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

Primitive elements for the Hopfalgebras

• f tableaux

Claudia Malvenuto

SapienzaUniversità diRoma

E SI

Vienna

12 16Oct 2020

Plan

• f the talk

Introduction

Some notations

Hopf algebras of permutations

Primitive elements of 25

Hopf algebras of tableaux

Primitive elements of 22

mms

Introduction

1995 C M

C Reutemann

in collaboration

Hopfstructures

• n permutations

inherited by concatenation

shuffle

Hopfalgebras con TN 1995 5 Poirier C Reutemann

M Aguiar F Sottile

12004

M Taskin

Kohl

Some notations

Sw

symmetric group

• n

1,2

n

resa

6111612

tn

6

• f letters of

Gem

l

6

e length in Coxeter

group

Inversion set for 6

Judo

Iii

i

sisi

le

Ex

6

251 7643

Inv

6

2,1

51

15,445,3

7,6

7,4

17,3

6,4

6,3 14,3

l

6

Judo

10

Right weak Bruhat order

• n Su

reflexive transitive

closure of therelation

U c

v

I

t

no

T

U EV

Juku

e Juri

standardisation

Vi

award

• n IN

without repetition

st

v

permutation replace letters by

Ex unique increasing

st

5713

3412

bijection from

Aleph

v

11,2

Ivi

61

• btainedby restrictionto I Eli

in

• tt
• btainedbyerasingtheletters

in

I

I

2

3

6

Ex

se

2117643

dee 5,14

• tt

263

5

U sn

Classical associative

ma o

products

• n

S

ne sp.ve Sq

v

add pt each

letter afro

right

shifted

concatenation

le

vie

UV

Ex 231

12

23145 C55

left

shifted

concatenation

V

U

iI

VU

Ex

12

231 45231

Facts

S

is

a free monoidwith

Free generators

are

the

indecomposable permutations

The weak order is compatible with

U En

E V

v u E v n

and

are conjugate under

want

v

u

Lun

rt

Hopf algebras of permutations

25

free module with

basis

s

the permutations

product

destandardised concatenation

de sp.pe So

a xp

E sp

• a p

E

un

resa

stolen stlpt.se

Ex

12

21

1221

1342

1432

2341 2431 3421

E sa

coproducts standardised

unshuffling

essi

S

È

11 il

stf liti

nl

i

1

Ex

f

312 4

E

St 3124

E

3124 1

St

324

1

x 213

12

St

34

12

x IL

312

St

4

312

1

3 124

E

3 124

E

25

s

is

a

gradedHopf algebra 25

S

is

a

gradedHopf algebra

shifted shuffle

S

standardised deconcatenation

Duality betweenthetwo Hopf

structures

• a

T

Lo

a SE

Conjugated

via

• a

T

Lo

a s'f 7

a

si

Aguiar Sottile

new

linear basis

No res

forks

r

ÈÉ

Theorem

s Mo

Max Mu

Lemma For m p

9

6

E

v u

6esp.iq

• ro 11 pl

a v and

ben

bests liti int

Remark

Proof

in Aguiar Sottile

uses

global descent

Def

si cSn has

a global descent in

i

e

1,2

n

1

if

V

jei.tk

i

dj

• li

7 8 46 5 213

12

132

213

Global Descents 2,5

12

132 213

indecomposatole

noglobal descents

5

53

Primitive elements of

KS

Corollary

The sub

moduleof the primitive

2g

elements

• f

is spanned by the M

such that

• has

no global descents

equini

• is indecomposablefor

i e

the generators of the freemonoid

s

Hopf algebras of

tableaux

Tn

standardYoung tableaux

1

U

Tu

with

n

cases

entries

no o

41,2

n f a

me

Risk

resa

Pio

Qlo

insertion

recording

witness

Knuthrelations

Jit

j Ki Plachi congruence Fai Ik

i'I'K

E

i

78465213

78645213

e

H E

i

76845213

74865213

no

I

76485213

7

e

74865271

i

78465231

784613

74762531

I

78462531

7846235T

Theorem

Knuth

si

Te

Pio

p

z

22

free module with basis 2

the tableaux

I

module

u

v uno

Theorem

inherited 25

1 22

product

PR

s

coproduct

by

6

1

Pt

permutations

Dbs

22

s

non commutative

not free associative algebra

22

i another Hopf structure

ter

lftp.foplaetic

riniti

class of t

Ex

t

raw

f

312

l

312 1132

312

132

Description of profits

via

jeudetaqu.in

c products

i

backward slides

Homomorphisms

• i

Pio

surjective Hopfmorphism

• f

s

d

in 122

d

Gym

s

122

d

si

Schurfunction

E

t

shh 1

ev

t

Schnitzerbergh evacuation

• f

tet

is

an

anti automorphism of

both Hopf algebras of tableaux

Primitive elements of 22

Weak order of

tableaux

Melnikov

2004

Ì Daft

• rder

E

weak M

Ta skin 2013

Jef

A

v

tableaux

A

U

3 nitido

dai

fa fu permutations

A

d

do ER

da

a Pan

dm

no

verra

The week

• rder
• n

Tu

µ

2,3

4,5

Lemma Plachi equivalence

is compatible with

un

n

ri

un

ri

n

Product

• n

tableaux

PIU

a

P

v

Pin

Pisa

homomorfism of the monoids

S and T

A simpler way to compute

• n

tableaux

EX

tetta

a

V

tata

ftp.go

µ

3

2

Lemma

The

weak order on tableaux

is

compatible with

VE

li

VE li v

u

Ev

li

Recall for permutations

Lemma For nep

9

6

E

v u

Gesu

Esp

ne so

a

• 11 pl

a v and

ben

b stf liti

int

LemmaFornaio.io

ftp

ea

EeTnveTpueTo

E E

V

u

A

E 11

p

BESTIE fin

nl

A

E V and B

EU

define

a

Aguiar Sottile

new linear basis

method

Me

ero

for 22

via Morbius inversion

in the poset of tableaux

2MW

EW

Theorem

SU

1 1

M

C R

E

v

u u

Corollary

The sub

moduleof the primitive

zz

elements

• f

is spanned by the M

such that

2

E

is indecomposablefor

i e

the generators of the free montai'd

22

Ex

FÉTTE

i

i i i Thai

decomprosable

indecomposatole

Final

remarks

Fa

s

right

shifted concatenation

left shifted concatenation

is compatible with

z

is

a monoid

A simpler way to compute

• n

tableaux

Ex

mai

te

i

Venire

i

mira

v

vomitasti

Ti

Theorem

a

v

6

u

ve 5

Loday Romeo

a

verso

a

2002

siete

v.v

a

intervaloftheweek

B

art of S

Theorem

Task.in

u

Et

v

vero

2005

U

e te v

v

t

tableau

in the interval of the

Task

in Buffo order of 2

Theorem In the linear basis

No

6 ES

a S

the structure constants

are positive

No

µ

ciMg ciao

A counterexample

Franco salio la

UQAM

Grazie

per

l'attenzione

Mao

ti

MATE

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3

te

n'Était

Mao

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mate

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tra i Mali

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3

Emily

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end

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1 3 2

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