Combinatorial Hopf algebras

Transcription

1 Combinatorial Hopf algebras Jean-Yves Thibon Université Paris-Est Marne-la-Vallée The Mathematical Legacy of Jean-Louis Loday Strasbourg, September 4-6, 2013

2 Combinatorial Hopf algebras Heuristic notion (no formal definition) Graded (bi-)algebras based on combinatorial objects Arise in various contexts: combinatorics, representation theory, operads, renormalization, topology, singularities sometimes with very different definitions. Example: Integer partitions; Sym =symmetric functions. Nontrivial product and coproduct for Schur functions (Littlewood-Richardson) I like to see combinatorial Hopf algebras as generalizations of the algebra of symmetric functions. Jean-Louis had a different point of view. This was the basis of our interactions.

3 Integer partitions 1, 1, 2, 3, 5, 7, 11,... Integer compositions 1, 1 2, 4, 8, 16, 32, 64,... Binary trees 1, 1, 2, 5, 14, 42, 132,... Plane trees 1, 1, 3, 11, 45, 197, 903,... Standard tableaux 1, 1, 2, 4, 10, 26, 76,... Permutations 1, 1, 2, 6, 24, 120, 720,... Set compositions 1, 1, 3, 13, 75, 541, 4683,... Set partitions 1, 1, 2, 5, 15, 52, 203,... Parking functions 1, 1, 3, 16, 125, 1296,

4 Why symmetric functions? I The algebra of symmetric functions contains interesting elements: Schur, Hall-Littlewood, zonal, Jack, Macdonald... solving important problems: Schur: character tables of symmetric groups, characters of GL(n, C), zonal spherical functions of (GL(n, C), U(n)), KP-hierarchy, Fock space, lots of combinatorial applications Hall-Littlewood (one parameter): Hall algebra, character tables of GL(n, F q ), geometry and toplogy of flag varieties, characters of affine Lie algebras, zonal spherical functions for p-adic groups, statistical mechanics Zonal polynomials: for orthogonal and symplectic groups Macdonald (two parameters): unification of the previous ones. Solutions of quantum relativistic models, diagonal harmonics, etc.

5 Why symmetric functions? II One may ask whether there are such things in combinatorial Hopf algebras... For our purposes, the example of Schur functions will be good enough Their product (LR-rule) solves a nontrivial problem (tensor products of representations of GL n ) This rule is now explained and generalized by the theory of crystal bases but it can also be interpreted in terms of a combinatorial Hopf algebra of Young tableaux, defined by means of the Robinson-Schensted correspondence and the Loday-Ronco Hopf algebra of binary trees admits a similar definition (LR-algebras?)

6 The Robinson-Schensted correspondence Insertion algorithm: w A P(w) (semi-standard tableau) (A a totally ordered alphabet) Example: P(132541) Bijection w (P(w), Q(w)) Q(w) standard tableau encoding the chain of shapes of P(x 1 ), P(x 1 x 2 ),..., P(w). Q(132541) = Clearly, Q(w) has the same shape as P(w).

7 The plactic monoid Equivalence relation on A u v P(u) = P(v) It is the congruence on A generated by the relations xzy zxy (x y < z) yxz yzx (x < y z) The plactic monoid on the alphabet A is the quotient A /, where is the congruence generated by the Knuth relations above.

8 Free Schur functions Tableau T monomial x T T = x T = x 1 x 3 2 x 2 4 x 5 Shape of T : partition sh(t ) = λ = (4, 2, 1) Schur functions: s λ = x T sh(t )=λ Free Schur functions (labeled by standard tableaux) S t = w Q(w)=t Goes to s λ (shape of t) by a i x i.

9 The Hopf algebra FSym t, t standard tableaux; k number of cells of t. S t S t = t Sh(t,t ) Sh(t, t ) set of standard tableaux in the shuffle of t (row reading) with the plactic class of t [k]. Thus, the S t span an algebra. It is also a coalgebra for the coproduct A A + A (ordinal sum): Hopf algebra FSym. [Littlewood-Richardson 1934; Robinson; Schensted; Knuth; Lascoux-Schützenberger; Poirier-Reutenauer; Lascoux-Leclerc-T.; Duchamp-Hivert-T. 2001] S t

10 Example t = t =

11 Binary search trees and the sylvester monoid The sylvester correspondence w (P(w), Q(w)) (binary search tree, decreasing tree) [Hivert-Novelli-T.] For w = bacaabca, Equivalence relation on A u v P(u) = P(v) It coincides with the sylvester congruence, generated by zxuy xzuy, x y < z A, u A.

12 The cosylvester algebra Flattening P(w) yields the nondecreasing rearrangement of w. Thus, the only nontrivial information is its shape T (w). Let P T = w Then, P T P T = T (w)=t T Sh(T,T ) P T, where Sh(T, T ) is the set of trees T in u v; (u = w T, v = w T [k] are words read from the trees). This is completely similar to the LRS rule. The P T span an algebra, and actually a bialgebra for the coproduct A + A as above. It is isomorphic to the Loday-Ronco algebra (free dendriform algebra on one generator): a polynomial realization of PBT.

13 Polynomial realizations Combinatorial objet polynomial in infinitely many variables (commuting or not) Combinatorial product ordinary product of polynomials Coproduct A A + A Can be found for most CHA. In the special case of PBT: Dendriform structure implied by trivial operations on words uv = u v + u v, u v = uv if max(u) > max(v) or 0 Easy computation of the dual (via Cauchy type identity) Open problem: FSym free P-algebra on one generator for some operad? (non-trivial implications: hook-length formulas)

14 Background on symmetric functions I functions : polynomials in an infinite set of indeterminates X = {x i i 1} λ t (X) or E(t; X) = (1 + tx i ) = e n (X)t n i 1 n 0 σ t (X) or H(t; X) = i 1 (1 tx i ) 1 = n 0 h n (X)t n e n = elementary symmetric functions h n = complete (homogeneous) symmetric functions Algebraically independent: Sym(X) = K [h 1, h 2,... ] With n variables: K [e 1, e 2,..., e n ]

15 Background on symmetric functions II Bialgebra structure: f = f (X + Y ) X + Y : disjoint union; u(x)v(y ) u v Graded connected bialgebra: Hopf algebra Self-dual. Scalar product s.t. f g, h = f g, h Linear bases: integer partitions λ = (λ 1 λ 2... λ r > 0) Multiplicative bases: e λ = e λ1 e λ2 e λr and h λ

16 Background on symmetric functions III Obvious basis: monomial symmetric functions m λ = Σx λ = distinct permutations Hall s scalar product realizes self-duality x µ h λ, m µ = δ λµ h and m are adjoint bases, and σ 1 (XY ) = i,j 1(1 x i y j ) 1 = λ m λ (X)h λ (Y ) (Cauchy type identity) Any pair of bases s.t. σ 1 (XY ) = λ u λ(x)v λ (Y ) are mutually adjoint

17 Background on symmetric functions IV Original Cauchy identity for Schur functions σ 1 (XY ) = λ s λ (X)s λ (Y ) where s λ = det(h λi +j i) = shape(t )=λ x T = A(x λ+ρ )/A(x ρ ) Schur functions encode irreducible characters of symmetric groups: χ λ µ = s λ, p µ (Frobenius) p n : power-sums p n (X) = i 1 i, σ t (X) = exp p m (X) tm m m 1 x n

18 Noncommutative Symmetric Functions I Very simple definition: replace the complete symmetric functions h n by non-commuting indeterminates S n, and keep the coproduct formula Realization: A = {a i i 1}, totally ordered set of noncommuting variables σ t (A) = ta i ) i 1(1 1 = S n (A)t n ( h n ) n 0 λ t (A) = + ta i ) = 1 i(1 Λ n (A)t n ( e n ) n 0 Coproduct: F = F(A + B) (ordinal sum, A commutes with B)

19 Noncommutative Symmetric Functions II Sym = n 0 Σ n, Σ n descent algebra of S n [Solomon,Malvenuto-Reutenauer, GKLLRT] Sym = n 0 K 0(H n (0)) (analogue of Frobenius) [Krob-T.] Topological interpretation: Sym = H (ΩΣ C P )) [Baker-Richter] Universal Leibniz Hopf algebra [Hazewinkel] Calling this algebra NCSF implies to look at it in a special way [GKLLRT] Find analogues of the classical families of symmetric functions and of the various interpretations of Sym

20 Noncommutative Symmetric Functions III Analogues of Schur functions: the ribbon basis A descent of w A n : an i s.t. w i > w i+1 w = Descent set Des(w) = {1, 3, 6} Descent composition C(w) = I = (1, 2, 3, 2) Des(I) = {1, 3, 6}

21 Noncommutative Symmetric Functions IV Analogue of the complete basis By inclusion exclusion R I := S I := S i1 S i2 S ir = C(w)=I w = Des(J) Des(I) Des(w) Des(I) w ( 1) l(i) l(j) S J goes to a skew Schur function under a i x i. It is a Schur like basis and the product rule is R I R J = R I J + R IJ Free As (2) -algebra on one generator.

22 Quasi-symmetric functions I Sym is cocommutative: S n = S n (A + A ) = i+j=n S i S j To find the dual, introduce an infinite set X of commuting indeterminates, and the Cauchy kernel K(X, A) := i 1 j 1(1 x i a j ) 1 = I M I (X)S I (A) = I F I (X)R I (A) The M I and F I are bases of a commutative Hopf algebra: Quasi-symmetric functions [Gessel 1984]. M I = x i1 k1 x i 2 k2 x ir k r k 1 <k 2 <...<k r (pieces of monomial symmetric functions).

23 Quasi-symmetric functions II QSym is the free commutative tridendriform algebra on one generator. The product rules for the M I and the F I have nontrivial multiplicities. F 11 F 21 = F F F 32 + F F F F F F 2111 M 11 M 21 = M M M M M M M M M 32 Their combinatorial understanding requires two larger Hopf noncommutative Hopf algebras, which can also be interpreted as operads: For the F I : FQSym, based on permutations For the M I : WQSym, based on packed words (surjections)

24 Standardization of a word: FQSym The descent set of a word is compatible with a finer invariant: the standardization word of length n permutation of S n w = a 1 a 2... a n σ = std(w) for all i < j set σ(i) > σ(j) iff a i > a j. Example: std(abcadbcaa) = a b c a d b c a a a 1 b 5 c 7 a 2 d 9 b 6 c 8 a 3 a

25 Free Quasi-Symmetric Functions Subspace of the free associative algebra K A spanned by G σ (A) := std(w)=σ w. It is a subalgebra, with product rule for α S m, β S n, G α G β = G γ. γ=u v std(u)=α, std(v)=β [Malvenuto-Reutenauer; Duchamp-Hivert-T.] G 21 G 213 = G G G G G G G G G G 21435

26 Morphisms and duality FQSym is a Hopf algebra for the coproduct (G σ ) = G σ (A + A ) The obvious embedding ι : Sym FQSym R I = C(σ)=I G σ is a morphism of Hopf algebras. FQSym is self-dual, the dual basis of G σ is F σ = G σ 1 Thus, ι : FQSym QSym is an epimorphism of Hopf algebras. It is given by a i x i (commutative image). Then, F σ F C(σ). The rule F α F β = γ α projects to the product rule of the F I. β[k] F γ

27 FQSym as Zinbiel FQSym is a dendriform (even bidendriform [Foissy]) algebra. It can also be interpreted as an operad. A rational mould is a sequence f = (f n (u 1,..., u n )) of rational functions. The mould product on single rational functions is f n g m = f (u 1,..., u n )g m (u n+1,..., u m+n ) Chapoton has defined an operad structure on these rational functions. The fractions f σ = 1 u σ(1) (u σ(1) + u σ(2) ) (u σ(1) + u σ(2) + + u σ(n) ) satisfy the product rule of FQSym f α f β = γ α and their linear span is stable under the i : a suboperad which can be recognized as Zinbiel. [Chapoton-Hivert-Novelli-T.] β[k] f γ

28 Packing of a word: WQSym One can refine standardization by giving an identical numbering to all occurences of the same letter: If b 1 < b 2 <... < b r are the letters occuring in w, u = pack(w) is the image of w by the homomorphism b i i. u is packed if pack(u) = u. The, we set [Hivert; Novelli-T.] For example, M u := pack(w)=u w. M = Under the abelianization a i x i, the M u M I where I = ( u i ). The M u span a subalgebra of K A, called WQSym

29 Structure of WQSym Hopf algebra for A A + A. It contains FQSym: G σ = std(u)=σ It is a tridendriform algebra. Again, the tridendriform structure is induced by trivial operations on words M u uv = u v + u v + u v (only one term is uv). The degree one element a = G 1 = F 1 = M 1 = i 1 a i generates a free tridendriform algebra (Schröder trees). WQSym is also an operad of rational functions (1 RatFct) F σ of FQSym can be interpreted a the characteristic function of a simplex σ (product rule for iterated integrals). Similarly, the ( 1) max(u) M u can be interpreted as characteristic functions of certain polyhedral cones [Menous-Novelli-T.].

30 Special words and equivalence relations I A whole class of combinatorial Hopf algebras whose operations are usually described in terms of some elaborated surgery on combinatorial objets are in fact just subalgebras of K A Sym: R I (A) is the sum of all words with the same descent set FQSym: G σ (A) is the sum of all words with the same standardization PBT: P T (A) is the sum of all words with the same binary search tree FSym: S t (A) is the sum of all words with the same insertion tableau WQSym: M u (A) is the sum of all words with the same packing It contains the free tridendriform algebra one one generator, based on sum of words with the same plane tree

31 Special words and equivalence relations II To these examples, one can add: PQSym: based on parking functions (sum of all words with the same parkization) In all cases, the product is the ordinary product of polynomials, and the coproduct is A + A.

32 Parking functions I A parking function of length n is a word over w over [1, n] such that in the sorted word w, the ith letter is i. Example w = OK since w = 12235, but not Parkization algorithm: sort w, shift the smallest letter if it is not 1, then if necessary, shift the second smallest letter of a minimal amount, and so on. Then put each letter back in its original place [Novelli-T.]. Example: w = (5, 7, 3, 3, 13, 1, 10, 10, 4), w = (1, 3, 3, 4, 5, 7, 10, 10, 13), p(w) = (1, 2, 2, 4, 5, 6, 7, 7, 9), and finally p(w) = (4, 6, 2, 2, 9, 1, 7, 7, 3).

33 Parking functions II PF n = (n + 1) n 1 Parking functions are related to the combinatorics of Lagrange inversion Also, noncommutative Lagrange inversion, antipode of noncommutative formal diffeomorphisms PQSym, Hopf algebra of (dual) Parking Quasi-Symmetric functions: G a = w p(w)=a Self dual in a nontrivial way. G a =: F a Many interesting quotients and subalgebras (WQSym, FQSym, Schröder, Catalan, 3 n 1...) Tridendriform. Operadic interpretation is unknown

34 The Catalan subalgebra I Natural: group the parking functions a according the the sorted word π = a (occurs in the definition and in the noncommutative Lagrange inversion formula) Then, the sums P π = a =π F a span a Hopf subalgebra CQSym of PQSym dimcqsym n = c n (Catalan numbers 1,1,2,5,14) P π is a multiplicative basis: P 11 P 1233 = P (shifted concatenation) Free over a Catalan set {1, 11, 111, 112,... } (start with 1) And it is cocommutative So it must be isomorphic to the Grossman-Larson algebra of ordered trees. However, this is a very different definition (no trees!)

35 The Catalan subalgebra II Duplicial algebras: two associative operations and such that (x y) z = x (y z) The free duplicial algebra D on one generator has a basis labelled by binary trees. and are \ (under) and / (over) CQSym is the free duplicial algebra on one generator: Let P α P β = P α P β and P α P β = P α β[max(α) 1] =: P α β For example, P 12 P 113 = P

36 Noncommutative Lagrange inversion I Can be formulated as a functional equation in Sym G = 1 + S 1 G + S 2 G 2 + S 3 G 3 + [Garsia-Gessel; Gessel; Pak-Postnikov-Retakh; Novelli-T.] Unique solution G 0 = 1, G 1 = S 1, G 2 = S 2 + S 11, G 3 = S 3 + 2S 21 + S 12 + S 111, G 4 = S 4 + 3S S 22 + S S S S S (notice the sums of coefficients)

37 Noncommutative Lagrange inversion II Sym embeds in PQSym by S n F 1 n, and the sum of all parking functions G = a PF F a solves the functional equation G = 1 + S 1 G + S 2 G 2 + S 3 G 3 + Actually, G belongs to CQSym, and solves the quadratic (duplicial) functional equation G = 1 + B(G, G) (B(x, y) = x P 1 y) and each term B T (1) of the tree expansion of the solution is a single P π, thus forcing a bijection beween binary trees and nondecreasing parking functions.

38 Tree expansion for x = a + B(x, x) By iterated substitution x = a + B(a, a) + B(B(a, a), a) + B(a, B(a, a)) + = a + For example, B a a + x = B B B a + a a a B T (a) T : Complete Binary Tree B a a =

39 Embedding of Sym in PBT I Sending S n to the left (or right) comb with n (internal) nodes is a Hopf embedding of Sym in PBT Under the bijection forced by the quadratic equation, nondecreasing parking functions with the same packed evaluation I form an interval of the Tamari order, whose cardinality is the coefficient of S I in G. The sum of the trees in this interval is the expansion of RĪ in PBT.

40 Duality I Interesting property of the (commutative) dual: CQSym contains QSym in a natural way Recall m λ = Σx λ (monomial symmetric functions) m λ = M I M I (X) = x i1 j1 x i 2 j2 x ir j r j 1 <j 2 <...<j r I =λ Let M π be the dual basis of P π. It can be realized by polynomials: M π = w p(w)=π where w means commutative image (a i x i )

41 Duality II Example: M 111 = i x 3 i M 112 = i x 2 i x i+1 M 113 = i,j;j i+2 M 122 = M 123 = i,j;i<j i,j,k;i<j<k x 2 i x j x i x 2 j x i x j x k

42 Duality III Then, M I = t(π)=i M π. where t(π) is the composition obtained by counting the occurences of the different letters of π. For example, M 3 = M 111, M 21 = M M 113, M 12 = M 122 In most cases, one knows at least two CHA structures on a given family of combinatorial objects: a self-dual one, and a cocommutative one. Sometimes one can interpolate between them (in general, only by braided Hopf algebras).

43 Other aspects of CHA s not discussed in this talk I Internal products. Analogues of the tensor product of S n representations. In Sym: given by the descent algebras. Application: Lie idempotents. Subalgebras, e.g., peak algebras In FQSym: just the product of Sn In WQSym: the Solomon-Tits algebra Also in WSym (invariants of S(A) in K A In PQSym and CQSym: does exist, but mysterious... In Sym (r) (colored multisymmetric functions, wreath products) Categorification Sym n R(SG n) (semisimple: R = G 0 = K 0 ) QSym n G 0(H n (0)) and Sym n K 0(H n (0)) (explains duality) Colored version for 0-Ariki-Koike-Shoji algebras [Hivert-Novelli-T.] Peak algebras for 0-Hecke-Clifford algebras [N. Bergeron-Hivert-T.] Supercharacters of U n (q) for WSym q

44 Other aspects of CHA s not discussed in this talk II (1) PBT: Tilting modules for A n -quiver [Chapoton] Other polynomial realizations. With bi-indexed letters a ij or x ij Commutative algebras on all objetcs from the diagram [Hivert-Novelli-T.] MQSym [Duchamp-Hivert-T.] Ordered forests, subalgebras and quotients, in particular Connes-Kreimer [Foissy-Novelli-T.]