115 research outputs found
Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices
This paper's aim is to present recent combinatorial considerations on r-Dyck
paths, r-Parking functions, and the r-Tamari lattices. Giving a better
understanding of the combinatorics of these objects has become important in
view of their (conjectural) role in the description of the graded character of
the Sn-modules of bivariate and trivariate diagonal coinvariant spaces for the
symmetric group.Comment: 36 pages, 12 figure
Tensorial square of the Hyperoctahedral group Coinvariant Space
The purpose of this paper is to give an explicit description of the trivial
and alternating components of the irreducible representation decomposition of
the bigraded module obtained as the tensor square of the coinvariant space for
hyperoctahedral groups.Comment: 27 page
Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables
We analyze the structure of the algebra N of symmetric polynomials in
non-commuting variables in so far as it relates to its commutative counterpart.
Using the "place-action" of the symmetric group, we are able to realize the
latter as the invariant polynomials inside the former. We discover a tensor
product decomposition of N analogous to the classical theorems of Chevalley,
Shephard-Todd on finite reflection groups.Comment: 14 page
New Formulas and Conjectures for the Nabla Operator
The operator nabla, introduced by Garsia and the author, plays a crucial role
in many aspect of the study of diagonal harmonics. Besides giving several new
formulas involving this operator, we show how one is lead to representation
theoretic explanations for conjectures about the effect of this operator on
Schur functions.Comment: 13 page
Multivariate Diagonal Coinvariant Spaces for Complex Reflection Groups
For finite complex reflexion groups, we consider the graded -modules of
diagonally harmonic polynomials in sets of variables, and show that
associated Hilbert series may be described in a global manner, independent of
the value of .Comment: 12 pages, Removed a (wrong) conjecture, and reformulated in
agreement. Also cleared up section on low degree term
Some remarkable new Plethystic Operators in the Theory of Macdonald Polynomials
In the 90's a collection of Plethystic operators were introduced in [3], [7]
and [8] to solve some Representation Theoretical problems arising from the
Theory of Macdonald polynomials. This collection was enriched in the research
that led to the results which appeared in [5], [6] and [9]. However since some
of the identities resulting from these efforts were eventually not needed, this
additional work remained unpublished. As a consequence of very recent
publications [4], [11], [19], [20], [21], a truly remarkable expansion of this
theory has taken place. However most of this work has appeared in a language
that is virtually inaccessible to practitioners of Algebraic Combinatorics.
Yet, these developments have led to a variety of new conjectures in [2] in the
Combinatorics and Symmetric function Theory of Macdonald Polynomials. The
present work results from an effort to obtain in an elementary and accessible
manner all the background necessary to construct the symmetric function side of
some of these new conjectures. It turns out that the above mentioned
unpublished results provide precisely the tools needed to carry out this
project to its completion
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