Plethysm and Kronecker Products

Transcription

1 Plethysm and Kronecker Products p. 1 Plethysm and Kronecker Products Richard P. Stanley University of Miami and M.I.T.

2 Plethysm and Kronecker Products p. 2 Intended audience Talk aimed at those with a general knowledge of symmetric functions but no specialized knowledge of plethysm and Kronecker product.

3 Plethysm and Kronecker Products p. 3 Introduction plethysm and Kronecker product: the two most important operations in the theory of symmetric functions that are not understood combinatorially Plethysm due to D. E. Littlewood Internal product of symmetric functions: the symmetric function operation corresponding to Kronecker product, due to J. H. Redfield and D. E. Littlewood We will give a survey of their history and basic properties.

4 Plethysm and Kronecker Products p. 4 Dudley Ernest Littlewood 7 September October 1979 tutor at Trinity College: J. E. Littlewood (no relation) : chair of mathematics at University College of North Wales, Bangor

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6 Plethysm and Kronecker Products p. 6 Plethysm introduced by D. E. Littlewood in 1936 name suggested by M. L. Clark in 1944 after Greek plethysmos (πληθυσµóς) for multiplication

7 Plethysm and Kronecker Products p. 7 Polynomial representations V, W : finite-dimensional vector spaces/c polynomial representation ϕ: GL(V) GL(W)(example) : [ ] a 2 2ab b 2 a b ϕ = ac ad+bc bd. c d c 2 2cd d 2

8 Plethysm and Kronecker Products p. 8 Definition of plethysm V, W, X: vector spaces/c of dimensions m, n, p ϕ: GL(V) GL(W): polynomial representation with character f Λ n, so trϕ(a) = f(x 1,...,x m ) if A has eigenvalues x 1,...,x m ψ: GL(W) GL(X): polynomial representation with character g Λ m

9 Plethysm and Kronecker Products p. 8 Definition of plethysm V, W, X: vector spaces/c of dimensions m, n, p ϕ: GL(V) GL(W): polynomial representation with character f Λ n, so trϕ(a) = f(x 1,...,x m ) if A has eigenvalues x 1,...,x m ψ: GL(W) GL(X): polynomial representation with character g Λ m ψϕ: GL(V) GL(X) is a polynomial representation. Let g[f] (or g f) denote its character, the plethysm of f and g.

10 Plethysm and Kronecker Products p. 8 Definition of plethysm V, W, X: vector spaces/c of dimensions m, n, p ϕ: GL(V) GL(W): polynomial representation with character f Λ n, so trϕ(a) = f(x 1,...,x m ) if A has eigenvalues x 1,...,x m ψ: GL(W) GL(X): polynomial representation with character g Λ m ψϕ: GL(V) GL(X) is a polynomial representation. Let g[f] (or g f) denote its character, the plethysm of f and g. if f = u I u (I = set of monomials) then g[f] = g(u: u I).

11 Plethysm and Kronecker Products p. 9 Extension of defintions Can extend definition of g[f] to any symmetric functions f, g using f[p n ] = p n [f] = f(x n 1,x n 2,...) (af +bg)[h] = af[h]+bg[h], a,b Q (fg)[h] = f[h] g[h], where p n = x n 1 +xn 2 +.

12 Plethysm and Kronecker Products p. 10 Examples Note. Can let m,n and define g[f] in infinitely many variables x 1,x 2,... (stabilization). h 2 = i j x ix j, so f[h 2 ] = f(x 2 1,x 1 x 2,x 1 x 3,...).

13 Plethysm and Kronecker Products p. 10 Examples Note. Can let m,n and define g[f] in infinitely many variables x 1,x 2,... (stabilization). h 2 = i j x ix j, so f[h 2 ] = f(x 2 1,x 1 x 2,x 1 x 3,...). By RSK, (1 x i x j ) 1 = s 2λ. Since i j λ (1 x i ) 1 = 1+h 1 +h 2 +, we get i h n [h 2 ] = λ n s 2λ, i.e., the character of S n (S 2 V).

14 Plethysm and Kronecker Products p. 11 Schur positivity ϕ: GL(V) GL(W): polynomial representation with character f Λ n ψ: GL(W) GL(X): polynomial representation with character g Λ m g[f]: character of ψ ϕ

15 Plethysm and Kronecker Products p. 11 Schur positivity ϕ: GL(V) GL(W): polynomial representation with character f Λ n ψ: GL(W) GL(X): polynomial representation with character g Λ m g[f]: character of ψ ϕ Theorem. If f,g are any Schur-positive symmetric functions, then g[f] is Schur-positive.

16 Plethysm and Kronecker Products p. 11 Schur positivity ϕ: GL(V) GL(W): polynomial representation with character f Λ n ψ: GL(W) GL(X): polynomial representation with character g Λ m g[f]: character of ψ ϕ Theorem. If f,g are any Schur-positive symmetric functions, then g[f] is Schur-positive. No combinatorial proof known, even for f = h m, g = h n.

17 Plethysm and Kronecker Products p. 12 Schur-Weyl duality for plethysm N(S m k ): normalizer of Sm k in S km, the wreath product S k S m, or order k! m m! ch(ψ): the Frobenius characteristic of the class function ψ of S n, i.e., ch(ψ) = λ n ψ,χ λ s λ.

18 Plethysm and Kronecker Products p. 12 Schur-Weyl duality for plethysm N(S m k ): normalizer of Sm k in S km, the wreath product S k S m, or order k! m m! ch(ψ): the Frobenius characteristic of the class function ψ of S n, i.e., ch(ψ) = λ n ψ,χ λ s λ. Theorem (Specht). Special case: ( ) ch 1 S km N(S m k ) = h m [h k ]

19 Plethysm and Kronecker Products p. 13 Main open problem Find a combinatorial interpretation of s λ [s µ ],s ν, especially the case h m [h n ],s ν. E.g., h 2 [h n ] = n/2 k=0 s 2(n k),2k. h 3 [h n ] known, but quickly gets more complicated.

20 Plethysm and Kronecker Products p. 14 Plethystic inverses Note p 1 = s 1 = x i and g[s 1 ] = s 1 [g] = g. We say that f and g are plethystic inverses, denoted f = g [ 1], if f[g] = g[f] = s 1. Note. f[g] = s 1 g[f] = s 1.

21 Plethysm and Kronecker Products p. 15 Lyndon symmetric function L n C n : cyclic subgroup of S n generated by (1,2,...,n) ζ: character of C n defined by ζ(1,2,...,n) = e 2πi/n Lyndon symmetric function: L n = 1 n d n µ(d)p n/d d = chind S n C n e 2πi/n

22 Plethysm and Kronecker Products p. 16 Cadogan s theorem f = e 1 e 2 +e 3 e 4 + g = L 1 +L 2 +L 3 + Theorem (Cadogan, 1971). g = f [ 1]

23 Plethysm and Kronecker Products p. 17 Lyndon basis Extend L n to a basis {L λ } for the ring Λ of symmetric functions: Let m,k 1, and k m = (k,k,...,k) (m times). Define L km = h m [L k ] L 1 m 1,2 m 2,... = L 1 m 1 L 2 m 2. Equivalently, for fixed m, k 0 L km t k = exp n 1 1 n L n(p i p mi )t i.

24 Plethysm and Kronecker Products p. 18 Cycle type Fix n 1. Let S [n 1]. F S : Gessel fundamental quasisymmetric function Example. n = 6,S = {1,3,4}: F S = x i1 x i6. 1 i 1 <i 2 i 3 <i 4 <i 5 i 6 Theorem (Gessel-Reutenauer, 1993). We have F D(w) = L λ. w S n type(w)=λ

25 Plethysm and Kronecker Products p. 19 An example Example. λ = (2,2): w D(w) , ,2,3

26 Plethysm and Kronecker Products p. 19 An example Example. λ = (2,2): w D(w) , ,2,3 L (2,2) = s (2,2) +s (1,1,1,1) = (F 1,3 +F 2 )+F 1,2,3

27 Plethysm and Kronecker Products p. 20 Free Lie algebras A: the alphabet x 1,...,x n C A : free associative algebra over C generated by A L[A]: smallest subalgebra of C A containing x 1,...,x n and closed under the Lie bracket [u,v] = uv vu (free Lie algebra)

28 Plethysm and Kronecker Products p. 21 Lie n Lie n : multilinear subspace of C A (degree one in each x i ) basis: [x 1,[x w(2),[x w(3),[ ] ]], w S [2,n] dimlie n = (n 1)!

29 Plethysm and Kronecker Products p. 21 Lie n Lie n : multilinear subspace of C A (degree one in each x i ) basis: [x 1,[x w(2),[x w(3),[ ] ]], w S [2,n] dimlie n = (n 1)! Theorem (Brandt, 1944). Action of S n on Lie n has Frobenius characteristic L n.

30 Plethysm and Kronecker Products p. 21 Lie n Lie n : multilinear subspace of C A (degree one in each x i ) basis: [x 1,[x w(2),[x w(3),[ ] ]], w S [2,n] dimlie n = (n 1)! Theorem (Brandt, 1944). Action of S n on Lie n has Frobenius characteristic L n. Note. Can be extended to L λ (decomposition of C A )

31 Plethysm and Kronecker Products p. 22 Partition lattices Π n : poset (lattice) of partitions of {1,...,n}, ordered by refinement Π n : Π n {ˆ0,ˆ1} (Π n ): set of chains of Π n (a simplicial complex) H i (Π n ): ith reduced homology group of (Π n ), say over C

32 Plethysm and Kronecker Products p. 23 Homology ands n -action Theorem. (a) H i (Π n ) = 0 unless i = n 3, and dim H n 3 (Π n ) = (n 1)!. (b) Action of S n on H n 3 (Π n ) has Frobenius characteristic ωl n.

33 Plethysm and Kronecker Products p. 24 Lower truncations of Π n Π n (r): top r levels of Π n

34 Plethysm and Kronecker Products p. 24 Lower truncations of Π n Π n (r): top r levels of Π n Π = Π (2) 4 4 Π (1) 4

35 Plethysm and Kronecker Products p. 25 S n -action on lower truncations Theorem (Sundaram, 1994) The Frobenius characteristic of the action of S n on the top homology of Π n (r) is the degree n term in the plethysm (ω(l r+1 L r + +( 1) r L 1 ))[h 1 + +h n ].

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37 Plethysm and Kronecker Products p. 27 Tensor product of characters χ,ψ: characters (or any class functions) of S n χ ψ (or χψ): tensor (or Kronecker) product of χ and ψ, i.e., (χ ψ)(w) = χ(w)ψ(w).

38 Tensor product of characters χ,ψ: characters (or any class functions) of S n χ ψ (or χψ): tensor (or Kronecker) product of χ and ψ, i.e., (χ ψ)(w) = χ(w)ψ(w). f: S n GL(V): representation with character χ g: S n GL(W): representation with character ψ χ ψ is the character of the representation f g: S n GL(V W) given by (f g)(w) = f(w) g(w). Plethysm and Kronecker Products p. 27

39 Plethysm and Kronecker Products p. 28 Kronecker coefficients Let λ,µ,ν n. g λµν = χ λ χ µ,χ ν = 1 χ λ (w)χ µ (w)χ ν (w) n! w S n

40 Plethysm and Kronecker Products p. 28 Kronecker coefficients Let λ,µ,ν n. g λµν = χ λ χ µ,χ ν = 1 χ λ (w)χ µ (w)χ ν (w) n! w S n Consequences: g λµν N = {0,1,...} g λµν is symmetric in λ, µ, ν.

41 Plethysm and Kronecker Products p. 29 Internal product Recall for λ,µ,ν n, g λµν = χ λ χ µ,χ ν. Define the internal product s λ s µ by s λ s µ,s ν = g λµν. Extend to any symmetric functions by bilinearity.

42 Plethysm and Kronecker Products p. 30 Tidbits (a) s λ s n = s λ, s λ s 1n = ωs λ

43 Plethysm and Kronecker Products p. 30 Tidbits (a) s λ s n = s λ, s λ s 1n = ωs λ (b) Conjecture (Saxl, 2012). Let δ n = (n 1,n 2,...,1) and λ ( n 2). Then s δn s δn,s λ > 0.

44 Plethysm and Kronecker Products p. 30 Tidbits (a) s λ s n = s λ, s λ s 1n = ωs λ (b) Conjecture (Saxl, 2012). Let δ n = (n 1,n 2,...,1) and λ ( n 2). Then s δn s δn,s λ > 0. (d) λ,µ,ν n g2 λµν = µ n z µ. Hence max logg λµν n λ,µ,ν n 2 logn. What λ,µ,ν achieve the maximum?

45 Plethysm and Kronecker Products p. 31 Generating function Theorem (Schur). i,j,k(1 x i y j z k ) 1 = λ,µ,ν g λµν s λ (x)s µ (y)s ν (z).

46 Plethysm and Kronecker Products p. 31 Generating function Theorem (Schur). i,j,k(1 x i y j z k ) 1 = λ,µ,ν g λµν s λ (x)s µ (y)s ν (z). Equivalent formulation: Write xy for the alphabet {x i y j } i,j 1. Thus f(xy) = f[s 1 (x)s 1 (y)]. Then f,g h = f(xy),g(x)h(y).

47 Plethysm and Kronecker Products p. 31 Generating function Theorem (Schur). i,j,k(1 x i y j z k ) 1 = λ,µ,ν g λµν s λ (x)s µ (y)s ν (z). Equivalent formulation: Write xy for the alphabet {x i y j } i,j 1. Thus f(xy) = f[s 1 (x)s 1 (y)]. Then f,g h = f(xy),g(x)h(y). What if we replace s 1 by s n, for instance?

48 Plethysm and Kronecker Products p. 32 Vanishing Vanishing of g λµν not well-understood. Sample result: Theorem (Dvir, 1993). Fix µ, ν n. Then max{l(λ) : g λµν 0} = µ ν (intersection of diagrams).

49 Plethysm and Kronecker Products p. 33 Example of Dvir s theorem s 41 s 32 = s 41 +s 32 +s 311 +s 221. Intersection of (4,1) and (3,2) = (2,2,1):

50 Plethysm and Kronecker Products p. 34 Combinatorial interpretation A central open problem: find a combinatorial interpretation of g λµν.

51 Plethysm and Kronecker Products p. 34 Combinatorial interpretation A central open problem: find a combinatorial interpretation of g λµν. Example. Let λ n. Then s j,1 n j s k,1 n k,s λ is the number of (u,v,w) S 3 n such that uvw = 1, D(u) = {j}, D(v) = {k}, and if w is inserted into λ from right to left and from bottom to top, then a standard Young tableau results.

52 Plethysm and Kronecker Products p. 35 Conjugation action S n acts on itself by conjugation, i.e., w u = w 1 uw. The Frobenius characteristic of this action is K n := λ n (s λ s λ ) = µ n p µ.

53 Plethysm and Kronecker Products p. 35 Conjugation action S n acts on itself by conjugation, i.e., w u = w 1 uw. The Frobenius characteristic of this action is K n := λ n (s λ s λ ) = µ n p µ. Combinatorial interpretation of K n,s ν not known. All known proofs that K n is Schur-positive use representation theory.

54 Plethysm and Kronecker Products p. 36 Stability Example. For n 8, s n 2,2 s n 2,2 = s n +s n 3,1,1,1 +2s n 2,2 +s n 1,1 +s n 2,1,1 +2s n 3,2,1 +s n 4,2,2 +s n 3,3 +s n 4,3,1 +s n 4,4.

55 Plethysm and Kronecker Products p. 36 Stability Example. For n 8, s n 2,2 s n 2,2 = s n +s n 3,1,1,1 +2s n 2,2 +s n 1,1 +s n 2,1,1 +2s n 3,2,1 +s n 4,2,2 +s n 3,3 +s n 4,3,1 +s n 4,4. λ[n] := (n λ,λ 1,λ 2,...) Theorem (Murnaghan, 1937). For any partitions α,β,γ, the Kronecker coefficient g α[n],β[n],γ[n] stabilizes. Vast generalization proved by Steven Sam and Andrew Snowden, 2016.

56 Plethysm and Kronecker Products p. 37 Reduced Kronecker coefficient ḡ αβγ : the stable value ḡ αβγ is called a reduced Kronecker coefficient. Combinatorial interpretation is not known.

57 Plethysm and Kronecker Products p. 37 Reduced Kronecker coefficient ḡ αβγ : the stable value ḡ αβγ is called a reduced Kronecker coefficient. Combinatorial interpretation is not known. Example. Recall that for n 8, s n 2,2 s n 2,2 = s n +s n 3,1,1,1 +2s n 2,2 +s n 1,1 +s n 2,1,1 +2s n 3,2,1 +s n 4,2,2 +s n 3,3 +s n 4,3,1 +s n 4,4. Hence ḡ 2,2, = 1, ḡ 2,2,111 = 1, ḡ 2,2,2 = 2, etc.

58 V P ws = V NP? Plethysm and Kronecker Products p. 38

59 Plethysm and Kronecker Products p. 39 Algebraic complexity Flagship problem: V P ws V NP. Determinantal complexity of f C[x 1,...,x n ]: smallest n N such that f is the determinant of an n n matrix whose entries are affine linear forms in the x i. Theorem (Valiant 1979, Toda 1992). TFAE: Determinantal complexity of an n n permanant is superpolynomial in n. V P ws V NP

60 Plethysm and Kronecker Products p. 40 Mulmuley and Sohoni 2001 Ω n : closure of the orbit of GL n 2 det n in Sym n C n2. padded permanent: x n m 11 per m Sym n C n2. Conjecture. For all c > 0 and infinitely many m, there exists a partition λ (i.e., an irreducible polynomial representation of GL n 2) occurring in the coordinate ring C[Z mc,m] but not in C[Ω m c].

61 Plethysm and Kronecker Products p. 41 Bürgisser, Ikenmeyer, and Panova Theorem (BIP 2016) The conjecture of Mulmuley and Sohoni is false.

62 Plethysm and Kronecker Products p. 41 Bürgisser, Ikenmeyer, and Panova Theorem (BIP 2016) The conjecture of Mulmuley and Sohoni is false. Proof involves Kronecker product coefficients g λµν in an essential way.

63 The last slide Plethysm and Kronecker Products p. 42

64 The last slide Plethysm and Kronecker Products p. 42