Motivated by the symmetric version of matrix multiplication we study the
plethysm $S^k(\mathfrak{sl}_n)$ of the adjoint representation $\mathfrak{sl}_n$
of the Lie group $SL_n$. In particular, we describe the decomposition of this
representation into irreducible components for $k=3$, and find highest weight
vectors for all irreducible components. Relations to fast matrix
multiplication, in particular the Coppersmith-Winograd tensor are presented.Comment: 5 page

Seynnaeve, Tim

English

arXiv

Tim Seynnaeve

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 356 (2018) 52–55Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comLie algebras/Computer sciencePlethysm and fast matrix multiplicationPléthysme et multiplication rapide des matricesTim SeynnaeveMax Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germanya r t i c l e i n f o a b s t r a c tArticle history:Received 10 October 2017Accepted after revision 20 November 2017Available online 6 December 2017Presented by Michèle VergneMotivated by the symmetric version of matrix multiplication we study the plethysm Sk(sln) of the adjoint representation sln of the Lie group SLn . In particular, we describe the decomposition of this representation into irreducible components for k = 3, and find highest-weight vectors for all irreducible components. Relations to fast matrix multiplication, in particular the Coppersmith–Winograd tensor, are presented.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éMotivés par la version symétrique de la multiplication des matrices, nous étudions le pléthysme Sk(sln) de la représentation adjointe sln du groupe de Lie SLn . En particulier, pour k = 3, nous décrivons la décomposition de cette représentation en composantes irréductibles, et nous trouvons les vecteurs de plus grand poids pour toutes ces dernières. Nous présentons les liens avec la multipliction rapide des matrices, notamment le tenseur de Coppersmith–Winograd.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionIn 1969, Strassen [18] presented his celebrated algorithm for matrix multiplication breaking for the first time the naive complexity bound of n3 for n × n matrices. Since then, the complexity of the optimal matrix multiplication algorithm is one of the central problems in computer science. In terms of algebra, we know that this question is equivalent to estimating the rank or the border rank of a specific tensor Mn,n,n ∈ Cn2 ⊗Cn2 ⊗Cn2 [1,8,9]. The current best lower and upper bounds are presented in [10,12–14,20].We recall that the constant ω is defined as the smallest number such that, for any ε > 0, the multiplication of n × nmatrices can be performed in time O (nω+ε). Further, recall that the Waring rank of a homogeneous polynomial P of degree d is the smallest number r of linear forms l1, . . . , lr such that P = ∑ri=1 ldi . Recently, Chiantini et al. [2] provided another equivalent interpretation of ω in terms of Waring (border) rank. Namely, let S Mn be a cubic in S3(sl∗n) given by E-mail address: tim.seynnaeve@mis.mpg.de.https://doi.org/10.1016/j.crma.2017.11.0121631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.T. Seynnaeve / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 52–55 53S Mn(A) = tr(A3). Then ω is the smallest number such that, for any ε > 0, the Waring rank (or Waring border rank) of S Mnis O (nω+ε). This observation was the initial motivation for our study of the plethysm S3(sln).The computations of plethysm are in general very hard, and explicit formulas are known only in specific cases [15]. For example, for a symmetric power S3(Sk), the decomposition was classically computed already in [17,19], but S4(Sk) and S5(Sk) were only recently explicitly obtained in [7]. As symmetric powers (together with exterior powers) are the simplest Schur functors, one could expect that the respective formulas for Sd(sln) are harder. In principle, one could use the methods in [6,7,16] to decompose this plethysm, but this requires a lot of nontrivial character manipulations. Instead, we present a very easy proof of explicit decomposition based on the Cauchy formula and the Littlewood–Richardson rule in Theorem 1. In fact, using our method, one can inductively obtain the formula for Sk(sln) for any k.While matrix multiplication is represented by the (unique) invariant in S3(sln) the aim of this article is to understand the other highest-weight vectors. A precise description of them is presented in Section 3. We plan to undertake a detailed study of ranks and border ranks of other highest-weight vectors in future work. Here we present just the first two nontrivial instances. It turns out that two of the highest-weight vectors are (isomorphic to) the (four and five dimensional) variants of the Coppersmith–Winograd tensor [4]. We recall that the best upper bounds for rank and border rank are based on a beautiful technique by Coppersmith and Winograd applied to a specific tensor T [20]. While T is extremely efficient for this technique, it is completely unclear which properties of T make it so useful and how to identify potentially better tensors. In fact, there are whole programs, see, e.g., [3], aimed at finding tensors similar to, but better than the Coppersmith–Winograd one. We hope that other highest-weight vectors will also reveal their importance.2. The plethysmIn this section, we describe a general procedure to decompose Sk(gln) and Sk(sln) into irreducibles. Recall that the irreducible representations of S Ln are precisely the representations Sλ(Cn), where λ = [λ1, . . . , λn−1] is a partition of length at most n − 1, and Sλ is the Schur functor associated with the partition λ (consult, for example, [5]).Theorem 1. For n ∈N, it holds thatSk(gln) ∼=⊕λk⊕νNνλλSν(Cn) (1)as S Ln-representations. Here the second summation is over all partitions ν of length at most n − 1, Nνλμ are the Littlewood–Richardson coefficients, and λ = [λ1 − λn−1, . . . , λ1 − λ2].Proof. Note that gln ∼= (Cn) ⊗ (Cn)∗ as S Ln-representations. SoSk(gln) ∼= Sk((Cn) ⊗ (Cn)∗) ∼= ⊕λkSλ(Cn) ⊗ Sλ(Cn)∗∼=⊕λkSλ(Cn) ⊗ Sλ(Cn) ∼=⊕λk⊕νNνλλSν(Cn).The second isomorphism holds by Cauchy’s formula; for the third one, see, for example, [5, 15.50]; the fourth isomorphism is the Littlewood–Richardson rule. To compute the decomposition of Sk(sln), we simply note thatSk(gln) ∼=Sk(sln ⊕C) ∼= C⊕k⊕i=1Si(sln).This allows us to compute the decomposition of Sk(sln) inductively.As a corollary we present an explicit decomposition in the case k = 3. Computing the Littlewood–Richardson coefficients in (1) gives us the decomposition of S3(gln) (resp. S3(sln)) into irreducibles. We present these in Table 1: the first column lists the highest weights λ of the occurring irreducible representations Sλ(Cn). To be more precise, the first column actually shows the highest weights when we view S3(gln) (resp. S3(sln)) as a GLn-representation. (Recall that weights of GLn are n-tuples [λ1, . . . , λn] ∈ Zn with λ1 ≥ . . . ≥ λn . The corresponding S Ln-weight is then [λ1 − λn, . . . , λn−1 − λn].) The second and third column list the multiplicities of the irreducibles in S3(gln) resp. S3(sln). We also list the dimensions of the occurring irreducible representations Sλ(Cn), as well as the dimensions of the projective homogeneous varieties contained in P(Sλ(Cn)) (see Subsection 2.1).54 T. Seynnaeve / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 52–55Table 1Irreducible components of S3(gln) and S3(sln).Highest weight S3(gln) S3(sln) Dimension Variety[0, . . . ,0] 3 1 1 0[1,0, . . . ,0,−1] 4 2 n2 − 1 2n − 3[2,0, . . . ,0,−2] 2 1 (n−1)n2(n+3)4 2n − 3[3,0, . . . ,0,−3] 1 1 (n−1)n2(n+1)2(n+5)36 2n − 3[1,1,0, . . . ,0,−1,−1] 2 1 (n−3)n2(n+1)4 4n − 12[2,0, . . . ,0,−1,−1] 1 1 (n−2)(n−1)(n+1)(n+2)4 3n − 7[1,1,0, . . . ,0,−2] 1 1 (n−2)(n−1)(n+1)(n+2)4 3n − 7[2,1,0, . . . ,0,−1,−2] 1 1 (n−3)(n−1)2(n+1)2(n+3)9 4n − 10[1,1,1,0, . . . ,0,−1,−1,−1] 1 1 (n−5)(n−1)2n2(n+1)36 6n − 27Table 2Highest-weight vectors of S3(gln).Weight Highest-weight vector[0, . . . ,0] I I I[0, . . . ,0] ∑i, j I Ei, j E j,i[0, . . . ,0] ∑i, j,k Ei, j E j,k Ek,i[1,0, . . . ,0,−1] I I E1,n[1,0, . . . ,0,−1] ∑i I E1,i Ei,n[1,0, . . . ,0,−1] ∑i, j E1,n Ei, j E j,i[1,0, . . . ,0,−1] ∑i, j E1,i Ei, j E j,n[2,0, . . . ,0,−2] I E1,n E1,n[2,0, . . . ,0,−2] ∑i E1,n E1,i Ei,n[1,1,0, . . . ,0,−2] ∑i E1,n E2,i Ei,n − E2,n E1,i Ei,n[2,0, . . . ,0,−1,−1] ∑i E1,n E1,i Ei,n−1 − E1,n−1 E1,i Ei,n[1,1,0, . . . ,0,−1,−1] I E1,n E2,n−1 − I E1,n−1 E2,n[1,1,0, . . . ,0,−1,−1] ∑i E1,n E2,i Ei,n−1 − E2,n E1,i Ei,n−1 −E1,n−1 E2,i Ei,n + E2,n−1 E1,i Ei,n[3,0, . . . ,0,−3] E1,n E1,n E1,n[2,1,0, . . . ,0,−1,−2] E1,n E1,n−1 E2,n − E1,n E1,n E2,n−1[1,1,1,0, . . . ,0,−1,−1,−1] ∑σ∈S3 sgnσ Eσ (1),n Eσ (2),n−1 Eσ (3),n−22.1. Homogeneous varietiesLet V be an irreducible representation of a semisimple Lie group G. Then PV has a unique closed G-orbit X , which is the orbit of the highest-weight vector in PV under the action of G . The projective variety X is isomorphic to G/P , where P is a parabolic subgroup. We call these varieties homogeneous varieties or partial flag varieties.In our case, G = S Ln , we can compute the dimension of X in the following way. Consider the Dynkin diagram of sln , which consists of n − 1 dots marked from 1 to n − 1, and the Young diagram λ associated with the representation V . For every j ∈ {1, . . . , n − 1}, if the Young diagram has at least one column of length j, we remove the dot j from the Dynkin diagram. After removing these dots, the Dynkin diagram splits in connected components of size ki . The dimension of our variety X is then given by12(n2 − n −∑i(k2i + ki)).This gives us the last column of Table 1.3. Highest-weight vectorsWe now describe highest-weight vectors for all irreducible components of S3(gln). We write Ei, j ∈ gln for the n × nmatrix with as only nonzero entry a 1 on position (i, j). Note that the vector Ei, j Ei′, j′ Ei′′, j′′ ∈ S3(gln) has weight ei + ei′ +ei′′ − e j − e j′ − e j′′ , where ei is the weight [0, . . . , 1, . . . , 0] with a 1 on the i-th position. Furthermore, to check that a weight vector v in some representation V of S Ln is a highest-weight vector, it suffices to view V as a representation of the Lie algebra sln and check that every matrix Ei,i+1 acts by zero. Using this, it is straightforward to check that the vectors listed in Table 2 are indeed highest-weight vectors.T. Seynnaeve / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 52–55 553.1. Waring rank and border Waring rankAs explained in the introduction (see also [2]), estimating the (border) Waring rank of the highest-weight vector ∑i, j,k Ei, j E j,k Ek,i is equivalent to determining the exponent ω of matrix multiplication. We will analyze the (border) Waring ranks of other highest-weight vectors. We start with the following surprising observation.Observation 1. Every highest-weight vector with weight different than [0, . . . , 0] has Waring rank O (n2). Furthermore, the weight space of [0, . . . , 0] is 3-dimensional: it has a basis consisting of two vectors of Waring rank O (n2), and the vector ∑i, j,k Ei, j E j,k Ek,i .Proof. Every highest-weight vector in Table 2, except ∑i, j,k Ei, j E j,k Ek,i , is a sum of at most n2 monomials, and every degree 3 monomial has Waring rank at most 4. We now study the highest-weight vectors I E1,n E2,n−1 − I E1,n−1 E2,n and E1,n E1,n−1 E2,n − E1,n E1,n E2,n−1, which we will rewrite as xyz − xwt and xzt − x2 y.Proposition 1. The cubics f1 = xyz − xwt and f2 = xzt − x2 y are two variants of the Coppersmith–Winograd tensor. Their ranks and border ranks (equal to Waring rank resp. Waring border rank) are given by rk( f1) = 9, rk( f1) = 6, rk( f2) = 7, rk( f2) = 4.Proof. After the change of basis x = x0, y = x1 + ix2, z = x1 − ix2, w = x3 + ix4, t = −x3 + ix4, our cubic f1 becomes x0x21 + x0x22 + x0x23 + x0x24, which is precisely the Coppersmith–Winograd tensor T4,C W (here we use the notation from [11, Section 7]). For f2, we can do a similar change of basis, or alternatively we can use the geometric characterization of the Coppersmith–Winograd tensor form [11, Theorem 7.4]. We find that f2 is isomorphic to T̃2,C W .The ranks and border ranks of Coppersmith–Winograd tensors are known: consult for example [4] for the border ranks and [11, Proposition 7.1] for the ranks. Remark 1. The highest-weight vectors that are monomials are easily understood: I I I and E1,n E1,n E1,n trivially have Waring rank equal to 1; I I E1,n and I E1,n E1,n agree with the Coppersmith–Winograd tensor T1,C W , hence they have Waring rank 3 and border Waring rank 2.AcknowledgementThe author would like to thank his advisor, Mateusz Michałek, for the many helpful comments and discussions.References[1] P. Bürgisser, M. Clausen, M.A. Shokrollahi, Algebraic Complexity Theory, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 315, Springer-Verlag, Berlin, 1997. With the collaboration of Thomas Lickteig.[2] L. Chiantini, J.D. Hauenstein, C. Ikenmeyer, J.M. Landsberg, G. Ottaviani, Polynomials and the exponent of matrix multiplication, arXiv preprint, arXiv:1706.05074, 2017.[3] H. Cohn, C. Umans, A group-theoretic approach to fast matrix multiplication, in: Foundations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on, IEEE, 2003, pp. 438–449.[4] D. Coppersmith, S. Winograd, Matrix multiplication via arithmetic progressions, J. Symb. Comput. 9 (3) (1990) 251–280.[5] W. Fulton, J. Harris, Representation Theory: A First Course, vol. 129, Springer Science & Business Media, 2013.[6] R. Howe, (GLn, GLm)-duality and symmetric plethysm, Proc. Indian Acad. Sci. Math. Sci. 97 (1–3) (1988) 85–109, 1987.[7] T. Kahle, M. Michałek, Plethysm and lattice point counting, Found. Comput. Math. 16 (5) (2016) 1241–1261.[8] J.M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, vol. 128, American Mathematical Society, Providence, RI, 2012.[9] J.M. Landsberg, Geometry and Complexity Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2017.[10] J.M. Landsberg, M. Michałek, On the geometry of border rank decompositions for matrix multiplication and other tensors with symmetry, SIAM J. Appl. Algebra Geom. 1 (1) (2017) 2–19.[11] J.M. Landsberg, M. Michałek, Abelian tensors, J. Math. Pures Appl. 108 (3) (2017) 333–371.[12] J.M. Landsberg, M. Michałek, A lower bound for the border rank of matrix multiplication, Int. Math. Res. Not. (2017), rnx025.[13] J.M. Landsberg, G. Ottaviani, New lower bounds for the border rank of matrix multiplication, Theory Comput. 11 (2015) 285–298.[14] F. Le Gall, Powers of tensors and fast matrix multiplication, in: ISSAC 2014—Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2014, pp. 296–303.[15] I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, 1998.[16] L. Manivel, M. Michałek, Secants of minuscule and cominuscule minimal orbits, Linear Algebra Appl. 481 (2015) 288–312.[17] S.P.O. Plunkett, On the plethysm of S-functions, Can. J. Math. 24 (1972) 541–552.[18] V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (4) (1969) 354–356.[19] R.M. Thrall, On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math. 64 (1942) 371–388.[20] V.V. Williams, Multiplying matrices faster than Coppersmith–Winograd [extended abstract], in: STOC’12—Proceedings of the 2012 ACM Symposium on Theory of Computing, ACM, New York, 2012, pp. 887–898.

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