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TBA

TBA. Nantel Bergeron (York University) CRC in mathematics. T otally interesting B i - A lgebras. 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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TBA

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  1. TBA Nantel Bergeron (York University) CRC in mathematics

  2. 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, ...

  3. Q‹x1, x2,..., xn› NCSymn NCQSymn Q[x1, x2,..., xn] QSymn 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

  4. Q‹x1, x2,..., xn› NCSym NCQSym Q[x1, x2,..., xn] Sym QSym outline of my talk Hopf algebras n  

  5. Q‹x1,..., xn;y1,..., yn› DNCSymn DNCQSymn very interesting [BRRZ] Q[x1,..., xn;y1,..., yn] DSymn DQSymn Q‹x1, x2,..., xn› NCSymn NCQSymn Q[x1, x2,..., xn] QSymn outline of my talk Symn n! Cn quotient: Temperley-Lieb covariants n! Sn-covariants

  6. Q[x1, x2,..., xn] QSym outline of my talk Diagonally Sn-covariants Haiman and others... (n+1)n-1 Diagonally Temperley Lieb covariants Aval Bergeron Bergeron Q[x1,..., xn;y1,..., yn] DSym DQSym Sym n! Cn n!

  7. QSym 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

  8. QSym outline of my talk n   Grothendick Hopf Algebra of the Representation: representations of all Hecke algebras at q=0 Geometry: ???? Sym

  9. DNCSym DNCQSym NCSym NCQSym DSym DQSym QSym outline of my talk n   DNSym NSym SSym D D Sym  

  10. 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)

  11. Some Bases for Sym • Monomial symmetric polynomials: ml(X) ml(X) = ∑ X orbit ofX l = x1 x2 ... xn    l1 l2 ln • Elementary symmetric polynomials: el(X) el = el1el2 ... elk and ∑ ei(X) ti = ∏ (1 + xit) Sym = Q[e1, e2,..., en] Newton • Schur symmetric polynomials: sl(X)

  12. Hivert’s Action • Compositions a = (a1,a2,...,ak), ai> 0 and k = (a) ≥ 0. • Monomials XI = xi1xi2xikI = {i1 < i2 < L < ik } example: x2x3x5I = {2, 3, 5} and a = (3, 1, 4) a a1 a2 ak 3 1 4

  13. Hivert’s action on monomials s.XI = Xs.I Hivert’s Action a a • Orbits of a monomial under this action for a fixed composition a: { XI : |I| = (a ) } a

  14. Why would this be a ring? It is, but one need to check that independently. QSym: Quasi-symmetric polynomials • Monomial quasi-symmetric polynomial indexed by a Ma(X) = ∑XI I Í {1, 2, ..., n} | I | =(a ) • Hivert’s 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

  15. a a s.XI = Xs.I a Ei,j,k . XI = 0 Temperley-Lieb polynomials invariants Hivert’s 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 Hivert’s action: ker =  Ei,j,k   QSn QSn / ker= TLnTemperley-Lieb Algebra. (spanned by 321-avoiding permutations) so far, this is a vector space... but it is closed under multiplication! QSymn = Q [x1, x2, ... , xn]TLn

  16. R = Q [x1, x2, ... , xn]/  h1,h2,... , hn  Sn - covariants h1(x1, x2, ... , xn) = x1 + x2 + ... +xn hk(x1, x2, ... , xn) = x1hk-1(x1, x2, ... , xn) + hk(x2, x3, ... , xn) (*) hk(x2, x3, ... , xn) = hk(x1, x2, ... , xn) - x1hk-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

  17. R = Q [x1, x2, ... , xn]/  h1,h2,... , hn  Basis of R is given by: Bn = { x1x2 ... xn : 0 ≤ k < k } 1 2 n Sn - covariants  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 dim(R) = n!

  18. R = Q [x1, x2, ... , xn]/  h1,h2,... , hn  Basis of R is given by: Bn = { x1x2 ... xn : 0 ≤ k < k } 1 2 n Sn - covariants n k dim(R) = n! 1

  19. R = Q [x1, x2, ... , xn]/  F  x1F 1-1, 2,...,k(x1, x2, ... , xn) + F(x2, x3, ... , xn) if 1 > 1 F(X) = x1F 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: = dim (TLn) ( ) 1 n+1 2n n TLn - covariants Aval-Bergeron-Bergeron

  20. X :=x2x42x6 c Weight on paths A Dick path c from (0,1) to (n,n+1) x6 6 Its weight: 5 x4 x4 4 3 x2 2 1

  21. R = Q [x1, x2, ... , xn]/  F  • Dimension: = dim (TLn) = number of 321-avoiding permutations ( ) 1 n+1 2n n TLn - covariants Aval-Bergeron-Bergeron Theorem {Xc | c Dyck path } is a basis of R

  22. R = Q [x1, x2, ... , xn]/  F  TLn - covariants Aval-Bergeron-Bergeron Open Problem: Find an action of TL on R? Study the underlines geometry?

  23. Interesting Properties of QSym [Hivert] • Temperley-Lieb “covariants” dim(TLn) = dim(Q[x1, x2, ..., xn] /‹QSym+›) • Temperley-Lieb invariants: QSym = Q[x1, x2, ..., xn]TLn [ABB] • Projective representation of Hn(0)[Hecke Algebra at q=0] [Krob Thibon] • Universal properties and much more... [Aguiar Bergeron Sottile] • Geometry ????

  24. Q[x1,..., xn;y1,..., yn] DSym DQSym Q[x1, x2,..., xn] QSym Sym

  25. XYI = xi1xi2xikyi1yi2yik a1a2ak b1b2bk I = {2, 3, 5} and a = ( ) 3 1 0 2 0 4 x2x3y2y5 3 1 2 4 Bi-compositions and Monomials • Bi-compositions • = ( ) where ai+ bi> 0 a1a2 ... akb1b2 ... bk Monomials example:

  26. Hiver’s diagonal action of symmetric group on polynomials s. XYI = XYs.I DQSym = { P(X; Y) : s.P =P } = Q[x1,..., xn;y1,..., yn] TLn  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 } = Q[x1,..., xn;y1,..., yn] QSn [Aval Bergeron Bergeron]

  27. Diagonally TL-covariants Dn := Q[x1, x2, ... , xn ;y1, … , yn]/< DQSym+ > [Aval Bergeron Bergeron] Conjectured bigraded Hilbert series: + degree in q n-1 0 degree in t 0 n-1

  28. Diagonally TL-covariants Dn := Q[x1, x2, ... , xn ;y1, … , yn]/< DQSym+ > [Aval Bergeron Bergeron] Conjectured explicit monomial basis: for example to build for n=4 and bidegree (1,1) Start withbasis forn=3 . x4 . x4y4 . y4 Build

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