TBA

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TBA

• Nantel Bergeron (York University) CRC in
  mathematics

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Totally interesting Bi - Algebras

• 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, ...

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outline of my talk
Non-commutative TL invariants Bergeron-Zabrocki
Non-commutative symmetric invariants Wolf,
Rosas/Sagan, BRRZ
Symn
The Ring of Symmetric Polynomials (Sn-invariants)
Temperley-Lieb invariants Hivert
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outline of my talk
Hopf algebras
n ? ?
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outline of my talk
Symn
n!
Cn
quotient
Temperley-Lieb covariants
n!
Sn-covariants
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outline of my talk
Diagonally Sn-covariants Haiman and others...
(n1)n-1
Diagonally Temperley Lieb covariants Aval
Bergeron Bergeron
Qx1,..., xny1,..., yn
DSym
DQSym
Sym
n!
Cn
n!
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outline of my talk
n ? ?
Grothendick Hopf Algebra of the Representation
representations of all symmetric groups
Geometry Cohomology Hopf algebra of
the equivariant Grassmanians
Sym
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outline of my talk
n ? ?
Grothendick Hopf Algebra of the Representation
representations of all Hecke algebras at
q0
Geometry ????
Sym
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outline of my talk
n ? ?
DNSym
SSym
NSym
D?
D?
Sym
?
?
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Sym Symmetric Polynomials

• Action of symmetric group on polynomials
• s.P(x1, x2, ..., xn) P(xs(1), xs(2), ...,
  xs(n))
• The Ring of Symmetric polynomials
• Sym P(X) s.P P
• X x1, x2, ..., xn

Symmetric group polynomial invariants form a ring
since s.(PQ) (s.P)(s.Q)
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Some Bases for Sym

• Monomial symmetric polynomials ml(X)

ml(X) ? X ? orbit of X l x1 x2
... xn ? ? ?
l1 l2 ln

• Elementary symmetric polynomials el(X)

el el1el2 ... elk and ? ei(X) ti ? (1
xit)
Sym Qe1, e2,..., en
Newton

• Schur symmetric polynomials sl(X)

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Hiverts Action

• Compositions
• a (a1, a2,..., ak), ai gt 0 and k
  ?(a) 0.
• Monomials
• X I xi1 xi2 ? xik
  I i1 lt i2 lt L lt ik
• example
• x2 x3 x5 I 2, 3, 5 and
  a (3, 1, 4)
•  

a
a1
a2
ak
3
1
4
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Hiverts Action

• Hiverts action on monomials
• s.XI Xs.I

a
a

• Orbits of a monomial under this action for a
  fixed composition a
• XI I ?(a )

a
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QSym Quasi-symmetric polynomials

• Monomial quasi-symmetric polynomial indexed by a
• Ma(X) ? XI
• I Í 1, 2, ..., n
• I ?(a )
• Hiverts Action on monomial (linear but not
  multiplicative)
• s.XI Xs.I
• The ring of Quasi-symmetric polynomials
• QSym P(X) s.P P

a
a
a
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Temperley-Lieb polynomials invariants
Hiverts action on monomials
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)
The kernel of Hiverts action ker ? Ei,j,k ?
? QSn
QSn / ker TLn Temperley-Lieb Algebra.
(spanned by
321-avoiding permutations)
so far, this is a vector space... but it is
closed under multiplication!
QSymn Q x1, x2, ... , xnTLn
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Sn - covariants
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 gt 1, then hk(x2, x3, ... , xn) is in ? h1,
h2, ... , hn ?. Repeating () we get
? h1, h2, ... , hn ? ? hk(xk, xk1, ... , xn)
1 k n ?
hk(xk, xk1, ... , xn) xkk lower lex-term
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Sn - covariants
? h1, h2, ... , hn ? ? hk(xk, xk1, ... , xn)
1 k n ?
hk(xk, xk1, ... , xn) xkk lower lex-term
xkk ? lower terms mod R
dim(R) n!
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Sn - covariants
n
k
dim(R) n!
1
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TLn - covariants
Aval-Bergeron-Bergeron
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Weight on paths
A Dick path c from (0,1) to (n,n1)
x6
6
Its weight
5
x4
x4
4
3
x2
2
1
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TLn - covariants
Aval-Bergeron-Bergeron
Theorem
Xc c Dyck path is a basis of R
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TLn - covariants
Aval-Bergeron-Bergeron
Open Problem Find an action of TL on R?
Study the underlines geometry?
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Interesting Properties of QSym

• Temperley-Lieb invariants QSym Qx1, x2, ...,
  xnTLn

Hivert

• Temperley-Lieb covariants
• dim(TLn) dim(Qx1, x2, ..., xn / QSym )

ABB

• Projective representation of Hn(0) Hecke
  Algebra at q0

Krob Thibon

• Universal properties and much more...

Aguiar Bergeron Sottile

• Geometry ????

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Sym
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Bi-compositions and Monomials

• Bi-compositions
• ( ) where ai bi gt 0

a1 a2 ... akb1 b2 ... bk
Monomials
example
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Diagonal actions of Symmetric group

• Classical diagonal action of symmetric group on
  polynomials
• s.P(x1,..., xn y1, ..., yn) P(xs(1), ...,
  xs(n) ys(1), ..., ys(n))
• DSym P(X Y) s.P P Qx1,...,
  xny1,..., yn QSn

Aval Bergeron Bergeron
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Dn Qx1, x2, ... , xn y1, , yn/lt DQSym gt
Diagonally TL-covariants
Aval Bergeron Bergeron
Conjectured bigraded Hilbert series

degree in q
n-1
0
degree in t
0
n-1
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Diagonally TL-covariants
Dn Qx1, x2, ... , xn y1, , yn/lt DQSym gt
Aval Bergeron Bergeron
Conjectured explicit monomial basis for
example to build for n4 and bidegree (1,1)
Start withbasis for n3
. x4
. x4y4
. y4
Build