Results 1  10
of
30
Representation theory and homological stability
, 2010
"... We introduce the idea of representation stability (and several variations) for a sequence of representations Vn of groups Gn. One main goal is to expand the important and wellstudied concept of homological stability so that it applies to a much broader variety of examples. Representation stability ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
(Show Context)
We introduce the idea of representation stability (and several variations) for a sequence of representations Vn of groups Gn. One main goal is to expand the important and wellstudied concept of homological stability so that it applies to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood–Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1) n−1 conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied in [CEF] to counting problems in
Higher Trivariate Diagonal Harmonics via Generalized Tamari Posets
 Journal of Combinatorics
, 2012
"... Abstract. We consider the graded Snmodules of higher diagonally harmonic polynomials in three sets of variables (the trivariate case), and show that they have interesting ties with generalizations of the Tamari poset and parking functions. In particular we get several nice formulas for the associa ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
Abstract. We consider the graded Snmodules of higher diagonally harmonic polynomials in three sets of variables (the trivariate case), and show that they have interesting ties with generalizations of the Tamari poset and parking functions. In particular we get several nice formulas for the associated Hilbert series and graded Frobenius characteristics. This also leads to entirely new combinatorial formulas. Contents
Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules
, 2004
"... ..."
Square q, tlattice paths and ∇(pn)
 TRANS. AMER. MATH. SOC
"... The combinatorial q, tCatalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, tCatalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, al ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
The combinatorial q, tCatalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, tCatalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n’th q, tCatalan number is the Hilbert series for the module of diagonal harmonic alternants in 2n variables; it is also the coefficient of s1n in the Schur expansion of ∇(en). Using q, tanalogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of ∇(en) and the Hilbert series of the diagonal harmonics modules. This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several q, tanalogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the q, tCatalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of ∇(pn), the “Hilbert series” 〈∇(pn), h1n〉, and the sign character 〈∇(pn), s1n〉.
Tensorial square of the hyperoctahedral group coinvariant space
, 2005
"... The purpose of this paper is to give an explicit description of the trivial and alternating component of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups. ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
The purpose of this paper is to give an explicit description of the trivial and alternating component of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups.
A COMPACTLY SUPPORTED FORMULA FOR EQUIVARIANT LOCALIZATION and SIMPLICIAL COMPLEXES OF BIAŁYNICKIBIRULA DECOMPOSITIONS
, 2008
"... The DuistermaatHeckman formula for their induced measure on a moment polytope is nowadays seen as the Fourier transform of the AtiyahBott localization formula, applied to the Tequivariant Liouville class. From this formula one does not see directly that the measure is positive, nor that it vanis ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
The DuistermaatHeckman formula for their induced measure on a moment polytope is nowadays seen as the Fourier transform of the AtiyahBott localization formula, applied to the Tequivariant Liouville class. From this formula one does not see directly that the measure is positive, nor that it vanishes outside the moment polytope. In [Knutson99] we gave a formula for the DuistermaatHeckman measure whose terms are all positive and compactly supported, using a Morse decomposition. Its derivation required that the stable and unstable Morse strata intersect transversely. In this paper, we remove this very restrictive condition, at the cost of working with an “iterated ” Morse (or BiałynickiBirula) decomposition. This leads in a natural way to a simplicial complex of “closure chains”, which in the toric variety case is just a pulling triangulation of the moment polytope. To handle the singularities of the closed strata we restrict to the projective algebraic setting. Conversely, this allows us to work from the beginning with singular projective schemes over algebraically closed ground fields.
MACDONALD POSITIVITY VIA THE HARISHCHANDRA DMODULE
"... Abstract. Using the HarishChandra Dmodule, we give a proof of Haiman’s theorem on the positivity of Macdonald polynomials. Ginzburg’s work on the connection between this Dmodule and the isospectral commuting variety is fundamental to this approach. 1. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. Using the HarishChandra Dmodule, we give a proof of Haiman’s theorem on the positivity of Macdonald polynomials. Ginzburg’s work on the connection between this Dmodule and the isospectral commuting variety is fundamental to this approach. 1.
On the commuting variety of a reductive Lie algebra AND OTHER RELATED VARIETIES.
, 2012
"... In this note, one discusses about some varieties which are constructed analogously to the isospectral commuting varieties. These varieties are subvarieties of varieties having very simple desingularizations. For instance, this is the case of the nullcone of any cartesian power of a reductive Lie a ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
In this note, one discusses about some varieties which are constructed analogously to the isospectral commuting varieties. These varieties are subvarieties of varieties having very simple desingularizations. For instance, this is the case of the nullcone of any cartesian power of a reductive Lie algebra and one proves that it has rational singularities. Moreover, as a byproduct of these investigations and the Ginzburg’s results, one gets that the normalizations of the isospectral commuting variety and the commuting variety have rational singularities.
Combinatorics of rDyck paths, rParking functions, and the rTamari lattices
, 2012
"... This is a surveylike paper presenting some of the recent combinatorial considerations on rDyck paths, rParking functions, and especially the rTamari lattices. Giving a better understanding of the combinatorial interplay between these objects has become important in view of their (conjectural) ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
This is a surveylike paper presenting some of the recent combinatorial considerations on rDyck paths, rParking functions, and especially the rTamari lattices. Giving a better understanding of the combinatorial interplay between these objects has become important in view of their (conjectural) role in the description of the graded character of the Snmodules of bivariate and trivariate diagonal coinvariant spaces for the symmetric group.