Jan 20051

TBATBA

Nantel Bergeron (York University) CRC in mathematics

Jan 20052

TTotally interesting otally interesting B Bi -i - A Algebraslgebras

Nantel Bergeron (York University) CRC in mathematics

M. Aguiar, J.C. Aval, F. Bergeron, F. Hivert, C. Hohlweg, C. Reutenauer, M. Rosas, F. Sottile, J.Y. Thibon, M. Zabrocki, ...

3

Jan 2005

Symn

NCSymn NCQSymn Q ‹x1, x2,..., xn›

QSymn Q[x1, x2,..., xn]

outline of my talkNon-commutative TL invariants

Bergeron-Zabrocki

The Ring of SymmetricPolynomials (Sn-invariants)

Temperley-Lieb invariantsHivert

Non-commutative symmetric invariants

Wolf, Rosas/Sagan, BRRZ

4

Jan 2005

Sym

NCSym NCQSym

QSym

Q ‹x1, x2,..., xn›

Q[x1, x2,..., xn]

outline of my talk

Hopf algebras

n

5

Jan 2005

DSymn

DNCSymn DNCQSymn Q‹x1,..., xn;y1,..., yn ›

DQSymn Q[x1,..., xn;y1,..., yn]

Symn

NCSymn NCQSymn Q ‹x1, x2,..., xn›

QSymn Q[x1, x2,..., xn]

n!

quotient: n! Cn

outline of my talk

Sn-covariantsTemperley-Lieb covariants

very interesting [BRRZ]

6

Jan 2005

DSym DQSym Q[x1,..., xn;y1,..., yn]

Sym QSym Q[x1, x2,..., xn]

n!

n! Cn

(n+1)n-1

outline of my talk

Diagonally Sn-covariantsHaiman and others...

Diagonally Temperley Lieb covariants Aval Bergeron Bergeron

7

Jan 2005

Sym QSym

outline of my talkn

Grothendick Hopf Algebra of theRepresentation: representations of all symmetric

groups

Geometry: Cohomology Hopf algebra of theequivariant Grassmanians

8

Jan 2005

Sym QSym

outline of my talkn

Grothendick Hopf Algebra of theRepresentation: representations of all Hecke

algebras at q=0

Geometry: ????

9

Jan 2005

DNSym

SSym

DSym

DNCSym DNCQSym

DQSym

Sym

NCSym NCQSym

QSym

outline of my talkn

NSym

D D

Jan 200510

Sym: Symmetric Polynomials

• Action of symmetric group on polynomials

.P(x1, x2, ..., xn) = P(x(1), x(2), ..., x(n))

• The Ring of Symmetric polynomials

Sym = { P(X) : .P =P }

X = x1, x2, ..., xn

Symmetric group polynomial invariantsform a ring since .(PQ) = (.P)(.Q)

Jan 200511

Some Bases for Sym

• Elementary symmetric polynomials: e(X)

• Schur symmetric polynomials: s(X)

e = e1e2

... ek and ∑ ei(X) ti = ∏ (1 + xit)

• Monomial symmetric polynomials: m(X)

m(X) = ∑ X orbit of X

= x1 x2 ... xn

1 2 n

Sym = Q[e1, e2,..., en] Newton

Jan 200512

• Compositions = (1, 2,..., k), i > 0 and k = () ≥ 0.

• Monomials

X I = xi1 xi2

xik I = {i1 < i2 < L < ik }

example:

x2 x

3 x

5 I = {2, 3, 5} and = (3, 1, 4)

 

Hivert’s Action

1 2 k

3 1 4

Jan 200513

• Hivert’s action on monomials

.XI = XI

Hivert’s Action

• Orbits of a monomial under this action for a fixed composition

{ XI : |I| = ( ) }

Jan 200514

QSym: Quasi-symmetric polynomials

• Monomial quasi-symmetric polynomial indexed by

M(X) = ∑ XI I {1, 2, ..., n} | I | = ( )

• Hivert’s Action on monomial (linear but not multiplicative)

.XI = XI

• The ring of Quasi-symmetric polynomials

QSym = { P(X) : .P =P }

Why would this be a ring?It is, but one need to check that independently.

Jan 200515

Temperley-Lieb polynomials invariants

Hivert’s action on monomials .XI = XI

In the symmetric group algebra QSn consider the elements

Ei,j,k = Id - (i j) - (i k) - (k j) + (i j k) + (i k j)

Ei,j,k . XI = 0

The kernel of Hivert’s action: ker = Ei,j,k QSn

QSn / ker = TLn Temperley-Lieb Algebra. (spanned by 321-avoiding permutations)

QSymn = Q [x1, x2, ... , xn]TLn

so far, this is a vectorspace... but it is closedunder multiplication!

Jan 200516

Sn - covariants

R = Q [x1, x2, ... , xn]/ h1, h2, ... , hn

h1(x1, x2, ... , xn) = x1 + x2 + ... + xn

hk(x1, x2, ... , xn) = x1 hk-1(x1, x2, ... , xn) + hk(x2, x3, ... , xn)

(*) hk(x2, x3, ... , xn) = hk(x1, x2, ... , xn) - x1 hk-1(x1, x2, ... , xn)

If k > 1, then hk(x2, x3, ... , xn) is in h1, h2, ... , hn . Repeating (*) we get

h1, h2, ... , hn = hk(xk, xk+1, ... , xn) : 1 ≤ k ≤ n

hk(xk, xk+1, ... , xn) = xkk + lower lex-term

Jan 200517

Sn - covariants

R = Q [x1, x2, ... , xn]/ h1, h2, ... , hn

h1, h2, ... , hn = hk(xk, xk+1, ... , xn) : 1 ≤ k ≤ n

hk(xk, xk+1, ... , xn) = xkk + lower lex-term

xkk lower terms mod R

Basis of R is given by: Bn = { x1 x2 ... xn : 0 ≤ k < k } 1

2

n

dim(R) = n!

Jan 200518

Sn - covariants

R = Q [x1, x2, ... , xn]/ h1, h2, ... , hn

Basis of R is given by: Bn = { x1 x2 ... xn : 0 ≤ k < k } 1

2

n

dim(R) = n!

n

k

1

Jan 200519

TLn - covariants

R = Q [x1, x2, ... , xn]/ F

x1 F 1-1, 2,...,k(x1, x2, ... , xn) + F(x2, x3, ... , xn) if 1 > 1

F(X) = x1 F 2,...,k

(x2, x3, ... , xn) + F(x2, x3, ... , xn) if 1 = 1

• R is a graded space.• Explicit description of a Gröbner basis of this ideal.• Explicit monomial basis of the quotient: Xc, c Dyck path • Dimension:

1n+1 ( )2n

n= dim (TLn)

Aval-Bergeron-Bergeron

Jan 200520

Weight on paths

x2

x4 x4

x6

cX :=x2x42x6

Its weight:

1

2

3

4

5

6

A Dick path c from (0,1) to (n,n+1)

Jan 200521

TLn - covariants

R = Q [x1, x2, ... , xn]/ F

• Dimension:

1n+1 ( )2n

n= dim (TLn) = number of 321-avoiding permutations

Aval-Bergeron-Bergeron

Theorem

{Xc | c Dyck path } is a basis of R

Jan 200522

TLn - covariants

R = Q [x1, x2, ... , xn]/ F

Aval-Bergeron-Bergeron

Open Problem: Find an action of TL on R?

Study the underlines geometry?

Jan 200523

Interesting Properties of QSym

• Temperley-Lieb invariants: QSym = Q[x1, x2, ..., xn]TLn

• Temperley-Lieb “covariants”

dim(TLn) = dim(Q[x1, x2, ..., xn] / ‹QSym+› )

• Universal properties and much more...

• Projective representation of Hn(0) [Hecke Algebra at q=0]

[Hivert]

[ABB]

[Krob Thibon]

[Aguiar Bergeron Sottile]

• Geometry ????

24

Jan 2005

DSym DQSym Q[x1,..., xn;y1,..., yn]

Sym QSym Q[x1, x2,..., xn]

Jan 200525

Bi-compositions

= ( ) where ai + bi > 0

Bi-compositions and Monomials

a1 a2 ... ak

b1 b2 ... bk

Monomials

XYI = xi1

xi2 xik

yi1 yi2

yik a1 a2 ak b1 b2 bk

example:

x2 x

3 y

2 y

53 1 2 4 I = {2, 3, 5} and = ( ) 3 1 0

2 0 4

Jan 200526

Diagonal actions of Symmetric group

• Classical diagonal action of symmetric group on polynomials

.P(x1,..., xn; y1, ..., yn) = P(x(1), ..., x(n); y(1), ..., y(n))

DSym = { P(X; Y) : .P =P } = Q[x1,..., xn;y1,..., yn] QSn

• Hiver’s diagonal action of symmetric group on polynomials

. XYI = XY.I

DQSym = { P(X; Y) : .P =P } = Q[x1,..., xn;y1,..., yn] TLn

[Aval Bergeron Bergeron]

Jan 200527

Dn := Q[x1, x2, ... , xn ;y1, … , yn]/< DQSym+ >

degree in q

0

n-1

degree in t0 n-1

+

[Aval Bergeron Bergeron]

Diagonally TL-covariants

Conjectured bigraded Hilbert series:

Jan 200528

. x4y4

. x4

. y4

Start withbasis for n=3

Build

Dn := Q[x1, x2, ... , xn ;y1, … , yn]/< DQSym+ >

[Aval Bergeron Bergeron]

Diagonally TL-covariants

Conjectured explicit monomial basis: for example to build for n=4 and bidegree (1,1)