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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)