Some closed formulas for canonical bases of Fock spaces

Abstract

We give some closed formulas for certain vectors of the canonical bases of the Fock space representation of U v ( s l n ) U_v(\mathfrak {sl}_n) . As a result, a combinatorial description of certain parabolic Kazhdan-Lusztig polynomials for affine type A A is obtained.

Euler characteristics of Hilbert schemes of points on simple surface singularities

“…We now introduce some standard combinatorics related to the type A root system, which will allow us to relate the generating series Z ∆ of (n + 1)-labelled partitions to the specialized series Z 0 ∆ of 0-generated partitions. We follow the notations of [29]. The abacus of type A n is the arrangement of the set of integers in (n + 1) columns according to the following pattern.…”

Character formulas and Bernstein–Gelfand–Gelfand resolutions for Cherednik algebra modules

“…A block with tight multiplicities need not be multiplicity-free. The RoCK blocks provide an example of this: [16] and [4] discovered a formula for decomposition numbers for RoCK blocks in terms of Littlewood-Richardson coefficients, and any Littlewood-Richardson coefficient appears as a decomposition number in some RocK block, so RoCK blocks in general are not multiplicity-free. ) has tight multiplicities.…”

“…For µ ≤ λ, let b n (µ, λ) be the number of chains of n arrows from µ to λ in Γ(Hom). Now we define a polynomial in N[v] which keeps track of all chains of arrows from µ to λ in Γ(Hom): 16. By Theorem 4.15, the number of chains of homs of length n from µ to λ is the number of paths in the v-dec matrix starting at (µ, µ), ending at (λ, λ), traveling up and left and bouncing off exactly n nonzero v-dec numbers and n − 1 times off the diagonal.…”

Representations of wreath products of algebras

“…Formula (1) has been independently discovered by Miyachi in [10]; we are following Leclerc-Miyachi's presentation of the formula [6].…”