Some closed formulas for canonical bases of Fock spaces, Representation Theory (Electronic：（一些福克公式规范基地关闭空间,代表(电子理论）.pdfVIP

Some closed formulas for canonical bases 1 of Fock spaces 0 0 2 Bernard LECLERC and Hyohe MIYACHI y April 2001 a M 1 1 ] Abstract A Q We give some closed formulas for certain vectors of the canonical bases of the Fock space rep- resentation of U (sl ). As a result, a combinatorial description of certain parabolic Kazhdan- . v n h Lusztig polynomials for affine type A is obtained. t a m [ 1 Introduction 2 v Let F be the Fock space representation of U (sl ) introduced by Hayashi [H] and further studied 7 v v n 0 by Misra and Miwa [MM], Stern [St], and Kashiwara, Miwa and Stern [KMS]. It has a standard 1 basis Σ = {s(λ) | λ ∈ P} indexed by the set P of all integer partitions. In [LT1] two canonical 4 bases B = {G(λ) | λ ∈ P} and B− = {G− (λ) | λ ∈ P} of Fv have been constructed. The 0 1 subset of B consisting of the G(λ)’s for which λ is n-regular coincides with Kashiwara’s lower 0 global basis (or Lusztig’s canonical basis) of the irreducible sub-representation of Fv generated / h by the highest weight vector s(∅). t − a The main motivation for introducing the bases B and B was their conjectural relation with m the decomposition matrices of the q-Schur algebras Sm (q) defined by Dipper and James [DJ] in : connection with the modular representation theory of the finite groups GL (F )