2,621 research outputs found
Effective Invariant Theory of Permutation Groups using Representation Theory
Using the theory of representations of the symmetric group, we propose an
algorithm to compute the invariant ring of a permutation group. Our approach
have the goal to reduce the amount of linear algebra computations and exploit a
thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at
http://www.springer.com
CHAMP: A Cherednik Algebra Magma Package
We present a computer algebra package based on Magma for performing
computations in rational Cherednik algebras at arbitrary parameters and in
Verma modules for restricted rational Cherednik algebras. Part of this package
is a new general Las Vegas algorithm for computing the head and the
constituents of a module with simple head in characteristic zero which we
develop here theoretically. This algorithm is very successful when applied to
Verma modules for restricted rational Cherednik algebras and it allows us to
answer several questions posed by Gordon in some specific cases. We could
determine the decomposition matrices of the Verma modules, the graded G-module
structure of the simple modules, and the Calogero-Moser families of the generic
restricted rational Cherednik algebra for around half of the exceptional
complex reflection groups. In this way we could also confirm Martino's
conjecture for several exceptional complex reflection groups.Comment: Final version to appear in LMS J. Comput. Math. 41 pages, 3 ancillary
files. CHAMP is available at http://thielul.github.io/CHAMP/. All results are
listed explicitly in the ancillary PDF document (currently 935 pages). Please
check the website for further update
Rational, Replacement, and Local Invariants of a Group Action
The paper presents a new algorithmic construction of a finite generating set
of rational invariants for the rational action of an algebraic group on the
affine space. The construction provides an algebraic counterpart of the moving
frame method in differential geometry. The generating set of rational
invariants appears as the coefficients of a Groebner basis, reduction with
respect to which allows to express a rational invariant in terms of the
generators. The replacement invariants, introduced in the paper, are tuples of
algebraic functions of the rational invariants. Any invariant, whether
rational, algebraic or local, can be can be rewritten terms of replacement
invariants by a simple substitution.Comment: 37 page
Rational invariants of even ternary forms under the orthogonal group
In this article we determine a generating set of rational invariants of
minimal cardinality for the action of the orthogonal group on
the space of ternary forms of even degree . The
construction relies on two key ingredients: On one hand, the Slice Lemma allows
us to reduce the problem to dermining the invariants for the action on a
subspace of the finite subgroup of signed permutations. On the
other hand, our construction relies in a fundamental way on specific bases of
harmonic polynomials. These bases provide maps with prescribed
-equivariance properties. Our explicit construction of these
bases should be relevant well beyond the scope of this paper. The expression of
the -invariants can then be given in a compact form as the
composition of two equivariant maps. Instead of providing (cumbersome) explicit
expressions for the -invariants, we provide efficient algorithms
for their evaluation and rewriting. We also use the constructed
-invariants to determine the -orbit locus and
provide an algorithm for the inverse problem of finding an element in
with prescribed values for its invariants. These are
the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application,
refinement of Definition 3.1. To appear in "Foundations of Computational
Mathematics
Discrete series representations and K multiplicities for U(p,q). User's guide
This document is a companion for the Maple program : Discrete series and
K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain
an algorithm to compute the multiplicities of an irreducible representation of
U(p)x U(q) in a discrete series of U(p,q). It is based on Blattner's formula.
We recall the general mathematical background to compute Kostant partition
functions via multidimensional residues, and we outline our algorithm. We also
point out some properties of the piecewise polynomial functions describing
multiplicities based on Paradan's results.Comment: 51 page
Recognising the small Ree groups in their natural representations
We present Las Vegas algorithms for constructive recognition and constructive
membership testing of the Ree groups 2G_2(q) = Ree(q), where q = 3^{2m + 1} for
some m > 0, in their natural representations of degree 7. The input is a
generating set X.
The constructive recognition algorithm is polynomial time given a discrete
logarithm oracle. The constructive membership testing consists of a
pre-processing step, that only needs to be executed once for a given X, and a
main step. The latter is polynomial time, and the former is polynomial time
given a discrete logarithm oracle.
Implementations of the algorithms are available for the computer algebra
system MAGMA
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