Positive Conjecture Research Articles

Tableau atoms and a new Macdonald positivity conjecture

Let $\lambda$ be the space of symmetric functions, and let $V\ sb k$ be the subspace spanned by the modified Schur functions $\{S\sb \lambda[X/(1-t)]\}\sb {\lambda\sb 1\leq k}$. We introduce a new family of symmetric polynomials, $\{A\sp {(k)}\sb \lambda[X;t]\}\sp {\lambda\sb 1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A\sp {(k)}\sb \lambda[X;t]$ form a basis for $V\sb k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but also substantially refine it. Our construction of the $A\sp {(k)}\sb \lambda[X;t]$ relies on the use of tableau combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A\sp {(k)}\sb \lambda[X;t]$ seem to play the same role for $V\sb k$ as the Schur functions do for $\lambda$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri-type and Littlewood-Richardson-type coefficients.

A dimension inequality for Cohen-Macaulay rings

The recent work of Kurano and Roberts on Serre’s positivity conjecture suggests the following dimension inequality: for prime ideals $\mathfrak {p}$ and $\mathfrak {q}$ in a local, Cohen-Macaulay ring $(A,\mathfrak {n})$ such that $e(A_{\mathfrak {p}})=e(A)$ we have $\dim (A/\mathfrak {p})+\dim (A/\mathfrak {q})\leq \dim (A)$. We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when $R/\mathfrak {p}$ is regular.

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study theisospectral Hilbert schemeXnX_{n}, defined as the reduced fiber product of(C2)n(\mathbb {C}^{2})^{n}with the Hilbert schemeHnH_{n}of points in the planeC2\mathbb {C}^{2}, over the symmetric powerSnC2=(C2)n/SnS^{n}\mathbb {C}^{2} = (\mathbb {C}^{2})^{n}/S_{n}. By a theorem of Fogarty,HnH_{n}is smooth. We prove thatXnX_{n}is normal, Cohen-Macaulay and Gorenstein, and hence flat overHnH_{n}. We derive two important consequences. (1) We prove the strong form of then!n!conjectureof Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficientsKλμ(q,t)K_{\lambda \mu }(q,t). This establishes theMacdonald positivity conjecture, namely thatKλμ(q,t)∈N[q,t]K_{\lambda \mu }(q,t)\in {\mathbb N} [q,t]. (2) We show that the Hilbert schemeHnH_{n}is isomorphic to theGG-Hilbert scheme(C2)n//Sn(\mathbb {C}^{2})^{n}{/\!\!/}S_nof Nakamura, in such a way thatXnX_{n}is identified with the universal family over(C2)n//Sn({\mathbb C}^2)^n{/\!\!/}S_n. From this point of view,Kλμ(q,t)K_{\lambda \mu }(q,t)describes the fiber of acharacter sheafCλC_{\lambda }at a torus-fixed point of(C2)n//Sn({\mathbb C}^2)^n{/\!\!/}S_ncorresponding toμ\mu. The proofs rely on a study of certain subspace arrangementsZ(n,l)⊆(C2)n+lZ(n,l)\subseteq (\mathbb {C}^{2})^{n+l}, calledpolygraphs, whose coordinate ringsR(n,l)R(n,l)carry geometric information aboutXnX_{n}. The key result is thatR(n,l)R(n,l)is a free module over the polynomial ring in one set of coordinates on(C2)n(\mathbb {C}^{2})^{n}. This is proven by an intricate inductive argument based on elementary commutative algebra.

Multiplicities and a Dimension Inequality for Unmixed Modules

We prove the following result, which is motivated by the recent work of Kurano and Roberts on Serre's positivity conjecture. Assume that (R,m) is a local ring with finitely generated module M such that R/Ann(M) is quasi-unmixed and contains a field, and that p and q are prime ideals in the support of M such that p is analytically unramified, p+q=m and e(Mp)=e(M). ThendimR/p+dimR/q≤dimM.We also prove a similar theorem in a special case of mixed characteristic. Finally, we provide several examples to explain our assumptions as well as an example of a noncatenary local domain R with prime ideal p such that e(Rp)>e(R)=1.

Test Modules to Calculate Dutta Multiplicities

Using Frobenius maps, the Dutta multiplicity χ∞(F) was first defined by Dutta for a complex F over a local ring of positive characteristic. Nowadays, using localized Chern characters or Adams operations, it is defined for a complex not only over a local ring of positive characteristic but also over a ring that is a homomorphic image of an arbitrary regular local ring. In the paper, we shall give another characterization to Dutta multiplicity using some Galois extension.Furthermore, we shall define the notion of test modules. If a test module for a local ring exists, then the positivity conjecture of Dutta multiplicities is true over the ring. In the paper, it is proved that, if the small Cohen–Macaulay modules conjecture is true, then test modules always exist for complete local domains whose field of fractions are of characteristic 0.

A Reader's Guide to Gacs's “Positive Rates” Paper

Peter Gacs's monograph, which follows this article, provides a counterexample to the important Positive Rates Conjecture. This conjecture, which arose in the late 1960's, was based on very plausible arguments, some of which come from statistical mechanics. During the long gestation period of the Gacs example, there has been a great deal of skepticism about the validity of his work. The construction and verification of Gacs's counterexample are unavoidably complex, and as a consequence, his paper is quite lengthy. But because of the novelty of the techniques and the significance of the result, his work deserves to become widely known. This reader's guide is intended both as a cheap substitute for reading the whole thing, as well as a warm-up for those who want to plumb its depths.

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.

Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions

Abstract. Let Jμ[X; q, t] be the integral form of the Macdonald polynomial and set Hμ[X; q, t] = tJμ[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 = tq ′)Hμ. This operator occurs naturally in the study of the Garsia-Haiman 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, t-analogue of Lagrange inversion. In particular, its specialization at t = 1 (or q = 1) reduces to the q-analogue 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.

Positivity properties for q-Schur algebras

We prove Du's positivity conjecture for the canonical basis of the q-Schur algebra, using elementary arguments and the positivity result for Lusztig's canonical basis for U+q(sln). We also describe a family of subalgebras of the q-Schur algebra, each of which is spanned by the canonical basis elements it contains.

Do most binary linear codes achieve the Goblick bound on the covering radius? (Corresp.)

The following two problems are dealt with: P1) finding the smallest rate, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> , of a binary code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> admitting a prescribed covering radius <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho n</tex> ; P2) discovering whether a majority of codes with any rate larger than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> admits the given covering radius. For the class of unrestricted (nonlinear) codes a solution to both problems is obtained by an elementary averaging argument. The solution to P1 is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R = 1 - H(\rho) + O(n^{-1} \log n)</tex> and the answer to P2 is positive. As for the more interesting class of linear codes, Goblick's extension method shows that the solution to P1 is the same as in the unrestricted case; in contrast, P2 seems to remain an open question. A simple derivation of Goblick's result is presented, and a discussion is made of the positive conjecture concerning P2 for linear codes.

On the consistency of conjectures with public goods

This note addresses Sugden's criticisms, levelled against our Non-Nash theory for public goods. In particular, we argue that our previous exercise was not to present a theory of public goods that could escape the voluntary contribution problem by relaxing Nash conjectures. Rather, our non-Nash model was merely an analytical exercise, meant to encompass a widevariety of behaviour. This note also indicates when positive conjectures might be realistic and, therefore, potentially consistent. Finally, we extend our specific example to include negative, zero, and positive conjectures.

Externalities, expectations, and pigouvian taxes

This article derives Pigouvian-type corrective measures for reciprocal externalities when non-Nash behavior characterizes the participants. These reciprocal externalities may involve various kinds of environmental pollutants, such as acid rain. A comparison between corrective measures for Nash and non-Nash behavior demonstrates that positive conjectures, regarding the other agent's externality-generating activity, have an expectation-internalizing influence that usually reduces the required corrective measures. Negative conjectures (e.g., free-riding expectations), however, have an expectation-externalizing effect that increases the required corrective measures. The article analyzes both two-person and n-person externalities.

Cosmology: Matters of moderate gravity

Cosmology: Matters of moderate gravity

The cosmic censorship hypothesis and the positive energy conjecture

The cosmic censorship hypothesis and the positive energy conjecture

Graph-theoretic properties in computational complexity

The extent to which a set of related graph-theoretic properties can be used to accont for the superlinear complexity of computational problems is explored. While a previously widely held positive conjecture is refuted, it is also shown that certain limited lower bounds can be proved by means of such properties.

Purity of critical cohomology and Kac’s conjecture

We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson-Thomas theory, dimensional reduction, and an easy purity result. These facts imply the purity of the cohomological Donaldson-Thomas invariants for a quiver with potential $(\tilde{Q},W)$ associated to $Q$, which in turn implies positivity of the Kac polynomials for $Q$.

Feynman motives of banana graphs

We consider the infinite family of Feynman graphs known as the “banana graphs” and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern–Schwartz–MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.

A positivity conjecture for Jack polynomials

We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley's ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for partitions having a rectangular shape.

Combinatorics, symmetric functions and Hilbert schemes

We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the n! and (n+1)n-1 conjectures relating Macdonald polynomials to the characters of doubly-graded Sn modules. To make the treatment self-contained, we include background material from combinatorics, symmetric function theory, representation theory and geometry. At the end we discuss future directions, new conjectures and related work of Ginzburg, Kumar and Thomsen, Gordon, and Haglund and Loehr.

Factorised solutions of Temperley-Lieb $qKZ$ equations on a segment

We study the q-deformed Knizhnik–Zamolodchikov (qKZ) equation in path representations of the Temperley–Lieb algebras. We consider two types of open boundary conditions, and in both cases we derive factorized expressions for the solutions of the qKZ equation in terms of Baxterized Demazurre–Lusztig operators. These expressions are alternative to known integral solutions for tensor product representations. The factorized expressions reveal the algebraic structure within the qKZ equation, and effectively reduce it to a set of truncation conditions on a single scalar function. The factorized expressions allow for an efficient computation of the full solution once this single scalar function is known. We further study particular polynomial solutions for which certain additional factorized expressions give weighted sums over components of the solution. In the homogeneous limit, we formulate positivity conjectures in the