The 144 symmetries of the Littlewood-Richardson coefficients of SL3

THE 144 SYMMETRIES OF THE LITTLEWOOD-RICHARDSON COEFFICIENTS OF SL3 EMMANUEL BRIAND AND MERCEDES ROSAS Abstract. We compute with SageMath the group of all linear symmetries for the Littlewood-Richardson associated to the representations of SL3. We find that there are 144 symmetries, more than the 12 symmetries known for the Littlewood-Richardson coefficients in general. 1. Introduction The Littlewood–Richardson coefficients cν λ,µ are among the most studied constants in geometry and representation theory. In geometry, they are the structure constants for the multiplication in the cohomology ring of the Grassmannian, in the basis of the Schubert cycles [1]. They can also be interpreted as cardinalities of the intersection of some triples of Schubert varieties [2]. In the representation theory of the general linear group, they are the multiplicities in the tensor product of irreducible representations, and also the dimensions of the subspace of invariants in the triple tensor products of irreducible representations [3]. In the representation theory of the symmetric groups, they are the multiplicities in the restrictions of the irreducible representations of a symmetric group Sm+nto its Young subgroup Sm×Sn. In the theory of symmetric functions, they are the structure constants for the ordinary multiplication in the basis of Schur functions. They receive also a number a combinatorial interpretations: they are known to count Littlewood-Richardson tableaux, hives, BZ triangles, . . . Together, these different interpretations of the Littlewood-Richardson coefficients make clear some symmetries they afford. For instance, the descriptions as structural constants for commutative products (of Schur functions, in the ring of symmetric functions; of Schubert cycles, in the cohomology ring of the Grassmannian) make clear the invariance cν λ,µ =cν µ,λ, under exchanging the two partitions λand µ. The descriptions as cardinalities of intersection of triples of Schubert varieties [4], or as dimensions of subspaces of invariants in triple tensor products [3] reveals the existence of a S3–symmetry of the Littlewood–Richardson Date: April 13, 2020. Both authors are partially supported by MTM2016-75024-P and FEDER, and Junta de Andalucia under grants P12-FQM-2696 and FQM-333. 1 2 EMMANUEL BRIAND AND MERCEDES ROSAS coefficients (that comprises the previous symmetry): (1) cν λ,µ =cν µ,λ =cλ µ,ν=cλ ν,µ =cµ ν,λ =cµ λ,ν where λ,µand νare obtained from the Young diagrams of λ,µ and νby “taking complements” in suitable rectangles (see Figure 1). λ λ Figure 1. New Young diagram obtained as complement of a Young diagram in a rectangle. When considering the Littlewood–Richardson coefficients associated to SLN. with λand µof length at most N−1and νof length at most N, one takes complements in rectangles of height N and: for (1), of length ν1; for (2), of length respectively λ1,µ1,λ1+µ1for λ,µand ν. Duality provides an additional symmetry (see [5]): (2) cν λ,µ =cν λ,µ. Together, these “known symmetries” generate a group of 12 symmetries, isomorphic to S2×S3. Note that it is not straightforward to understand these symmetries all together from the combinatorial descriptions of the Littlewood-Richardson coefficients (see [6, 4, 7]). Is it possible that the Littlewood–Richardson coefficients afford additional, unknown symmetries? To settle this question, we adopt an experimental approach. We consider the family of Littlewood-Richardson coefficients indexed by partitions λ,µand νwith restricted length. The combinatorial object they count can be interpreted as lattice points in a (rational convex polyhedral) cone. As a consequence (see [8]), cν λ,µ is a piecewise quasipolynomial function (actually polynomial, by an additional argument given again in [8]) of the parts of the partitions λ,µand ν. The domains of polynomiality are the maximal cones (“chamber”) of a fan (a complex of rational polyhedral convex cones, called the “Chamber Complex”). Explicit knowledge of such a description provides data making possible an exhaustive search of all linear symmetries. The Chamber Complex and the polynomial formulas for the Littlewood– Richardson coefficients cν λ,µ related to the representations of GL3(C) (i.e. where all three partitions have length at most 3) were explicitly given in [8]. We exploit and analyze these data with SageMath [9], and obtain: SYMMETRIES OF THE LITTLEWOOD–RICHARDSON COEFFICIENTS 3 Theorem 1. The group of linear symmetries of the Littlewood–Richardson coefficients cν λ,µ associated to SL3, has order 144. It is isomorphic to S2×(S3oS2), and acts transitively on the chambers of the Chamber Complex. It is generated by the well–known group of 12 symmetries, and the additional symmetry: c(ν1,ν2,ν3) (λ1,λ2),(µ1,µ2)=c(ν1,µ1,ν3) (λ1+µ1−ν2,λ2+µ1−ν2),(ν2,µ2) The calculations are presented in detail in a SageMath Notebook available online [10] and as an ancillary file. 2. The Littlewood–Richardson coefficients associated to SL3and their known symmetries 2.1. The Littlewood–Richardson coefficients associated to SL3. In this work we restrict our study to the family of Littlewood-Richardson coefficients related to SL3. These are the Littlewood–Richardson coefficients cν λ,µ such that λand µhave length at most 2and νhas length at most 3. Since cν λ,µ = 0 when |λ|+|µ| 6=|ν|, we have for any non–zero coefficient c(ν1,ν2,ν3) (λ1,λ2),(µ1,µ2)that (3) ν3=λ1+λ2+µ1+µ2−ν1−ν2. We thus consider the function Cdefined on Z6by C(λ1, λ2, µ1, µ2, ν1, ν2) = c(ν1,ν2,ν3) (λ1,λ2),(µ1,µ2) when λ1≥λ2≥0,µ1≥µ2≥0,ν1≥ν2≥ν3≥0(with ν3defined by (3)), and C(λ1, λ2, µ1, µ2, ν1, ν2) = 0 otherwise. Note that considering instead the Littlewood-Richardson coefficients associated to GL3(the c(ν1,ν2,ν3) (λ1,λ2,λ3),(µ1,µ2,µ3)) gives equivalent results (but hides symmetries). This is because of the invariance properties: c(ν1,ν2,ν3) (λ1,λ2,λ3),(µ1,µ2,µ3)=c(1+ν1,1+ν2,1+ν3) (1+λ1,1+λ2,1+λ3),(µ1,µ2,µ3) and c(ν1,ν2,ν3) (λ1,λ2,λ3),(µ1,µ2,µ3)=c(1+ν1,1+ν2,1+ν3) (λ1,λ2,λ3),(1+µ1,1+µ2,1+µ3) which imply c(ν1,ν2,ν3) (λ1,λ2,λ3),(µ1,µ2,µ3)=c(ν1−µ3−λ3,ν2−µ3−λ3,ν3−µ3−λ3) (λ1,λ2),(µ1,µ2). 2.2. Known symmetries. We now look for all linear symmetries of C, i.e. all invertible linear maps F:Z6→Z6such that C◦F=C. The group S3of symmetries (1) is generated by λ, µ, ν 7→ µ, λ, ν, and λ, µ, ν 7→ ν, µ, λ, that correspond in this context to: S: (λ1, λ2|µ1, µ2|ν1, ν2)7−→ (µ1, µ2|λ1, λ2|ν1, ν2), U: (λ1, λ2|µ1, µ2|ν1, ν2)7−→ (ν1−ν3, ν1−ν2|µ1, µ2|ν1, ν1−λ2). 4 EMMANUEL BRIAND AND MERCEDES ROSAS Chamber Generators Formula for C κ1b, c, d1, e2, d2, e11−λ2−µ2+ν1 κ2b, c, d1, g1, d2, g21 + ν2−ν3 κ3b, c, e2, g1, e1, g21 + λ1+µ1−ν1 κ4b, f, d1, e2, d2, e11 + ν1−ν2 κ5b, f, d1, g1, d2, g21 + λ2+µ2−ν3 κ6b, f, e2, g1, e1, g21−λ3−µ3+ν3 κ7b, c, d1, g1, d2, e11 + λ3+µ1−ν3 κ8b, c, d1, e2, d2, g21 + λ1+µ3−ν3 κ9b, c, d1, e2, e1, g21 + λ1−λ2 κ10 b, c, e2, g1, d2, e11 + µ1−µ2 κ11 b, c, d1, g1, e1, g21−λ2−µ3+ν2 κ12 b, c, e2, g1, d2, g21−λ3−µ2+ν2 κ13 b, f, d1, g1, d2, e11−λ1−µ3+ν1 κ14 b, f, d1, e2, d2, g21−λ3−µ1+ν1 κ15 b, f, d1, g1, e1, g21 + µ2−µ3 κ16 b, f, e2, g1, d2, g21 + λ2−λ3 κ17 b, f, d1, e2, e1, g21 + λ1+µ2−ν2 κ18 b, f, e2, g1, d2, e11 + λ2+µ1−ν2 Table 1. The Chamber Complex for C, from [8, Table 1]. Note that there is a typo in [8, Table 1]: there one should read ν1instead of ν3in the quasipolynomial formulas for chambers κ13 and κ14. The additional involution (2) corresponds to T: (λ1, λ2|µ1, µ2|ν1, ν2) 7−→ (λ1, λ1−λ2|µ1, µ1−µ2|λ1+µ1−ν3, λ1+µ1−ν2) where, again, ν3is given by (3). Rassart’s Chamber Complex and polynomial formulas for the GL3– Littlewood-Richardson coefficients restrict to a Chamber Complex and polynomial formulas for the function C(set λ3=µ3= 0). The rays of this Chamber Complex are given in Table 2, and the chambers and polynomial formulas are shown in Table 1. 3. Computation of the symmetries In this section, we prove Theorem 1. 3.1. Reduction to the symmetries of the Chamber Complex. In order to find all symmetries of the function C, we look for symmetries of simpler objects: first we check that any symmetry of Cmust be a SYMMETRIES OF THE LITTLEWOOD–RICHARDSON COEFFICIENTS 5 b= (2,1,2,1,3,2) c= (1,1,1,1,2,1) f= (1,0,1,0,1,1) d1= (1,1,1,0,1,1) e1= (1,1,0,0,1,1) g1= (1,0,0,0,1,0) d2= (1,0,1,1,1,1) e2= (0,0,1,1,1,1) g2= (0,0,1,0,1,0) Table 2. The minimal generators for the rays of the Chamber Complex for C. symmetry of the chamber complex. Afterwards, we will show that such a symmetry must be also a symmetry of an even simpler object: the set of the ray generators of the Chamber Complex. Lemma 1. Any linear symmetry of Cis necessarily a linear symmetry of the chamber complex, i.e. an invertible linear map from Z6to itself that permutes the cells of the chamber complex. Proof. We will call Cthe support of the Chamber Complex, i.e. the cone that is the union of all chambers. Let Fbe a symmetry of C. On any chamber κof the chamber complex, the function Ccoincides with a polynomial function Pκ. Let κbe a chamber of the chamber complex. Since C◦F=C, we have that C◦Falso coincides with Pκon κ. Therefore Ccoincides with Pκ◦F−1on F(κ). The cone F(κ)can’t meet the exterior of the cone C. Indeed, if F(κ)would meet the exterior of C, the intersection would contain a full-dimensional cone of R6. But two polynomial functions that coincide on the integer points of a full-dimensional cone must be equal. We would have Pκ◦F−1= 0, and thus Pκ= 0, which is false. Therefore F(κ)⊂ C. Again, it is impossible that F(κ)meet two chambers τ1and τ2full–dimensionally, because this would imply Pτ1=Pτ2, and there is no such coincidence in the list of formulas for C. As a consequence, there exists one cell τsuch that F(κ)⊂τ. Applying the same reasoning to F−1, we see that we must have also F−1(τ)⊂κ. As a conclusion, F(κ) = τ. Therefore, Fpermutes the chambers of the chamber complex. It follows that Fis a symmetry of the chamber complex.  3.2. Reduction to the Symmetries of the rays. After Lemma 1, any linear symmetry of Cis also a linear symmetry of the chamber complex. Obviously, any linear symmetry of the chamber complex induces also a linear symmetry of the set of its ray generators R= {b, c, f, d1, d2, e1, e2, g1, g2}(i.e. a permutation of Rinduced by a linear invertible map from Z6to itself). We will denote with S(R)the group of linear symmetries of R. We compute with SageMath the group S(R), thanks to the following lemma. Firstly, note that the group of symmetries of a set of vectors R={v1, v2, . . . , vn}embeds in the symmetric group Sn. 6 EMMANUEL BRIAND AND MERCEDES ROSAS g 1 c b d 1 d 2 e 1 f g 2 e 2 Figure 2. The edge–colored graph H(R). Its automorphism group is the group of linear symmetries of the ray generators of the chamber complex. Lemma 2 ([11, Proposition 3]).Let R={v1, v2, . . . , vn}be a set of vectors of Zm, spanning Zm. Then the linear symmetries of Rare the automorphisms of the edge–colored complete graph H(R)whose vertices are the vi, with the edge vi−vjcolored by the entry (i, j)of the matrix Q=Vt(V V t)−1V, where Vis the matrix of order m×nwhose columns are the vi. In the case under consideration, the edge–colored graph H(R)is shown in Figure 2, and its automorphism group is readily obtained: S(R)is the direct product of the group S2of permutations of {c, f}and of the wreath product S3oS2of the permutations of {d1, e2, g1, d2, e1, g2} that stabilize or swap the subsets {d1, e2, g1}and {d2, e1, g2}. In particular, S(R)is generated by v= (c, f),x= (e1, g2)and y= (e1, d2) that permute {e1, g2, d2}, and s= (d1, d2)(e1, e2)(g1, g2)that swaps {d1, e2, g1}and {d1, e1, g2}. Note that sis precisely the automorphism of Rinduced by the known symmetry Sof the Littlewood-Richardson coefficients. Let us consider the automorphisms t, u of the rays induced by the other two known symmetries T,U. It is easily calculated that t= (d1, d2)(e1, g1)(e2, g2)(c, f), u = (d1, g2)(d2, e2)(e1, g1). Again with SageMath, we check that s,t,uand xalready generate S(R). More precisely, we get that v=txsx and y=usxsu, which is easily checked by hand. 3.3. The symmetries of the rays are also symmetries of the Littlewood–Richardson coefficients. Note that any automorphism of the rays lifts uniquely to a linear automorphism of R6. Let Xbe the lifting of x. Let Gbe the group of all liftings of the elements of S(R). SYMMETRIES OF THE LITTLEWOOD–RICHARDSON COEFFICIENTS 7 After Lemmas 1 and 2, the group Gcontains all symmetries of C. We will check by explicit calculation that Gis actually equal to the group of symmetries of C. Since Gis generated by S,T,Uand X, and that S,Tand Uare known to be symmetries of C, it is enough to check that Xis a symmetry of C. To check that Xis a symmetry of Cwe proceed as follows: for each chamber κ, with generators w1, w2, w3, w4, w5, w6, we calculate x(w1),x(w2), . . . (This is immediate since xjust swaps e1and g2and leaves all other ray generators fixed). We check that they are the generators of a chamber g(κ)of the Chamber Complex. Then we check by inspection that gis indeed a permutation of the chambers. Last, we check that for each κ, the polynomials Pg(κ)and Pκ◦X−1are equal. The latter calculation is performed within SageMath, and gives the expected result: Xis indeed a symmetry of C. Finally, one calculates that Xis given by (λ1, λ2|µ1, µ2|ν1, ν2)7−→ (λ1+µ1−ν2, λ2+µ1−ν2|ν2, µ2|ν1, µ1). 3.4. Transitivity of the action on the chambers. By direct inspection of Table 1, one can check that the rays of each chamber are: b, one of {c, f}, two of {d1, e2, g1}and two of {d2, e1, g2}. This proves that Gpermutes transitively the chambers, since Gis exactly the stabilizer of ({c, f},{{d1, e2, g1},{d2, e1, g2}}). 4. Final remarks 4.1. Littlewood–Richardson coefficients associated to GL3.If we consider the Littlewood–Richardson coefficients associated to GL3, instead of SL3, we get one more generator for the group of symmetries, yielding in total 144 ×2 = 288 symmetries. The additional generator sends (λ1, λ2, λ3|µ1, µ2, µ3|ν1, ν2, ν3)to (λ1−m, λ2−m, λ3−m|µ1+m, µ2+m, µ3+m|ν1, ν2, ν3), where m=λ3−µ3. In Rassart’s description [8] of the chamber complex for the Littlewood-Richardson coefficients associated to GL3, this generator swaps the additional rays a1and a2while fixing all other rays. 4.2. The case of SLNfor N≥4.One finds that the linear symmetries of the Littlewood-Richardson coefficients associated to SLN, for Nin {4,5,6,7}, are only the 12 known symmetries. Indeed, the group of symmetries of these Littlewood–Richardson coefficients embeds in the group of linear symmetries of the ray generators of their support. But one computes that this group has only 12 elements[12]. 8 EMMANUEL BRIAND AND MERCEDES ROSAS 5. Perspectives There are two contributions in this work: the results (144 linear symmetries for the Littlewood-Richardson coefficients associated to the representations of SL3, a much bigger number than expected); and the method (embedding the group of the linear symmetries in the group of symmetries of a chamber complex, and the group of symmetries of its rays). This method may be applied to the study of the symmetries of other families of Littlewood-Richardson coefficients, and even more, to families of other representation–theoretic structural coefficients (such as Kostka coefficients, Kronecker coefficients, plethysm coefficients) as long as we have an explicit description of their piecewise quasipolynomial formulas. Explicit descriptions of piecewise quasipolynomial for such structural coefficients appear in the literature. For example, for the plethysm coefficients in the Schur expansion of sµ[sk]with µany fixed partition of 3,4or 5[13], for the Kronecker coefficients gλ,µ,ν with λand µof length ≤2[14] or with λ,µ,νof length ≤3[15]. References [1] William Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. [2] Sara Billey and Ravi Vakil. Intersections of Schubert varieties and other permutation array schemes. In Algorithms in algebraic geometry, volume 146 of IMA Vol. Math. Appl., pages 21–54. Springer, New York, 2008. [3] A. D. Berenstein and A. V. Zelevinsky. Triple multiplicities for sl (r+ 1) and the spectrum of the exterior algebra of the adjoint representation. J. Algebraic Combin., 1(1):7–22, 1992. [4] Hugh Thomas and Alexander Yong. An S3-symmetric Littlewood-Richardson rule. Math. Res. Lett., 15(5):1027–1037, 2008. 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Computation of dilated Kronecker coefficients. Journal of Symbolic Computation, 84:113 – 146, 2018. Emmanuel Briand, Departamento Matemática Aplicada I, Universidad de Sevilla E-mail address:[email protected] Mercedes Rosas, Departamento de Álgebra, Universidad de Sevilla E-mail address:[email protected]