1,880 research outputs found
Minimal generating sets of non-modular invariant rings of finite groups
It is a classical problem to compute a minimal set of invariant polynomial
generating the invariant ring of a finite group as an algebra. We present here
an algorithm for the computation of minimal generating sets in the non-modular
case. Apart from very few explicit computations of Groebner bases, the
algorithm only involves very basic operations, and is thus rather fast.
As a test bed for comparative benchmarks, we use transitive permutation
groups on 7 and 8 variables. In most examples, our algorithm implemented in
Singular works much faster than the one used in Magma, namely by factors
between 50 and 1000. We also compute some further examples on more than 8
variables, including a minimal generating set for the natural action of the
cyclic group of order 11 in characteristic 0 and of order 15 in characteristic
2.
We also apply our algorithm to the computation of irreducible secondary
invariants.Comment: 14 pages v3: Timings updated. One example adde
Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras
We study the representation theory of three towers of algebras which are
related to the symmetric groups and their Hecke algebras. The first one is
constructed as the algebras generated simultaneously by the elementary
transpositions and the elementary sorting operators acting on permutations. The
two others are the monoid algebras of nondecreasing functions and nondecreasing
parking functions. For these three towers, we describe the structure of simple
and indecomposable projective modules, together with the Cartan map. The
Grothendieck algebras and coalgebras given respectively by the induction
product and the restriction coproduct are also given explicitly. This yields
some new interpretations of the classical bases of quasi-symmetric and
noncommutative symmetric functions as well as some new bases.Comment: 12 pages. Presented at FPSAC'06 San-Diego, June 2006 (minor
explanation improvements w.r.t. the previous version
Some relational structures with polynomial growth and their associated algebras II: Finite generation
The profile of a relational structure is the function which
counts for every integer the number, possibly infinite, of
substructures of induced on the -element subsets, isomorphic
substructures being identified. If takes only finite values, this
is the Hilbert function of a graded algebra associated with , the age
algebra , introduced by P.~J.~Cameron.
In a previous paper, we studied the relationship between the properties of a
relational structure and those of their algebra, particularly when the
relational structure admits a finite monomorphic decomposition. This
setting still encompasses well-studied graded commutative algebras like
invariant rings of finite permutation groups, or the rings of quasi-symmetric
polynomials.
In this paper, we investigate how far the well know algebraic properties of
those rings extend to age algebras. The main result is a combinatorial
characterization of when the age algebra is finitely generated. In the special
case of tournaments, we show that the age algebra is finitely generated if and
only if the profile is bounded. We explore the Cohen-Macaulay property in the
special case of invariants of permutation groupoids. Finally, we exhibit
sufficient conditions on the relational structure that make naturally the age
algebra into a Hopf algebra.Comment: 27 pages; submitte
On the theta number of powers of cycle graphs
We give a closed formula for Lovasz theta number of the powers of cycle
graphs and of their complements, the circular complete graphs. As a
consequence, we establish that the circular chromatic number of a circular
perfect graph is computable in polynomial time. We also derive an asymptotic
estimate for this theta number.Comment: 17 page
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