ ( A ⊗ B ), whendecomposed into irreducible representations for GL ( A ) × GL ( B ). By the Borel-Weil theorem, S λ ( A ⊗ B ) = H ( F l ( A ⊗ B ) , L λ )for a suitable linearized line bundle L λ on the variety F l ( A ⊗ B ) of complete flags in A ⊗ B . Astandard tableau T defines an embedding ι T : F l ( A ) × F l ( B ) ֒ → F l ( A ⊗ B ) . ι ∗ T : P ic ( F l ( A ⊗ B )) ≃ Z ab → P ic ( F l ( A ) × F l ( B )) ≃ Z a × Z b is precisely our map λ ( a T ( λ ) , b T ( λ )) when expressed in natural basis. In particular, restric-tion gives a nonzero map H ( F l ( A ⊗ B ) , L λ ) −→ H ( F l ( A ) × F l ( B ) , L a T ( λ ) ⊗ L b T ( λ ) ) = S a T ( λ ) A ⊗ S b T ( λ ) B, (1)implying that the Kronecker coefficient g ( λ, a T ( λ ) , b T ( λ )) is positive. Then we can define afiltration of H ( F l ( A ⊗ B ) , L λ ) by the order of vanishing on F l ( A ) × F l ( B ). This allows todefine an injective map H ( F l ( A ⊗ B ) , L λ ) ֒ → H ( F l ( A ) × F l ( B ) , L a T ( λ ) ⊗ L b T ( λ ) ⊗ S ∗ N ∗ ) , (2)where N denotes the normal bundle of the embedding ι T , and S ∗ N ∗ is the symmetric algebraof the dual bundle, the conormal bundle. This map must be thought of as taking a sectionof L λ , to its Taylor expansion in the normal directions to F l ( A ) × F l ( B ). Moreover, if λ is strictly decreasing, the line bundle L λ is very ample. By the usual properties of amplebundles, the previous map becomes surjective onto every finite part of S ∗ N ∗ if L λ is sufficientlyample (that is, if the differences λ i − λ i +1 are large enough). This shows that the multiplicitiesin S λ ( A ⊗ B ) = H ( F l ( A ⊗ B ) , L λ ), otherwise said, the Kronecker coefficients, are somehowcontrolled by the normal bundle.This works particularly well when the embedding ι T is convex , in the sense that the weights ofthe normal bundle are contained in a strictly convex cone. Combinatorially, this exactly meansthat the tableau T is additive. Then the Kronecker coefficient g ( α + kλ, β + ka T ( λ ) , γ + kb T ( λ ))is bounded by the multiplicity of L β − a T ( α ) ⊗ L γ − b T ( α ) inside S ∗ N ∗ , and the latter is finite byconvexity. This implies that we can focus on a finite part of this algebra, independantly of k .But then, if λ is strict and k is large enough, the surjectivity of (2) in finite degrees implies thatwe have equality between the latter multiplicity, and the Kronecker coefficient. In particularthis coefficient does not depend on k , when big enough: this is the stability phenomenon. Butof course we get much more information since we are in principle able to compute the stableKronecker coefficients, directly from the normal bundle.Combinatorially, the weights of the conormal bundle are determined as follows. Denote by e , . . . , e a and f , . . . , f b basis of the character lattices of maximal tori in GL ( A ) and GL ( B ).If T ( i, j ) = k (ie the box ( i, j ) is numbered k in T ), let g k = e i + f j . Then the weights of thenormal bundle are the differences g ℓ − g k for ℓ > k . Among those weights, the horizontal andvertical ones are those of the form e p − e q and f p − f q . They will appear repeatedly, in fact in theconormal bundle their multiplicities are b − a − a, b >
2. Allthe other weights have multiplicity one. Of course, the multiplicity of L β − a T ( α ) ⊗ L γ − b T ( α ) inside S ∗ N ∗ , which gives the stable Kronecker coefficient, can be obtained as the number of ways toexpress the weight ( β − a T ( α ) , γ − b T ( α )) as a non negative linear combination of the weightsof the conormal bundle, considered with their multiplicities.Of course this is possible only when ( β − a T ( α ) , γ − b T ( α )) belongs to the cone generated bythose weights, which we call the conormal cone . This cone gives a local picture of the Kroneckerpolyhedron locally around f T . In particular any face of the latter containing f T , can be identifiedwith a face of the conormal cone, and conversely. Now consider a triple of the form ( λ, a T ( λ ) + σ, b T ( λ ) + θ ) , where σ and θ are not necessarilypartitions, but sequences (or weights) such that a T ( λ ) + σ and b T ( λ ) + θ are partitions.4y the injectivity of (2), the Kronecker coefficient g ( kλ, k ( a T ( λ )+ σ ) , k ( b T ( λ )+ θ )) is boundedby the multiplicity of ( kσ, kθ ) as a weight of S ∗ N ∗ . If we suppose that the line generated by( σ, θ ) belongs to the conormal cone, this multiplicity will eventually become positive, and weexpect it to grow to infinity with k . But this is not necessarily the case: the multiplicity willremain bounded if ( σ, θ ) belongs to a face of the cone which is strongly simplicial . By thiswe mean that the weights of the conormal bundle contained in the face, considered with theirmultiplicities, define a basis of the linear space generated by the face. Then the multiplicity willbe 0 or 1, the second possibility occuring exactly when ( kσ, kθ ) belongs to the lattice generatedby the latter weights.Let us insist on the definition of strongly simplicial faces. Definition . A face F of the Kronecker polytope is strongly simplicial if it contains an additiveface f T such that locally around f T , the face F corresponds to a face in the cone generated bythe conormal bundle which is strongly simplicial in the sense that:1. it is a face of dimension d generated by d vectors g k +1 − g k , . . . , g k d +1 − g k d ,2. none of these vectors is horizontal (unless b = 2 ) or vertical (unless a = 2 ),3. no other vector of the form g p − g q belongs to the face,4. in particular the pairs ( k , k + 1) , . . . , ( k d , k d + 1) do not intersect. The structure of a strongly simplicial face is not difficult to describe. Recall that an additivetableau T is defined by parameters x < · · · < x a and y < · · · < y b such that T ( i, j ) < T ( k, l )if and only if x i + y j < x k + y l . Of course these parameters are not unique. In fact the tableau T really corresponds to a connected component C T of the complement of the collection ofhyperplanes defined by the equations x i + y j = x k + y l inside the parameter space.Locally around the additive face f T , the Kronecker polyhedron is, by hypothesis, the simpli-cial cone over the vectors g k +1 − g k , . . . , g k d +1 − g k d . Let us choose one of them, say g k s +1 − g k s .Since it is neither horizontal nor vertical, we can exchange the entries k s and k s + 1 in T andobtain another standard tableau T s . We claim that T s is again additive. Indeed, if the entries k s and k s + 1 of T appear in boxes ( i, j ) and ( k, l ), the fact that g k s +1 − g k s is an extremal vectorof the cone implies that the hyperplane x i + y j = x k + y l is really a facet of C T . Crossing thisfacet we get into a component corresponding to T s , which is therefore additive.Iterating the process, we deduce that the 2 d standard tableaux obtained by considering allthe possibilities to exchange the entries ( k , k + 1) . . . ( k d , k d + 1), are all additive. Moreoverthe Kronecker polyhedron, around each of the corresponding additive faces, is described by thesame cone, up to a change of signs for the generators. This implies that our strongly simplicialface is contained in the set of triples( λ, µ, ν ) = ( λ, a T ( λ ) , b T ( λ )) + d X i =1 u s ( g k s +1 − g k s ) , (3)with 0 ≤ u s ≤ λ k s − λ k s +1 . Note that the coefficients u , . . . , u d need to be integers. But it is apriori possible that we get a triple of partitions ( λ, µ, ν ) given by the same expression but withrational coefficients u , . . . , u d , not all integral. In this case, the Kronecker coefficient g ( λ, µ, ν )would certainly be zero.Otherwise said, the identity (3) defines a lattice L F , which could be a proper sublattice ofthe intersection of the weight lattice with the linear span of F . In this lattice, F is simplydefined by the inequalities 0 ≤ u s ≤ λ k s − λ k s +1 for 1 ≤ s ≤ d . Recall that d is the number ofgenerators of the face in the normal directions of an additive face it contains. In particular thecodimension of F is the codimension of an additive face (that is, a + b −
2) minus d . We get thefollowing description of strongly simplicial faces.5 roposition 3 A strongly simplicial face F of codimension δ in the Kronecker polyhedron is acone over a hypercube of dimension a + b − − δ . The main result of this paper is the following.
Theorem 2
A strongly simplicial face F of the Kronecker polyhedron is stable. More precisely,any point in F is a stable triple if it belongs to L F , and the corresponding Kronecker coefficientis zero otherwise.Proof. Consider a triple of the form ( kλ + α, k ( a T ( λ ) + σ ) + β, k ( b T ( λ ) + θ )) + γ ) , where as before( σ, θ ) belongs to the simplicial face corresponding to F in the conormal cone of the additive face f T . As we have seen, the corresponding Kronecker coefficient is bounded by the multiplicity of( kσ + β − a T ( α ) , kθ + γ − b T ( α )) as a weight of S ∗ N ∗ . Suppose that we have expressed thisweight as a non negative integer linear combination t η + · · · + t N η N of the weights η , . . . , η N of the conormal bundle, considered with their multiplicities. Suppose these weights are indexedin such a way that the first d generate our simplicial face. By projection along the direction ofthis face, we get a relation of the form p F ( β − a T ( α ) , γ − b T ( α )) = t d +1 p F ( η d +1 ) + · · · + t N p F ( η N ) , (4)where p F denotes the projection. But the projected weights p F ( η d +1 ) , . . . , p F ( η N ) generate astrictly convex cone, so the latter equation has only finitely many non negative integer solutions( t d +1 , . . . , t N ). These solutions do not depend on k , and for each of these, the original equationhas at most one solution in ( t , . . . , t d ), since it can be considered as an equation in the simplicialface F . This proves that the Kronecker coefficient g ( kλ + α, k ( a T ( λ ) + σ ) + β, k ( b T ( λ ) + θ )) + γ ) , is bounded independently of k .This is precisely the definition of stability, up to the fact that the Kronecker coefficient g ( λ, a T ( λ ) + σ, b T ( λ ) + θ ) must be equal to one. Recall that by [6], Propoition 3.2, the onlyalternative is that it is equal to zero. So what remains to prove is that if ( λ, µ, ν ) is a point of F that also belongs to the lattice L F , the Kronecker coefficient g ( λ, µ, ν ) cannot be zero.To check this we will use that ( λ, µ, ν ) is given by (3) for some integer coefficients u , . . . , u d such that 0 ≤ u s ≤ λ k s − λ k s +1 for all s . Denote by ω t the partition of t with t parts equal toone. Recall that we denoted by T s the standard tableau obtained by exchanging the entries k s and k s + 1 in T . It is straightforward to check that( ω k s , a T s ( ω k s ) , b T s ( ω k s )) − ( ω k s , a T ( ω k s ) , b T ( ω k s )) = g k s +1 − g k s . This allows to rewrite (3) as( λ, µ, ν ) = ( θ, a T ( θ ) , b T ( θ )) + d X s =1 u s ( ω k s , a T s ( ω k s ) , b T s ( ω k s )) , where θ = λ − P ds =1 u s ω k s . Since u s ≤ λ k s − λ k s +1 for all s , this θ is again a partition. Since T and the T s are all additive, we know that g ( θ, a T ( θ ) , b T ( θ )) = 1 and g ( ω k s , a T s ( ω k s ) , b T s ( ω k s )) = 1for all s . In particular all these Kronecker coefficients are non zero, and from the semigroupproperty we deduce that g ( λ, µ, ν ) is positive. ✷ Corollary 2
A strongly simplicial face is the non negative integral span of the additive faces itcontains. Moreover any additive face is properly contained in some strongly simplicial face.Proof.
The first statement means that any stable triple in F can be obtained as a linear com-bination with positive integer coefficients, of some stable triples in the additive faces containedin F . This is what we established in the proof of the Theorem.6or the second statement, simply observe that at least one face of the Kronecker polyhedron,that contains f T and has dimension one more, must be simplicial. Indeed, these faces corre-spond to the minimal generators of the conormal cone, and they are simplicial exactly for thosegenerators that are neither horizontal nor vertical. But the generators cannot be all horizontalor vertical, since otherwise inside T , the integer k + 1 would always be South-East of k , whichis absurd. ✷ Remarks.
One can wonder if there can be non trivial arithmetic conditions on the stronglysimplicial faces, for the Kronecker coefficients to be non zero? This would mean that L F isreally a proper sublattice of the intersection of F with the weight lattice. This seems a prioripossible but we have no example of such a phenomenon.One can also wonder if stable Kronecker coeficients, when one considers strongly simplicialfaces, count points in some polytopes, as they do on additive faces [3]. In the proof above weindeed bounded the stretched Kronecker coefficients by numbers of points in some polytopes, butit is not clear that this bound coincides with the stable Kronecker coefficient. In the additive casethis follows from an ampleness argument, which does not apply in this more general situation. In [3] we gave a combinatorial description of the facets of the Kronecker polytope containing agiven additive face f T . These facets are in bijection with the maximal relaxations compatiblewith T , where a maximal relaxation R is given by an additive (non standard) tableau defined bysequences x ≤ · · · ≤ x a and y ≤ · · · ≤ y b such that the sums R ( i, j ) = x i + y j are not necessarilydistinct. What we require is that the set of vectors e i + f j − e k − f l , for R ( i, j ) = R ( k, l ), hasmaximal rank r = a + b −
3. Such a family of vectors being given, the sequences x ≤ · · · ≤ x a and y ≤ · · · ≤ y b are uniquely defined up to translation, and multiplication by the same positivenumber. It is convenient to define the tableau R uniquely by letting x = y = 0, and askingthe two sequences to be made of integers, with no common divisor. The compatibility conditionwith a standard tableau T is that R ( i, j ) < R ( k, l ) implies T ( i, j ) < T ( k, l ). Otherwise said, R defines a partial order on the boxes in the rectangle a × b , which is refined by the total orderdefined by T . The equation of the facet F R is then given by a X i =1 x i µ i + b X j =1 y j ν j = a X i =1 b X j =1 ( x i + y j ) λ T ( i,j ) , where T is any standard tableau compatible with R .Can such a maximal relaxation R define a strongly simplicial facet? This would mean that R is defined by strictly increasing sequences, and that there exists exactly r = a + b − R appearing twice, the corresponding difference vectors being independent. In terms of thehyperplanes of equations x i + y j − x k − y l = 0, and the arrangement they define in the opencone defined by 0 = x < · · · < x a and 0 = y < · · · < y b , such an R corresponds to a pointwhere exactly r hyperplanes meet transversaly. Recall that this transversality property impliesthat any of the 2 r standard tableaux T compatible with R is additive.Another unexpected fact is that in general, there exist surprisingly many strongly simplicialfacets! Proposition 4
P Kron ( a, b, ab ) contains at least (cid:0) a + b − b − (cid:1) strongly simplicial facets.Proof. One can construct tableaux defining strongly simplicial facets by a simple induction:suppose that a tableau S defines a simplicial facet for the format a × ( b − a × b by adding a column defined by y b = x a + y b − . Of course this also works for7ows. So starting from the tableau defining the unique simplicial face in format 2 ×
2, we canconstruct (cid:0) a + b − b − (cid:1) strongly simplicial facets in format a × b by choosing to apply the previousprocess on rows or columns successively, in all possible orders. ✷ Let us examine in more detail the low dimension cases.
Example 1.
For a = b = 2 there is exactly one additive face (up to symmetry). This additiveface is the intersection of two facets, one of which is strongly simplicial. On the additive facewe get g (( λ , λ , λ , λ ) , ( λ + λ , λ + λ ) , ( λ + λ , λ + λ )) = 1 , and for the strongly simplicial facet we get the more general statement that g (( λ , λ , λ , λ ) , ( µ , µ ) , ( ν , ν )) = 1when µ − ν = λ − λ and λ + λ ≤ µ ≤ λ + λ . Moreover all these triples are stable. Example 2.
For a = b = 3 there exist 42 standard tableaux fitting in a square of size three,among which 36 are additive. The number of maximal relaxations is 17. They are encoded inthe following tableaux: F +1 F +2 F +3 F +4 F +5 F F +7 F F F +10 F + i there is another one denoted F − i obtained by diagonal symmetry.Recall that additive faces have dimension four. A detailed analysis yields the following result: Proposition 5
For a = b = 3 , the maximal strongly simplicial faces are, up to diagonal sym-metry:1. in codimension one, F +5 and F +10 ,2. in codimension two, F +3 ∩ F +4 , F +3 ∩ F , F +4 ∩ F ,3. in codimension three, F ∩ F +7 ∩ F and F +7 ∩ F ∩ F . Let us describe the sets of triples ( λ, µ, ν ) on these strongly simplicial faces. We will use thenotations λ ij = λ i + λ j and λ ijk = λ i + λ j + λ k . F +5 is defined by the equation 2 µ + 3 µ + ν + 2 ν = λ + 2 λ + 2 λ + 3 λ + 3 λ + 4 λ + 4 λ + 5 λ and the inequalities λ ≤ µ ≤ λ , λ + λ ≤ µ + ν ≤ λ + λ , λ − λ ≤ µ − ν ≤ λ − λ .F +10 is defined by the equation µ + 3 µ + ν + 2 ν = λ + λ + 2 λ + 2 λ + 3 λ + 3 λ + 4 λ + 5 λ and the inequalities λ − λ ≤ ν − µ ≤ λ − λ , λ + λ ≤ µ + ν ≤ λ + λ ,λ ≤ µ ≤ λ .F +3 ∩ F +4 is defined by the two equations ν = λ and µ + 2 µ + ν = λ + λ + 2 λ andthe inequalities λ ≤ µ ≤ λ , λ ≤ µ ≤ λ . +3 ∩ F is defined by the two equations ν = λ and µ − µ + ν = λ + λ − λ andthe inequalities λ ≤ µ ≤ λ , λ ≤ µ ≤ λ .F +4 ∩ F is defined by the two equations ν = λ and µ − µ − ν = λ − λ − λ andthe inequalities λ ≤ µ ≤ λ , λ ≤ µ ≤ λ .F ∩ F +7 ∩ F is defined by the three equalities µ + ν = λ + λ , µ = λ , ν = λ andthe inequalities λ ≤ µ ≤ λ .F +7 ∩ F ∩ F is defined by the three equalities µ + ν = λ + λ , µ = λ , ν = λ andthe inequalities λ ≤ µ ≤ λ . There are no arithmetic constraints on these stongly simplicial faces, so the Kronecker coef-ficients are always equal to one and all these triples are stable.Note also that the additive face defined by the standard tableau1 2 34 5 76 8 9is contained in both F +5 and F +3 ∩ F +4 , showing that an additive face can be contained in twomaximal strongly simplicial faces of different dimensions! This indicates that the structure ofthe set of additive triples must be quite intricate in general. Example 3.
For a = b = 4 there are 6660 additive tableaux and 457 maximal relaxations,according to [1, 8]. Among these, we know 43 strongly simplicial ones, among which:0 1 2 31 2 3 44 5 6 77 8 9 A C A D
A D B A C A C B D
A C
B DB C G I B A D (The symbols
A, B and so on stand for 10 ,
11 and so on.) It would be interesting to havethe complete list.
There exist two natural involutions on the set of additive tableaux. Recall that an additivetableau can be defined by increasing sequences x < · · · < x a and y < · · · < y b such thatthe sums x i + y j are distinct. Then we can replace each of this sequence by the opposite one,reordered increasingly. Since this preserves the set of hyperplanes of equations x i + y j = x k + y l ,this defines two commuting involutions on the set of additive tableaux, and then also on the setof maximal relaxations, and on the subset of simplicial relaxations.9 Non simplicial faces
Recall that a stretched Kronecker coefficient g ( kλ, kµ, kν ) is quasipolynomial: there exists acollection of polynomials P , . . . , P p − , such that g ( kλ, kµ, kν ) = P i ( k ) for k = i (mod p) . By the monotonicity property, P i + j ( k + ℓ ) ≥ P i ( k ) as soon as P j ( ℓ ) = 0. This implies thatamong the polynomials P , . . . , P p − , those that are not identically zero have the same degree d , and the same leading term as well. We call d = d ( λ, µ, ν ) the degree of the triple ( λ, µ, ν ).For example weakly stable triples have degree zero, and a triple of degree zero is one that has aweakly stable multiple.Another straightforward consequence of the monotonicity property, and of the convexity ofthe faces, is the following statement. Proposition 6
Let F be a face of the Kronecker polyhedron. The degree is constant on theinterior of F , and can only decrease, or remain the same, on its boundary faces. Definition.
Let F be a face of the Kronecker polyhedron. We define its degree as the degree ofits interior points. For example, any additive face, more generally any strongly simplicial face,has degree zero. For a non simplicial face, containing an additive face, we will show that the degree can be readoff directly on the normal bundle.
Definition.
A face F of the Kronecker polyhedron will be called δ -simplicial if there existsan additive face f T in F , such that the face f of the conormal cone corresponding to F is δ -simplicial. By this we mean that f has dimension d , but contains d + δ weights of the conormalbundle, counted with their multiplicity. Strongly simplicial is therefore the same as 0-simplicial. Note also that starting from a face F of the Kronecker polyhedron, the integer δ will not depend on the minimal face f T containedin F . This is a consequence of the following statement: Theorem 3 A δ -simplicial face F of the Kronecker polyhedron has degree δ .Proof. We consider F with the additive face f T , and we identify F with the corresponding face ofthe conormal cone. We consider stretched Kronecker coefficients g ( kλ, k ( a T ( λ )+ σ ) , k ( b T ( λ )+ θ )),where the weight ( σ, θ ) belongs to the linear span of the face. Denote this Kronecker coefficientby g k . It may a priori happen that the lattice generated by the weights of the conormal bundlebelonging to the face does not contain ( σ, θ ). In general there exists a minimal integer e ,depending on ( σ, θ ), such that e ( σ, θ ) belongs to this lattice. If k is not divisible by e , then k ( σ, θ ) does not belong to the lattice and g k = 0. If we restrict to those k that are divisible by e , then the number of ways to express k ( σ, θ ) as a non negative integer linear combination ofweights of the conormal bundle certainly grows like k δ . By the injectivity of (2), this impliesthat the growth of g k is at most in k δ .To get to the required conclusion, we must control the surjectivity of (2). The key point isthe following general statement. 10 emma 1 Let L be an ample line bundle, M a globally generated line bundle on a smoothcomplex projective variety X . Let ι : Y ֒ → X be the embedding of a smooth subvariety, anddenote by N the normal bundle. Then there exists integers m and n , not depending on M , suchthat the natural map H ( X, I dY ⊗ L a ⊗ M ) −→ H ( Y, S d N ∗ ⊗ ι ∗ ( L a ⊗ M )) is surjective when a ≥ md + n . If we apply this statement to ι T , we deduce that there exists integers m T , n T such that (2)is surjective up to degree d as soon as λ i − λ i +1 ≥ m T d + n T for each i . Replacing λ by kλ we get the surjectivity up to degree ( k − n T ) /m T . This yields a lower bound for g k of order( k − n T /m T ) δ , and the claim follows. ✷ Proof of the Lemma.
To get the surjectivity it is enough to prove that H ( X, I d +1 Y ⊗ L a ⊗ M ) = 0 . Let π : Z → X be the blow-up of Y , and E the exceptional divisor. Since π ∗ O Z ( − iE ) = I iY andthere are no higher direct images, we are reduced to proving that H ( Z, O Z ( − ( d + 1) E ) ⊗ π ∗ ( L a ⊗ M )) = 0 . The canonical line bundle of Z is K Z = π ∗ K X ⊗ O Z (( c − E ), if c denotes the codimension of Y in X . So we can rewrite the previous condition as H ( Z, K Z ⊗ O Z ( − ( d + c ) E ) ⊗ π ∗ ( L a ⊗ M ⊗ K − X )) = 0 . We can find an a such that L a ⊗ K − X is ample. Moreover the exists b such that I Y ⊗ L b isgenerated by global sections, hence also O Z ( − E ) ⊗ π ∗ L b . Then O Z ( − ( d + c ) E ) ⊗ π ∗ ( L a ⊗ M ⊗ K − X ) is nef and big as soon as a ≥ a + b ( d + c ), and the required vanishing follows from theKawamata-Viehweg vanishing theorem. ✷ Acknowledgements.
This paper was in Montr´eal at the Centre de Recherches Math´ematiques(Universit´e de Montr´eal) and the CIRGET (UQAM). The author warmly thanks these institu-tions for their generous hospitality. We also thank Mateusz Michalek for his help regarding thecombinatorics of simplicial facets. In particular Proposition 6 is due to him.
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