Schubert polynomials Functors asked by Lascoux

Issai Schur’s dissertation (Berlin, 1901): classification of irreducible polynomial representations of GLn: Homomorphisms GLn → GLN sending X to a matrix [Pij(X)], where Pij is a polynomial in the entries of X. Two actions on E⊗n (E vector space over a field K of char. 0). – of the symmetric group Sn via permutations of the factors, – the diagonal action of GL(E). Irreductible representations Sλ of the symmetric group Sn are labeled by partitions of n. Piotr Pragacz (IM PAN, Warszawa) joint with Witold Kraśkiewicz with results of Masaki WatanabeOn certain family of B-modules Schur functors Schubert polynomials Functors asked by Lascoux Filtrations of weight modules Partition of n: λ = (λ1 ≥ · · · ≥ λk ≥ 0) s.t. λ1 + · · ·+ λk = n. Graphical presentation for 8742: Schur module: Vλ(E): = HomZ[Sn](S λ,E⊗n) Vλ(−) is a functor: if E,F are R-modules, f: E → F is an R-homomorphism, then f induces an R-homomorphism Vλ(E) → Vλ(F). In this way, we get all irreducible polynomial representations of GLn. Piotr Pragacz (IM PAN, Warszawa) joint with Witold Kraśkiewicz with results of Masaki WatanabeOn certain family of B-modules Schur functors Schubert polynomials Functors asked by Lascoux Filtrations of weight modules Let us label the boxes of the diagram with 1,..., n