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We define oriented Temperley–Lieb algebras for Hermitian symmetric spaces. This allows us to explain the existence of closed combinatorial formulae for the Kazhdan–Lusztig polynomials for these spaces.

We show a precise formula, in the form of a monomial, for certain families of parabolic Kazhdan–Lusztig polynomials of the symmetric group. The proof stems from results of Lapid–Mínguez on irreducibility of products in the Bernstein–Zelevinski ring. By quantizing those results into a statement on quantum groups and their canonical bases, we obtain identities of coefficients of certain transition matrices that relate Kazhdan–Lusztig polynomials to their parabolic analogues. This affirms some basic cases of conjectures raised recently by Lapid.

Wir entwickeln eine Strategie zum Beweis der Positivität der Koeffizienten von Kazhdan-Lusztig-Polynomen für beliebige Coxeter-Gruppen.

We develop a strategy to prove the positivity of coefficients of Kazhdan–Lusztig polynomials for arbitrary Coxeter groups.

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Oriented Temperley–Lieb algebras and combinatorial Kazhdan–Lusztig theory

AN IDENTITY OF PARABOLIC KAZHDAN–LUSZTIG POLYNOMIALS ARISING FROM SQUARE-IRREDUCIBLE MODULES

KAZHDAN-LUSZTIG-POLYNOME UND UNZERLEGBARE BIMODULN ÜBER POLYNOMRINGEN

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3 results

Oriented Temperley–Lieb algebras and combinatorial Kazhdan–Lusztig theory

AN IDENTITY OF PARABOLIC KAZHDAN–LUSZTIG POLYNOMIALS ARISING FROM SQUARE-IRREDUCIBLE MODULES

KAZHDAN-LUSZTIG-POLYNOME UND UNZERLEGBARE BIMODULN ÜBER POLYNOMRINGEN