Hopf structures on binary trees (variations on a theme)

Transcription

1 Hopf structures on binary trees (variations on a theme) Aaron Lauve Texas A&M University joint work with: Stefan Forcey Tennessee State University Frank Sottile Texas A&M University AMS Sectional Meeting, Raleigh, NC Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

2 The (known) cast of characters Polytopes and their Hopf Algebras Permutahedra: vertices are permutations. Malvenuto-Reutenauer, 95 Hypercubes: vertices are compositions. Gessel, 84 SSym YSym QSym Associahedra: vertices are planar binary trees. Stasheff, 6 Loday-Ronco, 98 Hivert-Novelli-Thibon, 0 Reading, 05 Aguiar-Sottile, 06 Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

3 The new players Another Family of Polytopes SSym YSym QSym SSym MSym YSym Multiplihedra: a family nestled between permutahedra and associahedra, vertices are bi-leveled trees. Stasheff, 70 Iwase-Mimura, 89 Forcey, 07 Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

4 The new players Another Hopf Algebra? SSym YSym QSym SSym MSym YSym Multiplihedra: Main Results Forcey-Lauve-Sottile a graded algebra an SSym-module a YSym Hopf module and, in an unrelated way, a Hopf algebra! Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

5 The Multiplihedra Stasheff s Multiplihedra Fix a map f between tensor categories (not necessarily associative): (C, ) f (D, ) How many ways to multiply and map three objects a, b, c? f (a (b c)) f ((a b) c) f (a) (f (b) f (c)) (f (a) f (b)) f (c) f (a) f (b c) f (a b) f (c) Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

6 The Multiplihedra The Multiplihedra (Poset) M... f (a (b c)) f ((a b) c) f (a) f (b c) f (a b) f (c) f (a) (f (b) f (c)) (f (a) f (b)) f (c)... of bi-leveled trees Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

7 The map S n M n Y n Permutations as Ordered Trees permutations 4 45 Read between the leaves. ordered trees (total orders) planar binary trees (partial orders) Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

8 The map S n M n Y n Understanding the map S n M n Y n 4 β φ Here, β forgets most of the total ordering (remembering only relation to the leftmost node); the map φ forgets even that information. Write τ = φ β. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

9 Hopf structures on trees Hopf Structures on Binary Trees The Hopf algebra YSym = n 0 span{ Ft t Y n } is defined via splitting and grafting operations on trees. ( ) Definition (Splitting) A p-splitting of a tree t is a forest (t 0,...,t p ), gotten by choosing p leaves of t and splitting their branchings down to the root. t = ( ),,, = (t 0,t,t,t ). Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

10 Hopf structures on trees Hopf Structures on Binary Trees The Hopf algebra YSym = n 0 span{ Ft t Y n } is defined via splitting and grafting operations on trees. ( ) Definition (Grafting) A grafting of a forest (t 0,...,t p ) onto a tree s with p nodes is the tree gotten by grafting the root of t i to the i th leaf of s. ( ) /,,, =. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

11 Hopf structures on trees Hopf Structures on Binary Trees The Hopf algebra YSym = n 0 span{ Ft t Y n } is defined via splitting ( ) and grafting ( / ) operations on trees. Definition (Multiplication and Comultiplication) Given two binary trees s and t (s having p nodes), put F t F s = F (t0,t,...,t p )/s and (F t ) = F t0 F t. t (t 0,t,...,t p ) t (t 0,t ) (Loday-Ronco, 98, 0) Analogous operations on ordered trees give the Hopf algebra SSym of permutations. (Loday-Ronco, 98, 0) The map τ : SSym YSym (F t F τ(t) ) is a Hopf algebra map. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

12 Hopf structures on trees Hopf Structure on Bi-leveled Trees? Let MSym = n 0 span{ Ft t M n } be the space of bi-leveled trees. Proposition (F-L-S) Splitting & grafting may be extended to bi-leveled trees, making MSym into an algebra... but not a coalgebra. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

13 Hopf structures on trees (Hopf) Structures on Bi-leveled Trees Let MSym = n 0 span{ Ft t M n } be the algebra of bi-leveled trees. Theorem (F-L-S) MSym is a right (and left) module over YSym and SSym. The maps S structures above. M Y extend to algebra maps respecting the MSym and MSym + are right YSym Hopf modules......in several ways. Neither is a SSym Hopf module. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

14 Change of basis (Möbius inversion) MSym in the Monomial Basis Define new basis { M t t M } via Möbius Inversion in the posets M. (Aguiar-Sottile, 05, 06) hidden structure was revealed in analogous bases for SSym and YSym. Main Theorem (F-L-S) (ρ : MSym + MSym + YSym): ρ(m s ) is the sum of ways to prune along right branch without clipping between circled nodes. ρ ( ) = + + ρ ( ) = Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

15 Change of basis (Möbius inversion) Corollary (F-L-S) MSym in the Monomial Basis The Fundamental Theorem of Hopf Modules states that M M co H, Where M co := {m : ρ(m) = m } are the coinvariants. In the Hopf module structure on MSym +, a basis of coinvariants is given by M s, with s ranging over bi-leveled trees with no un-circled nodes on the rightmost branch. Enumeration of coinvariants [OEIS, A76] Hilb(MSym + ) Hilb(YSym) = t + t + t + t t t 6 + How Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

16 Questions & references Future Questions Coinvariants for the other YSym Hopf module structure? (Much harder; seems to involve a restricted Möbius function.) What happened to QSym? (We expect to find many new Hopf modules for YSym and QSym.) Connections between the algebra and the geometry? (See also graph multiplihedra in arxiv: ) References Our data: sottile/pages/hopfalgebras/msym/index.html An extended abstract: lauve/papers/msym fpsac.pdf Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

17 Curious aspects of the proof Towards a Proof It would be nice to embed M n in S n. A similar idea was used in (Aguiar-Sottile, 06). They found a section ι : Y n S n of the natural map τ : S n Y n. They showed that the pair gives a Galois connection between the posets, i.e., a pair of order-preserving maps (τ, ι) satisfying The rest was easy. τ(w) t w ι(t) for all t Y n, w S n. Q&R Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

18 Curious aspects of the proof Towards an embedding M n S n An interval in M 7 This choice doesn t give an embedding.... and its preimage in S This choice works. This choice doesn t either Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

19 Curious aspects of the proof Towards an embedding M n S n ( max min ) = = Proposition (F-L-S) The section ι : M S given by ι(t) = ( max min) (t) is an embedding of posets. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

20 Curious aspects of the proof (β, ι) does not form a Galois connection β(6745) t...but 6745 ι(t) = t Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

21 Curious aspects of the proof...but (β, ι) does form an interval retract Two posets P and Q are related by an interval retract if there are order-preserving maps β : P Q and ι : Q P satisfying β(ι(t)) = t and β (t) is an interval t. Theorem (F-L-S) If P and Q are two posets related by an interval retract (β,ι), then the Möbius functions µ P and µ Q are related by s < t Q v β (s) w β (t) µ P (v,w) = µ Q (s,t). Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

22 Curious aspects of the proof...but (β, ι) does form an interval retract Two posets P and Q are related by an interval retract if there are order-preserving maps β : P Q and ι : Q P satisfying β(ι(t)) = t and β (t) is an interval t. Proposition (F-L-S) The maps β : S n M n and ι : M n S n form an interval retract between S n and M n. In particular, ( ) β M σ = M t. σ β (t) This is good enough to prove our main theorem. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6 Q&R

23 The future A host of new polytopes waiting to be explored. SSym YSym QSym The Future The polytopes J G d and J G r also appear in Devadoss- Forcey, 08. (See that paper for notations.) We focus on J (4) here. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

24 The future SSym YSym QSym 4 (½,γ) (τ,½) 4 (½,τ) (τ,½) (½,γ) (τ,½) (½,τ) (γ,½) 4 The vertices of each polytope are indexed by certain types of planar binary trees. (γ,½) (½,τ) 4 (γ,½) (½,γ) We focus on bi-leveled trees here. Q&R Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

25 The future Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

26 The future (½,τ) (τ,½) 4 (½,γ) (τ,½) 4 4 (τ,½) (½,γ) (½,τ) (γ,½) (γ,½) (½,τ) Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6 4

27 The future Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

28 (½,γ) (τ,½) The future (½,τ) (γ,½) 4 (τ,½) (½,γ) (γ,½) (½,τ) 4 (γ,½) (½,γ) Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

29 Appendix The weak order on S 4, highlighting the fibers of the maps β and τ, respectively. Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

30 Appendix Coinvariant Enumeration for the two separate YSym Hopf module structure on MSym. Hilb(M + ) Hilb(Y ) = 0 + t + t + t + t t t 6 + Hilb(M ) Hilb(Y ) = + 0t + 0t + t + 6t 4 + 0t 5 + 4t 6 + Aaron Lauve (TAMU) Hopf structures on binary trees April 5, / 6

31 Appendix From Painted Trees to Bi-leveled Trees An f -mapping A painted tree f (a) `f (b c) f (d) Add a branch on the left, marking first application of f. Circle new node, and all nodes corresponding to multiplication in D. Forget the branch paintings. A bi-leveled tree Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6

32 Appendix The Multiplihedra (Posets) M & M 4 Aaron Lauve (TAMU) Hopf structures on binary trees April 5, 009 / 6