72 research outputs found
Deformed diagonal harmonic polynomials for complex reflection groups
We introduce deformations of the space of (multi-diagonal) harmonic
polynomials for any finite complex reflection group of the form W=G(m,p,n), and
give supporting evidence that this space seems to always be isomorphic, as a
graded W-module, to the undeformed version.Comment: 11 pages, 1 figur
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
The biHecke monoid of a finite Coxeter group
The usual combinatorial model for the 0-Hecke algebra of the symmetric group
is to consider the algebra (or monoid) generated by the bubble sort operators.
This construction generalizes to any finite Coxeter group W. The authors
previously introduced the Hecke group algebra, constructed as the algebra
generated simultaneously by the bubble sort and antisort operators, and
described its representation theory.
In this paper, we consider instead the monoid generated by these operators.
We prove that it has |W| simple and projective modules. In order to construct a
combinatorial model for the simple modules, we introduce for each w in W a
combinatorial module whose support is the interval [1,w] in right weak order.
This module yields an algebra, whose representation theory generalizes that of
the Hecke group algebra. This involves the introduction of a w-analogue of the
combinatorics of descents of W and a generalization to finite Coxeter groups of
blocks of permutation matrices.Comment: 12 pages, 1 figure, submitted to FPSAC'1
Spectral gap for random-to-random shuffling on linear extensions
In this paper, we propose a new Markov chain which generalizes
random-to-random shuffling on permutations to random-to-random shuffling on
linear extensions of a finite poset of size . We conjecture that the second
largest eigenvalue of the transition matrix is bounded above by
with equality when the poset is disconnected. This Markov
chain provides a way to sample the linear extensions of the poset with a
relaxation time bounded above by and a mixing time of . We conjecture that the mixing time is in fact as for the
usual random-to-random shuffling.Comment: 16 pages, 10 figures; v2: typos fixed plus extra information in
figures; v3: added explicit conjecture 2.2 + Section 3.6 on the diameter of
the Markov Chain as evidence + misc minor improvements; v4: fixed
bibliograph
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