On the growth of the Kronecker coefficients: accompanying appendices

ON THE GROWTH OF THE KRONECKER COEFFICIENTS: ACCOMPANYING APPENDICES. EMMANUEL BRIAND, AMARPREET RATTAN, AND MERCEDES ROSAS This text is an appendix to our work ”On the growth of Kronecker coefficients” [1]. Here, we provide some complementary theorems, remarks, and calculations that for the sake of space are not going to appear into the final version of our paper. We follow the same terminology and notation. External references to numbered equations, theorems, etc. are pointers to [1]. This file is not meant to be read independently of the main text. Appendix A. Bounds We prove here the assertions made in Remarks 5.3 and 6.2 about the values of the constants ki(in Theorem 5.2 ) and k′ i(in Theorem 6.1). These technical and less central results do not appear in the printed version of this work. A.1. Hook stability, reduced Kronecker coefficients. In this section, we find explicitly bounds for the quantities k1,k2,k3appearing in Theorem 5.2 Theorem A.1. In Theorem 5.2 , one can take k1=|α|+α1+β′ 1+γ′ 1, k2=|β|+β1+α′ 1+γ′ 1, k3=|γ|+γ1+α′ 1+β′ 1. Proof. From the proof of Theorem 5.2 , one can take ki= maxω∈Ωℓi(ω), where Ω is the support of Q−=Q− α,β,γ (x, y, z). Let us perform the change of variables x=vw u,y=uw v,z=uv w, so that the identity xaybzc=uℓ1vℓ2wℓ3holds. Then k1(resp. k2,k3), as defined above, is the degree of Pwith respect to the variable u(resp. v,w). Date: August 2, 2018. E. Briand and M.Rosas are partially supported by projects MTM2013–40455–P, FQM–333, P12–FQM–2696 and FEDER. 1 2 BRIAND, A. RATTAN, AND M. ROSAS After this change of variables, H(−εx, −εy, −εz) equals XY Z +XY +XZ +Y Z −ε·uv wXY +uw vXZ +vw uY Z +u2X+v2Y+w2Z+ε1 vw u2−uv2−uw2X +ε1 uw v2−vu2−vw2Y+ε1 uv w2−wu2−wv2Z. We reorder the terms as follows: H(−εx, −εy, −εz) = u2X+εuXH1−εu v wY+w vZ+H0, where H0is a sum of monomials with non-positive degree in u, and H1 is free of uand X. We now factorize σ[H(−εx, −εy, −εz)] as σ[u2X]·σ[εuXH1]·σh−εu v wYi·σh−εuw vZi·σ[H0] and expand each series, except σ[H0]. We get that σ[H(−εx, −εy, −εz)] is equal to Xu2ihi[X]ujej[XH1]v wukek[Y]w vuℓeℓ[Z]σ[H0], where the sum ranges over all nonnegative integers i,j,k,ℓ. Therefore, Q−=hσ[H(−εx, −εy, −εz)] |sα[X]sβ[Y]sγ[Z]i =Xu2i+j+k+ℓvk−ℓwℓ−kej[XH1]σ[H0](h⊥ isα)[X](e⊥ ksβ)[Y](e⊥ ℓsγ[Z]). We have h⊥ isα= 0 unless i≤α1, and that e⊥ ksβ= 0 (resp. e⊥ ℓsγ= 0) unless j≤β′ 1(resp. k≤γ′ 1). Finally, since ej[XH1] is homogeneous of degree jin Xand h⊥ isαhas degree |α|−i, the summand corresponding to i,j,k,ℓcan be non zero only if i≤α1,j≤ |α| − i,k≤β′ 1and ℓ≤γ′ 1. Therefore 2i+j+k+ℓ≤ |α|+α1+β′ 1+γ′ 1. This proves that in Theorem 5.2 one can take k1=|α|+α1+β′ 1+γ′ 1. By symmetry, it follows that one can also take k2=|β|+β1+α′ 1+γ′ 1 and k3=|γ|+γ1+α′ 1+β′ 1. Remark A.2.More detailed computations show that the coefficient of u|α|+α1+β′ 1+γ′ 1in Q− α,β,γ is sα′1 + v2+w2 vw ·sβ′[1]sγ′[1] where αis the partition obtained from αby removing its first row (and α′is the conjugate of α). This is non–zero if and only if βand γhave at most one column, and αhas at most three columns. This is the only case when the bound is reached. KRONECKER COEFFICIENTS 3 A.2. First row for reduced Kronecker coefficients. We give bounds for the constants k′ 1, k′ 2and k′ 3appearing in Theorem 6.1. Theorem A.3. In Theorem 6.1, one can take k′ 1=|α|+|β|+|γ|+β1, k′ 2=|α|+|β|+|γ|+γ1, k′ 3=|α|+|β|+|γ|+α1+β1+γ1. Proof. From the proof of Theorem 6.1 one can take k′ 1= maxω∈Ωℓ1(ω), k′ 2= maxω∈Ω(ℓ1(ω)−ℓ2(ω)) , k′ 3= maxω∈Ω(ℓ1(ω)−ℓ3(ω)) . Let us perform the change of variables x=uvw,y=vw,z=uw, so that xaybzc=uℓ1−ℓ2vℓ1−ℓ3wℓ1. Then the constants k′ 1,k′ 2,k′ 3are the degrees of Q+ α,β,γ in the variables, respectively, u,vand w. Let us bound the degree in uof Q+. After the change of variables, we obtain that H(x, y, z) = u2vw2Y+uH1+H0where H1is free of u and has all its terms of degree at least 1 in X,Yand Z, and H0has all its terms of degree ≤0 in u. Thus, σ[H] = σ[u2vw2Y]σ[uH1]σ[H0] = X i,j u2i+j(vw2)ihi[Y]hj[H1]σ[H0], and, therefore, Q+=X i,j u2i+j(vw2)jhhi[Y]hj[H1]σ[H0]|sα[X]sβ[Y]sγ[Z]i =X i,j u2i+j(vw2)jhj[H1]σ[H0]sα[X](h⊥ isβ)[Y]sγ[Z]. Note that h⊥ isβ= 0 unless i≤β1. Moreover the left–hand side of each scalar product in the sum is now a sum of homogeneous symmetric functions all of total degree at least j, while the right–hand side has degree |α|+|β|+|γ| − i. Thus, the non-zero summands fulfill j≤ |α|+|β+|γ|−i. We conclude that for all non–zero summands, 2i+j≤ |α|+|β|+|γ|+β1. Appendix B. Another approach to the hook stability property, derived from Murnaghan’s stability and conjugation Here we detail the arguments sketched in Section 5.3. B.1. Hook Stability. Recall that the Kronecker coefficients are invariant under conjugation of any two of their arguments: gλ,µ,ν =gλ′,µ′,ν =gλ′,µ,ν′=gλ,µ′,ν′. 4 BRIAND, A. RATTAN, AND M. ROSAS Assume that (λ′, µ′, ν) is stable. That is, that the value of the Kronecker coefficient gλ′,µ′,ν does not change by adding one to the first parts of the three indexing partitions: (A1) gλ′,µ′,ν =gλ′⊕(1|0),µ′⊕(1|0),ν⊕(1|0). Conjugating the partitions in position 1 and 2, we obtain that the Kronecker coefficient gλ′⊕(1|0),µ′⊕(1|0),ν⊕(1|0) is equal to gλ⊕(0|1),µ⊕(0|1),ν⊕(1|0). We conclude that under our stability assumption gλ,µ,ν =gλ⊕(0|1),µ⊕(0|1),ν⊕(1|0). Similarly, under the corresponding stability hypothesis for the triples (λ′, µ, ν′) and (λ, µ′, ν′), we have gλ,µ,ν =gλ⊕(0|1),µ⊕(1|0),ν⊕(0|1) and gλ,µ,ν =gλ⊕(1|0),µ⊕(0|1),ν⊕(0|1). Let λ,µand νbe three non–empty partitions with the same weight N. A sufficient condition for the stability of the triple (λ, µ, ν) is N≥ N0(λ, µ, ν) (see (9), in Section 3.1). As a consequence of Lemma 5.6, (A2) N≥N0(b λ, bµ, bν) + λ′ 1+µ′ 1+ν′ 1 2 is also a sufficient condition for a triple of non–empty partitions (λ, µ, ν), all with weight N, to be stable. Let Lbe the set of triples of partitions of the same weight that fulfill (A2). Let L4be the set of triples of partitions (λ, µ, ν) such that all four triples (λ, µ, ν),(λ′, µ′, ν),(λ′, µ, ν′),(λ, µ′, ν′) are in L. Then L4is defined by inequalities (A3) N≥N0(b λ, bµ, bν) + (λ′ 1+µ′ 1+ν′ 1)/2, N≥N0(b λ, bµ, bν) + (λ1+µ1+ν′ 1)/2, N≥N0(b λ, bµ, bν) + (λ1+µ′ 1+ν1)/2, N≥N0(b λ, bµ, bν) + (λ′ 1+µ1+ν1)/2, where, again, Nis the weight of the partitions λ,µand ν. Fix partitions λ,µ,νof the same weight N. Let Dbe the set of all (a, b, c, m)∈N4such that m≥a, b, c and (λ⊕(m−a|a), µ⊕(m−b|b), ν⊕ (m−c|c)) belongs to L4. Then, from (A3), it is straightforwardly calculated that Dis defined by the system of inequalities: ℓi(a, b, c)≥δifor all i∈ {1,2,3}, m−(a+b+c)/2≥δ, m≥a, b, c. with δ1= 2 (N0−N) + λ′ 1+µ1+ν1, δ2= 2 (N0−N) + λ1+µ′ 1+ν1, δ3= 2 (N0−N) + λ1+µ1+ν′ 1, δ=N0−N+ (λ′ 1+µ′ 1+ν′ 1)/2, KRONECKER COEFFICIENTS 5 and N0=N0(b λ, bµ, bν). Inequalities (A3) are just the inequalities (21) with δiand δfor diand d. So, proving that the Kronecker coefficients gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(m−c|c)take only one value for (a, b, c, m)∈ D, would prove again Theorem 5.6. What we will get simply is that these Kronecker coefficients take at most two values. Note that Dis stable under addition of u1= (1,1,0,1), u2= (1,0,1,1), u3= (1,1,0,1) and u4= (0,0,0,1). The lattice spanned by these vectors is the set of all (a, b, c, m)∈Z4such that a+b+c≡0 (mod 2). Consider (a, b, c, m) and (a′, b′, c′, m′) in D, such that a+b+ c≡a′+b′+c′(mod 2). Their difference is a linear combination with integer coefficients of the vectors ui: (a, b, c, m)−(a′, b′, c′, m′) = 4 X i=1 xiui. Set (a′′, b′′, c′′, m′′) = (a, b, c, m) + X i:xi<0 (−xi)ui= (a′, b′, c′, m′) + X i:xi>0 xiui. Then gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(m−c|c)=gλ⊕(m′′ −a′′ |a′′ ),µ⊕(m′′ −b′′ |b′′),ν⊕(m′′ −c′′|c′′ ) and gλ⊕(m′−a′|a′),µ⊕(m′−b′|b′),ν⊕(m′−c′|c′)=gλ⊕(m′′ −a′′ |a′′ ),µ⊕(m′′ −b′′ |b′′),ν⊕(m′′ −c′′|c′′ ). This shows that gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(m−c|c)=gλ⊕(m′−a′|a′),µ⊕(m′−b′|b′),ν⊕(m′−c′|c′). We conclude that the Kronecker coefficients gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(m−c|c) for (a, b, c, m)∈ D take at most two values, one for each value of a+b+c (mod 2). To recover fully Theorem 5.6, it would now be enough to show that for some (a, b, c, m)∈ D such that a+b+c≡0 (mod 2), we have that the Kronecker coefficients gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(m−c|c) and gλ⊕(m−a+1|a+1),µ⊕(m−b+1|b+1),ν⊕(m−c+1|c+1) are equal. This would follow from Conjecture 5.10. Indeed, assume Conjecture 5.10 holds. Let M0(resp. M1) be the value taken by the Kronecker coefficients gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(m−c|c)for (a, b, c, m)∈ D with a+b+ceven (resp. odd). Choose arbitrarily (a, b, c, m)∈ D such that a+b+cis even. Set α=λ⊕(m−a|a), β=µ⊕(m−b|b) and γ=ν⊕(n−c|c). We have gα,β,γ ≤gα⊕(1|1),β⊕(1|1),γ⊕(1|1) ≤gα⊕(2|2),β⊕(2|2),γ⊕(2|2). But this means M0≤M1≤M0. Therefore M0=M1. 6 BRIAND, A. RATTAN, AND M. ROSAS B.2. Monotonicity conjecture. Let us recall the two conjectures of Section 5.3. Conjecture (Conjecture 5.10 restated).For any three partitions λ,µ and νof the same weight, gλ,µ,ν ≤gλ⊕(1|1),µ⊕(1|1),ν⊕(1|1). Conjecture (Conjecture 5.11 restated).For any three partitions λ,µ and νof the same weight, and any (a, b, c, m)fulfilling (24), gλ,µ,ν ≤gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(m−c|c). Here we prove that Conjecture 5.10 implies the more general Conjecture 5.11. The proof of this implication is based, once again, on the invariance of the Kronecker coefficients under conjugating two arguments, and on Murnaghan’s stability (see Section 3.1): for any three partitions λ,µ, νof the same weight, (A4) gλ,µ,ν ≤gλ⊕(1|0),µ⊕(1|0),ν⊕(1|0). Assume that Conjecture 5.10 holds. Let λ,µand νbe three partitions of the same weight. We have the identity gλ,µ,ν =gλ′,µ′,ν. Using (A4) we get gλ′,µ′,ν ≤gλ′⊕(1|0),µ′⊕(1|0),ν⊕(1|0).Conjugating again the arguments in position 1 and 2, we have that gλ′⊕(1|0),µ′⊕(1|0),ν⊕(1|0) is equal to gλ⊕(0|1),µ⊕(0|1),ν⊕(1|0).Therefore, gλ,µ,ν ≤gλ⊕(0|1),µ⊕(0|1),ν⊕(1|0).Likewise gλ,µ,ν ≤gλ⊕(0|1),µ⊕(1|0),ν⊕(0|1) and gλ,µ,ν ≤gλ⊕(1|0),µ⊕(0|1),ν⊕(0|1). Using these 3 identities, together with (A4), we see that gλ,µ,ν ≤ gλ⊕(m−a|a),µ⊕(m−b|b),ν⊕(n−c|c)for all (a, b, c, m) in the semigroup Sgenerated by (1,1,0,1), (1,0,1,1), (0,1,1,1) and (0,0,0,1). This semigroup Sis easily determined: it is the set of points (a, b, c, m)∈N4that fulfill (24) and a+b+c≡0 (mod 2). The set of integer points fulfilling (24) splits in two classes: Sin the one hand, and (1,1,1,2) +Sin the other hand. The implication follows now straightforwardly from this. Appendix C. The generating function for the coefficients Bα,β,γ C.1. Expression involving Schur functions indexed by hooks. It is proved in Theorem 7.3 that the Schur generating function for the coefficients Bα,β,γ is σ[XY Z + 2W]·3 4+1 4σ[(ε−1)W]−1 2χ[W] + χ[Y Z −X] The following result is stated in Remark 7.4, with no proof. KRONECKER COEFFICIENTS 7 Proposition C.1. Fix partitions α,β,γ. The coefficient Bα,β,γ in Theorem 6.1 is the coefficient of sα[X]sβ[Y]sγ[Z]in the expansion in the Schur basis of σ[XY Z +2W]·1−X aeven ,b (−1)bs(a|b)[W] + X a,b (−1)bs(a|b)[Y Z −X]!. Proof. From Cauchy’s Formula, (A5) σ[(ε−1)W] = σ[(1 −ε)(−W)] = X λ sλ[1 −ε]sλ[−W] = X λ sλ[1 −ε](−1)|λ|sλ′[W]. From [3, Ex. 7.43 with t= 1], sλ[1 −ε] is 1 if λis the empty partition, 2 if λis a hook and 0 otherwise. Therefore, σ[(ε−1)W] = 1 + 2 X a,b≥0 (−1)1+a+bs(a|b)[W]. Thus, 3 4+1 4σ[(ε−1)W] = 1 + 1 2X a,b (−1)1+a+bs(a|b)[W]. From [2, I.§3. Ex. 11 (2) with µ=∅], we have (A6) χ=X a,b (−1)bs(a|b), where χis the sum of the power sum symmetric functions as defined in Proposition 7.3. Therefore, 3 4+1 4σ[(ε−1)W]−1 2χ[W] = 1 + 1 2X a,b (−1)1+a+bs(a|b)[W]−1 2X a,b (−1)bs(a|b) = 1 −X aeven,b (−1)bs(a|b)[W]. Using again (A6) to rewrite χ[Y Z −X], we get the following formula for the generating function of the coefficients of B: σ[XY Z + 2W]·1−X aeven ,b (−1)bs(a|b)[W] + X a,b (−1)bs(a|b)[Y Z −X]!.  8 BRIAND, A. RATTAN, AND M. ROSAS C.2. Toolbox for other expressions. In order to write in other ways the generating function for the coefficients Bα,β,γ, the following formulas may be useful: σ[X]·χ[X] = X k k hk[X], σ[2X]·χ[X] = X λ:ℓ(λ)≤2 (λ1−λ2+ 1)(λ1+λ2) 2sλ[X]. They follow from the fact that σ[tX]χ[tX] is the derivative of σ[tX] (for the first one), and that σ[2tX]χ[2tX] is the derivative of σ[2tX]. Last, by Cauchy’s formula, σ[2tX] = X λ sλ[2]sλ[X]t|λ|, and sλ[2] = (λ1−λ2+ 1) if λhas at most two parts, and is equal to 0 otherwise. Appendix D. Table of coefficients Tables 1 and 2 display the coefficients gα,β,γ,Aα,β,γ,Bα,β,γ and Cα,β,γ for all partitions α,βand γwith weight at most 3. Note that gα,β,γ, Aα,β,γ and Cα,β,γ are invariant under permutation of their three indices. This is why the table gives their values only for α≥β≥γ, where the order ≥is the degree lexicographic ordering. The coefficients Bα,β,γ is only invariant under permuting its last two indices. These coefficients where calculated by series expansion of the generating series and using SAGE [4]. References [1] E. Briand, A. Rattan, and M. Rosas. On the growth of the Kronecker coefficients. ArXiv e-prints, 2016, 1607.02887. [2] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications. [3] Richard P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. [4] W. A. Stein et al. Sage Mathematics Software (Version 6.4.1). The Sage Development Team, 2014. http://www.sagemath.org. KRONECKER COEFFICIENTS 9 α β γ gα,β,γ Aα,β,γ Bα,β,γ Bβ,α,γ Bγ,α,β Cα,β,γ ∅ ∅ ∅ 1 1 1 1 1 1 (1) ∅ ∅ 2 2 0 1 1 0 (1) (1) ∅6 6 0 0 3 0 (1) (1) (1) 21 21 0 0 0 1 (2) ∅ ∅ 2 3 −2 1 1 1 (2) (1) ∅8 10 −7−2 3 0 (2) (1) (1) 34 40 −25 −5−5 0 (2) (2) ∅14 20 −14 −14 6 2 (2) (2) (1) 66 86 −57 −57 −14 0 (2) (2) (2) 145 203 −133 −133 −133 5 (2) (2) (1,1) 144 150 −84 −84 −84 −4 (2) (2) (1,1,1) 204 134 −54 −54 −121 0 (2) (1,1) ∅14 12 −8−8 4 −2 (2) (1,1) (1) 66 62 −33 −33 −2 0 (2) (1,1) (1,1) 145 131 −55 −55 −55 5 (2) (1,1,1) ∅16 6 −3−6 3 0 (2) (1,1,1) (1) 84 46 −19 −42 4 0 (2) (1,1,1) (1,1) 206 144 −45 −117 −45 0 (2) (1,1,1) (1,1,1) 326 240 −48 −168 −168 0 (1,1) ∅ ∅ 2 1 −1 0 0 −1 (1,1) (1) ∅8 6 −3 0 3 0 (1,1) (1) (1) 34 28 −13 1 1 0 (1,1) (1,1) ∅14 12 −4−4 8 2 (1,1) (1,1) (1) 66 54 −21 −21 6 0 (1,1) (1,1) (1,1) 144 110 −38 −38 −38 −4 (3) ∅ ∅ 2 4 −6 0 0 0 (3) (1) ∅8 14 −20 −6 1 0 (3) (1) (1) 38 59 −78 −19 −19 1 (3) (2) ∅16 30 −42 −27 3 0 (3) (2) (1) 84 138 −178 −109 −40 0 (3) (2) (2) 206 348 −435 −261 −261 0 (3) (2) (1,1) 204 258 −299 −170 −170 0 (3) (2) (1,1,1) 320 250 −250 −125 −250 0 (3) (1,1) ∅16 18 −24 −15 3 0 (3) (1,1) (1) 84 98 −118 −69 −20 0 (3) (1,1) (1,1) 206 220 −235 −125 −125 0 (3) (3) ∅22 50 −72 −72 3 0 (3) (3) (1) 122 240 −321 −321 −81 2 (3) (3) (2) 326 640 −820 −820 −500 0 (3) (3) (1,1) 320 478 −574 −574 −335 0 (3) (3) (3) 565 1243 −1597 −1597 −1597 5 (3) (3) (2,1) 1056 1632 −1888 −1888 −1888 0 (3) (3) (1,1,1) 544 506 −521 −521 −521 −4 Table 1. Table of the coefficients of the paper, for three indexing partitions with weight at most 3 (part 1 of 2).