Results 1  10
of
85
Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
 J. Amer. Math. Soc
"... The Hilbert scheme of points in the plane Hn = Hilb n (C2) is an algebraic variety which parametrizes finite subschemes S of length n in C2. To each such subscheme S corresponds an nelement multiset, or unordered ntuple with possible repetitions, σ(S) =[P1,...,Pn] of points in C2,wherethePiare the ..."
Abstract

Cited by 105 (4 self)
 Add to MetaCart
The Hilbert scheme of points in the plane Hn = Hilb n (C2) is an algebraic variety which parametrizes finite subschemes S of length n in C2. To each such subscheme S corresponds an nelement multiset, or unordered ntuple with possible repetitions, σ(S) =[P1,...,Pn] of points in C2,wherethePiare the points of S, repeated with
Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
 Invent. Math
, 2001
"... Abstract. In an earlier paper [13], we showed that the Hilbert scheme of points in the plane Hn = Hilb n (C 2) can be identified with the Hilbert scheme of regular orbits C 2n //Sn. Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspond ..."
Abstract

Cited by 62 (2 self)
 Add to MetaCart
Abstract. In an earlier paper [13], we showed that the Hilbert scheme of points in the plane Hn = Hilb n (C 2) can be identified with the Hilbert scheme of regular orbits C 2n //Sn. Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author in [9]. In particular we prove that the dimension of the space of diagonal harmonics is equal to (n + 1) n−1. 1.
Ysystems and generalized associahedra
 Ann. of Math
"... Root systems and generalized associahedra 1 Root systems and generalized associahedra 3 ..."
Abstract

Cited by 61 (8 self)
 Add to MetaCart
Root systems and generalized associahedra 1 Root systems and generalized associahedra 3
A combinatorial formula for the characters of the diagonal coinvariants
 Duke Math. J
"... Abstract. Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of Rn as a doublygraded Sn module can be expressed using the Frobenius characteristic map as ∇en, where en is the nth elementary symmetric function, and ∇ is an operator f ..."
Abstract

Cited by 48 (15 self)
 Add to MetaCart
Abstract. Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of Rn as a doublygraded Sn module can be expressed using the Frobenius characteristic map as ∇en, where en is the nth elementary symmetric function, and ∇ is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for ∇en and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc and Thibon. We also show that a variety of earlier conjectures and theorems on ∇en are special cases of our conjecture. Finally, we extend our conjectures on ∇en and several of the results supporting them to higher powers ∇men. 1.
The cyclic sieving phenomenon
 J. Combin. Theory Ser. A
"... Abstract. The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridge’s q = −1 phenomenon. The phenomenon is shown to appear in various situations, involving qbinomial coefficients, PólyaRedfield theory, polygon dissections, n ..."
Abstract

Cited by 43 (11 self)
 Add to MetaCart
Abstract. The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridge’s q = −1 phenomenon. The phenomenon is shown to appear in various situations, involving qbinomial coefficients, PólyaRedfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field qanalogues. 1.
Identities and Positivity Conjectures for Some Remarkable Operators
 in the Theory of Symmetric Functions, Methods and Applications of Analysis
, 1999
"... Abstract. Let Jµ[X; q, t] be the integral form of the Macdonald polynomial and set ˜ Hµ[X; q, t] = t n(µ) Jµ[X/(1 − 1/t); q, 1/t], where n(µ) = ∑ i (i − 1)µi. This paper focusses on the linear operator ∇ defined by setting ∇ ˜ Hµ = tn(µ) qn(µ ′) Hµ. ˜ This operator occurs naturally in the study of ..."
Abstract

Cited by 39 (14 self)
 Add to MetaCart
Abstract. Let Jµ[X; q, t] be the integral form of the Macdonald polynomial and set ˜ Hµ[X; q, t] = t n(µ) Jµ[X/(1 − 1/t); q, 1/t], where n(µ) = ∑ i (i − 1)µi. This paper focusses on the linear operator ∇ defined by setting ∇ ˜ Hµ = tn(µ) qn(µ ′) Hµ. ˜ This operator occurs naturally in the study of the GarsiaHaiman modules Mµ. It was originally introduced by the first two authors to give elegant expressions to Frobenius characteristics of intersections of these modules (see [3]). However, it was soon discovered that it plays a powerful and ubiquitous role throughout the theory of Macdonald polynomials. Our main result here is a proof that ∇ acts integrally on symmetric functions. An important corollary of this result is the Schur integrality of the conjectured Frobenius characteristic of the Diagonal Harmonic polynomials [11]. Another curious aspect of ∇ is that it appears to encode a q, tanalogue of Lagrange inversion. In particular, its specialization at t = 1 (or q = 1) reduces to the qanalogue of Lagrange inversion studied by Andrews [1], Garsia [7] and Gessel [17]. We present here a number of positivity conjectures that have emerged in the few years since ∇ has been discovered. We also prove a number of identities in support of these conjectures and state some of the results that illustrate the power of ∇ within the Theory of Macdonald polynomials.
The Flag Major Index and Group Actions on Polynomial Rings
 EUROP. J. COMBIN
, 2001
"... A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra. ..."
Abstract

Cited by 35 (7 self)
 Add to MetaCart
A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.
Conjectured statistics for the (q, t)Catalan numbers
 Adv. Math
"... � � � We introduce the distribution function Fn(q, t) of a pair of statistics on Catalan words and conjecture Fn(q, t) equals Garsia and Haiman’s q, tCatalan sequence Cn(q, t), which they defined as a sum of rational functions. We show that Fn,s(q, t), defined as the sum of these statistics restri ..."
Abstract

Cited by 30 (8 self)
 Add to MetaCart
� � � We introduce the distribution function Fn(q, t) of a pair of statistics on Catalan words and conjecture Fn(q, t) equals Garsia and Haiman’s q, tCatalan sequence Cn(q, t), which they defined as a sum of rational functions. We show that Fn,s(q, t), defined as the sum of these statistics restricted to Catalan words ending in s ones, satisfies a recurrence relation. As a corollary we are able to verify that Fn(q, t) = Cn(q, t) when t = 1/q. We also show the partial symmetry relation Fn(q, 1) = Fn(1,q). By modifying a proof of Haiman of a qLagrange inversion formula based on results of Garsia and Gessel, we obtain a qanalogue of the general Lagrange inversion formula which involves Catalan words grouped according to the number of ones at the end of the word. 1.
Parking Functions and Noncrossing Partitions
 Electronic J. Combinatorics
, 1997
"... this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions ..."
Abstract

Cited by 29 (5 self)
 Add to MetaCart
this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions