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A normal form algorithm for the Brieskorn lattice
This article describes a normal form algorithm for the Brieskorn lattice of
an isolated hypersurface singularity. It is the basis of efficient algorithms
to compute the Bernstein-Sato polynomial, the complex monodromy, and
Hodge-theoretic invariants of the singularity such as the spectral pairs and
good bases of the Brieskorn lattice. The algorithm is a variant of Buchberger's
normal form algorithm for power series rings using the idea of partial standard
bases and adic convergence replacing termination.Comment: 23 pages, 1 figure, 4 table
q-Analogs of symmetric function operators
For any homomorphism V on the space of symmetric functions, we introduce an
operation which creates a q-analog of V. By giving several examples we
demonstrate that this quantization occurs naturally within the theory of
symmetric functions. In particular, we show that the Hall-Littlewood symmetric
functions are formed by taking this q-analog of the Schur symmetric functions
and the Macdonald symmetric functions appear by taking the q-analog of the
Hall-Littlewood symmetric functions in the parameter t. This relation is then
used to derive recurrences on the Macdonald q,t-Kostka coefficients.Comment: 17 pages - minor revisions to appear in Discrete Mathematics issue
for LaCIM'200
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