Introduction in collaboration 1995 C M C Reutemann Hopfstructures on - PowerPoint PPT Presentation

Primitive elements for the Hopf algebras of tableaux Claudia Malvenuto SapienzaUniversità di Roma E SI 12 16 Oct 2020 Vienna

of the talk Plan Introduction Some notations Hopf algebras of permutations Primitive elements of 25 Hopf algebras of tableaux Primitive elements of 22 mms

Introduction in collaboration 1995 C M C Reutemann Hopfstructures on permutations inherited by concatenation shuffle TN Hopfalgebras con C Reutemann 1995 5 Poirier

M Aguiar F Sottile 12004 M Task in 0 Kohl

Some notations symmetric group Sw on 1,2 n tn 6111612 resa of letters of Gem 6 e length in Coxeter l 6 group

Inversion set for 6 Iii le Judo sisi i 251 7643 Ex 6 15,445,3 51 2,1 Inv 6 17,3 7,4 7,6 14,3 6,3 6,4 Judo 10 l 6

on Su Right weak Bruh at order closure of the relation reflexive transitive I T no v t U c U EV e Juri Juku

standardisation without repetition on IN award Vi permutation st v replace letters by Ex st unique increasing 5713 3412 bijection from Ivi 11,2 Aleph v obtained by restriction to I Eli 61 in I in obtained by erasing theletters ott 3 6 2 I Ex 2117643 se ott dee 5,14 263

Classical associative U sn 5 S products on ma o add pt each ne sp.ve Sq v letter afro concatenation shifted right 23145 C 55 12 Ex 231 UV le vie concatenation left shifted 45231 VU Ex 12 231 iI V U

Facts a free monoid with S is the Free generators are indecomposable permutations weak order is compatible with The u E E V v v n U En are conjugate under want and L un rt v u

Hopf algebras of permutations free module with s basis 25 the permutations destandardised concatenation product a xp E sp de sp.pe So o a p E stolen stlpt.se un resa Ex 1432 1221 1342 12 21 E sa 3421 2431 2341

coproducts standardised un shuffling È stf liti nl 11 il S essi i 1 f 312 4 Ex St 3124 E E 3124 St 324 1 1 x 213 St 34 x IL 12 12 St 4 1 312 312 E E 3 124 3 124

graded Hopf algebra 25 s is a graded Hopf algebra 25 S is a shifted shuffle standardised de concatenation S structures Duality between the two Hopf a SE Conjugated Lo a T o via a s'f 7 si Lo a a T o

linear basis Aguiar Sottile new No res forks r ÈÉ Max Mu Theorem s Mo Lemma For m p 9 6 v u E 6esp.iq oro 11 pl v and ben a int bests liti

in Aguiar Sottile Remark Proof global descent uses a global descent in si c Sn has Def if 1 i 1,2 n e oli i dj jei.tk V 213 132 12 7 8 46 5 213 Global Descents 2,5 indecomposatole 213 12 132 noglobal descents 5 53

Primitive elements of KS Corollary module of the primitive of 2g The sub elements such that is spanned by the M no global descents equini has o i e is indecomposable for o the generators of the free monoid s

Hopf algebras of tableaux standard Young tableaux Tn Tu U 1 entries with n cases no o n f 41,2 me a Risk Qlo Pio resa insertion recording witness j Ki Jit Knuth relations i'I'K Ik Plachi congruence Fai

E i 78645213 e 78465213 E i H 76845213 74865213 no I 76485213 7 e 74865271 i 78465231 784613 74762531 I 7846235T 78462531

Knuth Theorem p Pio z si Te free module with basis 2 22 the tableaux module I uno v u inherited Theorem product 22 25 1 by coproduct PR s permutations Pt 6 1 non commutative Dbs 22 s not free associative algebra

another Hopf structure i 22 l ftp.foplaetic ter class of t riniti Ex l 1132 312 312 f raw t 132 312 Description of profits jeudetaqu.in via c products i backward slides

Homomorphisms surjective Hopf morphism Pio o i in 122 of s d d 122 s Gym d t E si Schur function shh 1 Schnitzerbergh evacuation t ev tet of anti automorphism of is an both Hopf algebras of tableaux

Primitive elements of 22 Melnikov Weak order of 2004 tableaux Ì Daft order E weak Ta skin M 2013 A U tableaux A v Jef fu permutations 3 nitido fa dai A d a Pan do ER da dm no verra

The week order on Tu 2,3 4,5 µ

Lemma Plachi equivalence is compatible with n ri un n ri un tableaux Product on Pin P PIU v a homomorfism of the Pisa and T monoids S

tableaux A simpler way to compute on EX tetta ftp.go V tata a 3 µ 2 weak order on tableaux The Lemma compatible with is Ev v li u VE li VE li

Recall for permutations Lemma For nep 9 6 v u E ne so Esp Gesu o 11 pl a v and ben a b stf liti int ftp ea Lemma Fornaio.io V u E E EeTnveTpueTo E 11 A p E V and B EU A BESTIE fin nl

define a Aguiar Sottile new linear basis method ero for 22 Me via Morbius inversion in the poset of tableaux 2MW EW 1 1 Theorem SU u E v u M C R

Corollary module of the primitive of zz The sub elements such that is spanned by the M 2 i e is indecomposable for E montai'd the generators of the free 22 Ex i Thai i FÉTTE i i indecomposatole decomprosable

Final remarks shifted concatenation right s Fa left shifted concatenation a monoid is compatible with z is tableaux A simpler way to compute on Ex te mai v vomitasti Venire i i mira Ti

ve 5 6 u Theorem v a Loday Romeo verso a a 2002 interval of the week art of S siete v.v a B Theorem Task.in Et u vero v 2005 e te v v U in the interval of the tableau t in Buffo order of 2 Task

6 ES Theorem In the linear basis No are positive the structure constants a S ci Mg ciao µ No A counterexample Franco salio la UQAM

Grazie per l'attenzione

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