Highest Weight Categories Research Articles

Proof of Varagnolo–Vasserot conjecture on cyclotomic categories $${\mathcal {O}}$$ O

We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category \({\mathcal {O}}\) for a cyclotomic rational Cherednik algebra and a suitable truncation of an affine parabolic category \({\mathcal {O}}\) that, in particular, implies Rouquier’s conjecture on the decomposition numbers in the former. Our proof uses two ingredients: an extension of Rouquier’s deformation approach as well as categorical actions on highest weight categories and related combinatorics.

Equivalence of Blocks for the General Linear Lie Superalgebra

We develop a reduction procedure which provides an equivalence (as highest weight categories) from an arbitrary block (defined in terms of the central character and the integral Weyl group) of the BGG category O for a general linear Lie superalgebra to an integral block of O for (possibly a direct sum of) general linear Lie superalgebras. We also establish indecomposability of blocks of O.

Highest weight categories for Dedekind domains

This paper examines the concept of a stratified exact category in the context of Dedekind domains and corresponding finite Galois groups. BGG reciprocity and duality are proven for these categories making them highest weight categories. The strong connections between the structure of the category and ramification in the ring are explored.

Highest weight $$\mathfrak{sl }_2$$ -categorifications I: crystals

We define highest weight categorical actions of \(\mathfrak{sl }_2\) on highest weight categories and show that basically all known examples of categorical \(\mathfrak{sl }_2\)-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories \(\mathcal O \) of type \(A\), categories \(\mathcal O \) for cyclotomic Rational Cherednik algebras) are highest weight in our sense. Our main result is an explicit combinatorial description of (the labels of) the crystal on the set of simple objects. A new application of this is to determining the supports of simple modules over the cyclotomic Rational Cherednik algebras starting from their labels.

Weighing the Pediatric Patient During Trauma Resuscitation and Its Concordance With Estimated Weight Using Broselow Luten Emergency Tape

Obtaining an accurate weight is crucial during pediatric trauma/medical resuscitation. Currently, length-based weight estimations are used. Study objective was to assess feasibility of obtaining actual weights of children during trauma resuscitation and study its concordance with length-based estimated weight using the Broselow Pediatric Emergency Tape. Pediatric trauma patients 0 to 14 years old presenting to a tertiary care pediatric trauma center between November 2008 and October 2009 were enrolled prospectively. Length-based weight estimation was done on patient arrival using the Broselow tape; in addition, an actual patient weight was recorded using the trauma stretcher integrated weighing scale. Two hundred thirty-one patients were eligible and enrolled. Weights were recorded in 145 children (63.2%). In 27 patients (18.6%) whose body length exceeded Broselow tape range, weight was measured using stretcher scale only. The remaining 118 patients (mean age, 5.0 [SE ± 0.3] years; 67% male) were used for correlation analysis. There was good correlation (Pearson correlation coefficient, r = 0.86) between estimated weight and measured weight. However, Bland-Altman analysis showed mean bias +2.6 kg (95% confidence interval [CI], 1.6-3.6 kg); lower/upper limits of agreement were -8.3 kg (CI, -10.0 to -6.6 kg) and 13.5 kg (CI, 11.7-15.2 kg). It is possible to obtain an actual patient weight during pediatric trauma resuscitation. Length-based estimated weight using Broselow tape underestimated weight by 2.6 kg; the mean error was greatest in the highest weight category.

Causes of obesity

The prevalence of obesity has been rising steadily over the last several decades and is currently at unprecedented levels: more than 68% of US adults are considered overweight, and 35% are obese (Flegal et al., JAMA 303:235-241, 2010). This increase has occurred across every age, sex, race, and smoking status, and data indicate that segments of individuals in the highest weight categories (i.e., BMI > 40 kg/m(2)) have increased proportionately more than those in lower BMI categories (BMI < 35 kg/m(2)). The dramatic rise in obesity has also occurred in many other countries, and the causes of this increase are not fully understood (Hill and Melanson, Med Sci Sports Exerc 31:S515-S521, 1999).

Relationship between body mass index and perceived insufficient sleep among U.S. adults: an analysis of 2008 BRFSS data

BackgroundOver the past 50 years, the average sleep duration for adults in the United States has decreased while the prevalence of obesity and associated outcomes has increased. The objective of this study was to determine whether perceived insufficient sleep was associated with body mass index (BMI) in a national sample.MethodsWe analyzed data from the 2008 Behavioral Risk Factor Surveillance System (BRFSS) survey (N = 384,541) in which respondents were asked, "During the past 30 days, for about how many days have you felt you did not get enough rest or sleep?" We divided respondents into six BMI categories and used multivariable linear regression and logistic regression analyses to assess the association between BMI categories and days of insufficient sleep after adjusting for sociodemographic variables, smoking, physical activity, and frequent mental distress.ResultsAdjusted mean days of insufficient sleep ranged from 7.9 (95% confidence interval [CI]: 7.8, 8.0) days for people of normal weight to 10.5 (95% CI: 10.2, 10.9) days for those in the highest weight category (BMI ≥ 40). Days of perceived insufficient sleep followed a linear trend across BMI categories. The likelihood of reporting ≥14 days of insufficient sleep in the previous 30 days was higher for respondents in the highest weight category than for those who were normal weight (34.9% vs. 25.2%; adjusted odds ratio = 1.7 (95% CI: 1.5, 1.8]).ConclusionAmong U.S. adults, days of insufficient rest or sleep strongly correlated with BMI. Sleep sufficiency should be an important consideration in the assessment of the health of overweight and obese people and should be considered by developers of weight-reduction programs.

Nonzero Infinitesimal Blocks of Category $\boldsymbol{\mathcal{O}}_{\textbf{\emph{S}}}$

Category \(\mathcal{O}\) is a nice category of modules for a finite dimensional semisimple Lie algebra \(\mathfrak{g}\) It was first introduced by Bernstein–Gelfand–Gelfand in 1976. In the early 1980’s, Rocha–Caridi introduced parabolic category \(\mathcal{O}_S\), where S is a subset of simple roots. Category \(\mathcal{O}_S\) is a generalization of ordinary category O. These are highest weight categories that decompose into certain subcategories, called infinitesimal blocks. An infinitesimal block contains at most finitely many simple modules, and some contain only the zero module. The representation type of the infinitesimal blocks of category \(\mathcal{O}_S\) has been studied by Futorny–Nakano–Pollack, Brustle–Konig–Mazorchuk, and Boe–Nakano. The representation type of the singular blocks of category \(\mathcal{O}_S\) is still generally unknown, though Boe–Nakano classified certain of these. Understanding when a singular block is zero is an important step to understanding the singular blocks in general. In this work, we will answer this question. It is given in terms of nilpotent orbits of \(\mathfrak{g}\).

Functoriality of the BGG Category 𝒪††

This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category 𝒪. We study functorial properties of 𝒪 across various setups. The first setup is over a skew group ring, involving a finite group Γ acting on a regular triangular algebra A. We develop Clifford theory for A⋊Γ, and obtain results on block decomposition, complete reducibility, and enough projectives. 𝒪 is shown to be a highest weight category when A satisfies one of the “Conditions (S);” the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories 𝒪; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup, and which yield information about 𝒪.

Coinvariants for a coadjoint action of quantum matrices

Let K be a (algebraically closed ) field. A morphism A ⟼ g −1 Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational GL q × GL q -modules is a highest weight category.

Semistandard filtrations in highest weight categories

This paper is dedicated to the memory of Donald Higman. Don was especially attracted to methods with some kind of fundamental simplicity but with sufficient substance to be useful and illuminating. This paper is written in the spirit of this lofty aspiration and also somewhat in the direction of Don’s early interests in homological algebra. The paper begins in Section 1 with a foundational discussion of a new notion, that of a semistandard filtration in a highest weight category C. The latter categories [CPS1] axiomatize features found in many Lie-theoretic module settings. For simplicity we stick to the case of a finite weight poset , to which many considerations reduce. Here, the irreducible objects L(λ) and projective indecomposable objects P(λ) are indexed by , and there are standard objects (λ) with head L(λ) and all other composition factors indexed by a smaller weight. The objects P(λ) have finite filtrations with sections as standard objects, the top section being (λ) and all others of the form (ν) with ν > λ. In particular, these conditions imply ExtC( (λ), (μ)) = 0 unless λ > μ. Thus, whenever an object M in C has a finite length filtration of subobjects 0 = F0 ⊆ F1 ⊆ · · · ⊆ Fn = M

On Moduli Spaces for Abelian Categories

We show that if 𝒜 is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.

Current algebras, highest weight categories and quivers

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametrized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on the affine weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and star-shaped type, as well as tame quasi-hereditary algebras, that arise from our study.

Some properties of general linear supergroups and of Schur superalgebras

It is shown that the category of rational supermodules over a general linear supergroup is a highest weight category. More exactly, we construct superanalogs of the theory of modules with good filtration and of the dual theory of modules with Weyl’s. Using these, we show that indecomposable injective supermodules have good filtration of a certain kind.

AUSLANDER–REITEN COMPONENTS FOR G1T-MODULES

In this paper we study the Auslander–Reiten quiver of the highest weight category of finite-dimensional G1T-modules, associated to a smooth reductive algebraic group G. By relating properties of stable Auslander–Reiten components to those of their rank varieties we show that there are at most three isomorphism types of these components. For the Frobenius kernels G1Tr the maximal ranks of tubes are determined.

Category [formula omitted] over a deformation of the symplectic oscillator algebra

We discuss the representation theory of H f , which is a deformation of the symplectic oscillator algebra sp(2n)⋉ h n , where h n is the ((2 n+1)-dimensional) Heisenberg algebra. We first look at a more general algebra with a triangular decomposition. Assuming the PBW theorem, and one other hypothesis, we show that the BGG category O is abelian, finite length, and self-dual. We decompose O as a direct sum of blocks O(λ) , and show that each block is a highest weight category. In the second part, we focus on the case H f for n=1, where we prove all these assumptions, as well as the PBW theorem.

Projective modules in the category [formula omitted] for the Cherednik algebra

We study projective objects in the category O c of the rational Cherednik algebra introduced recently by Berest, Etingof and Ginzburg. We prove that it has enough projectives and that it is a highest weight category in the sense of Cline, Parshall and Scott, and therefore satisfies an analog of the BGG-reciprocity formula for a semisimple Lie algebra.

Highest weight categories of Lie algebra modules

We study a certain class of categories of Lie algebra modules which include the well-known categories O and O S . We show that all these categories are highest weight categories.

Representation Theory of Reductive Normal Algebraic Monoids

New results in the representation theory of “semisimple” algebraic monoids are obtained, based on Renner’s monoid version of Chevalley’s big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as “polynomial” representations of the associated reductive group of units in the monoid, and this representation category splits into a direct sum of subcategories by “homogeneous” degree. We show that each of these homogeneous subcategories is a highest weight category, in the sense of Cline, Parshall, and Scott, and so equivalent with the module category of a certain finite-dimensional quasihereditary algebra, which we show is a generalized Schur algebra in S. Donkin’s sense.

Complex Representations of Finite Monoids II. Highest Weight Categories and Quivers

In this paper we continue our study of complex representations of finite monoids. We begin by showing that the complex algebra of a finite regular monoid is a quasi-hereditary algebra and we identify the standard and costandard modules. We define the concept of a monoid quiver and compute it in terms of the group characters of the standard and costandard modules. We use our results to determine the blocks of the complex algebra of the full transformation semigroup. We show that there are only two blocks when the degree ≠3. We also show that when the degree ≥5, the complex algebra of the full transformation semigroup is not of finite representation type, answering negatively a conjecture of Ponizovskii.