Principal Block Research Articles

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

AbstractLet be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient. For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Two new coordination polymers: crystal structures and photodegradation of the organic dye RhB

The solvothermal assemblies of a silicon-centered tetrahedral ligand as a principal building block, an N-donor ancillary ligand, and the corresponding metal salts yield two new transition metal coordination polymers (CPs), [Zn3(OH)(H2O)3L(ttb)]·3H2O (1) and [Co3(OH)(H2O)2L(ttb)(DMF)2]·2H2O (2) [H4L = 4,4',4'',4'''-silanetetrayltetrabenzoic acid, ttp = 1-(tetrazo-5-yl)-4-(triazo-1-yl)benzene, DMF = N,N-dimethylformamide]. Photocatalytic test results indicate that 1 and 2 can serve as excellent photocatalysts for Rhodamine B (RhB) degradation.

The Steinberg character in the principal block of a finite reductive group

We give a generic description of the non-defining primes l , for which the Steinberg character lies in the principal l -block of a finite reductive group.

A neutron diffraction study of the hydrous borate inderborite, CaMg[B3O3(OH)5]2(H2O)4·2H2O

Abstract The crystal chemistry of inderborite, a B-rich mineral (B2O3 ~41 wt%) with ideal formula CaMg[B3O3(OH)5]2·6H2O or CaMg[B3O3(OH)5]2(H2O)4·2H2O from the Inder Deposit, Kazakhstan, was re-investigated by a multi-methodological approach (single-crystal X-ray and neutron diffraction, electron probe micro-analysis in wavelength-dispersive mode, laser ablation multi-collector inductively mass spectrometry). The experimental findings show that the real chemical formula of inderborite from the Inder Deposit is virtually identical to the ideal one: the fraction of potential isomorphic substituents is insignificant. Boron is, therefore, the only industrially relevant element occurring in this mineral. The in situ B isotope composition of the Inder inderborite shows enrichment in the heavy 11B isotope, giving a weighted mean δ11BNIST951 of +35.15 ± 0.49 ‰ (2σ, N = 6). Such a positive δ11B value falls within the range of values in which the source of boron is ascribable to marine reservoirs rather than to terrestrial ones. X-ray (at 293 K) and neutron (at 20 K) structure refinements confirm that the principal building block unit of the structure is a [B3O3(OH)5]2– ring, consisting of two BO2(OH)2 tetrahedra (B-ion in sp3 electronic configuration) and one planar-triangular BO2OH group (B-ion in sp2 electronic configuration). In the [B3O3(OH)5]2– ring, all the oxygen atoms that are not shared between two boron atoms are protonated. The building units share corners with the CaO2(OH)4(OH2)2 polyhedra and Mg(OH)4(OH2)2 octahedra, forming hetero-polyhedral sheets parallel to (100). Subsequent hetero-polyhedral sheets are mutually connected only by H-bonding interactions, even mediated by the zeolitic (“interstitial”) H2O molecules. Ten out of 11 independent oxygen sites in the structure of inderborite are involved in H-bonds as donors or acceptors, and this reflects the pervasive effect of the H-bonding network. The role played by the complex H-bond network is expected to be substantial on the stability of the crystalline edifice, having effects within the single hetero-polyhedral sheet, between subsequent sheets, and in the bonding with the interstitial zeolitic H2O molecules. Finally, the potential utilizations of inderborite, as a B-bearing mineral, are discussed.

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

AbstractLet k(B0) and l(B0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B0 of a finite group G. We prove that, if k(B0)−l(B0) = 1, then l(B0) ≥ p − 1 or else p = 11 and l(B0) = 9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B0) ≥ p − 1 or else p = 11 and $$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ G / O p ′ ( G ) ≅ C 11 2 ⋊ SL ( 2 , 5 ) . These results are useful in the study of principal blocks with few characters.We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least $$2\sqrt {p - 1} + 1 - {k_p}(G)$$ 2 p − 1 + 1 − k p ( G ) , where kp(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

Characters of prime power degree in principal blocks

We describe finite groups whose principal block contains only characters of prime power degree.

Inverse images of block varieties

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. Kawai’s result is a tool to calculate the cohomology variety of a module in a block B of a finite group algebra kG restricted to subgroups of a defect group P, provided that P is abelian. Kawai’s result coincides with a Theorem of Avrunin and Scott specialized to modules in the principal block and their restrictions to p-subgroups. J. Rickard raised the question whether Kawai’s result can be extended to modules in blocks with arbitrary defect groups. We show that this is indeed the case for modules whose corresponding module over some almost source algebra is fusion stable. We show that this fusion stability hypothesis is automatically satisfied for principal blocks and blocks with abelian defect groups.

Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability

Abstract We discuss representations of finite groups having a common central 𝑝-subgroup 𝑍, where 𝑝 is a prime number. For the principal 𝑝-blocks, we give a method of constructing a relative 𝑍-stable equivalence of Morita type, which is a generalization of stable equivalence of Morita type and was introduced by Wang and Zhang in a more general setting. Then we generalize Linckelmann’s results on stable equivalences of Morita type to relative 𝑍-stable equivalences of Morita type. We also introduce the notion of relative Brauer indecomposability, which is a generalization of the notion of Brauer indecomposability. We give an equivalent condition for Scott modules to be relatively Brauer indecomposable, which is an analog of that given by Ishioka and the first author.

Hecke category actions via Smith–Treumann theory

Let $\textbf {G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block $\operatorname {Rep}_0(\textbf {G})$ of $\operatorname {Rep}(\textbf {G})$ by wall-crossing functors. This action was conjectured to exist by Riche and Williamson. Our method uses constructible sheaves and relies on Smith–Treumann theory.

Principal blocks with six ordinary irreducible characters

We classify Sylow p-subgroups of finite groups whose principal p-blocks have precisely six ordinary irreducible characters.

Principal blocks with distinct character degrees

Let G be a finite group, p a prime dividing | G | . In this paper, we study the structure of G, which satisfies the condition that any two irreducible characters in its principal p-block have distinct degrees. We provide a reduction of this question to simple groups and determine the structure of G for p = 2. Furthermore, we describe the structure of G in which any two characters of the principal p-block of G have coprime degrees.

Least Squares Monte Carlo and Pathwise Optimization for Merchant Energy Production

Modeling as real options the operations of energy production companies that operate in wholesale markets gives rise to a challenging Markov decision process. In “Least Squares Monte Carlo and Pathwise Optimization for Merchant Energy Production,” Yang, Nadarajah, and Secomandi study the performance of two reinforcement learning techniques that can be used to determine feasible operating policies and optimality bounds for this model, namely least squares Monte Carlo and pathwise optimization, extending the applicability of the latter method beyond optimal stopping by using principal component analysis and block coordinate descent. They find that both approaches lead to near optimal policies, but pathwise optimization outperforms least squares Monte Carlo in terms of dual bounds at the expense of more sizable computational requirements. These findings have potential relevance for managers of energy production assets that use analytics to optimize their operations and researchers interested in broadening the scope of pathwise optimization.

Positivity preservers forbidden to operate on diagonal blocks

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at “opposite ends”, and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.

On the derived category of the Iwahori–Hecke algebra

We state a conjecture that relates the derived category of smooth representations of a $p$-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of ${\rm GL}_n$ by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and of Emerton and Helm.

A HECKE ACTION ON -MODULES

AbstractWe construct an action of the affine Hecke category on the principal block $\mathrm {Rep}_0(G_1T)$ of $G_1T$ -modules where G is a connected reductive group over an algebraically closed field of characteristic $p> 0$ , T a maximal torus of G and $G_1$ the Frobenius kernel of G. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

The principal block of a ℤℓ-spets and Yokonuma type algebras

The principal block of a ℤℓ-spets and Yokonuma type algebras

The principal block of a ℤℓ-spets and Yokonuma type algebras

We formulate conjectures concerning the dimension of the principal block of a ${\mathbb Z}_\ell$-spets (as defined in our earlier paper), motivated by analogous statements for finite groups. We show that these conjectures hold in certain situations. For this we introduce and study a Yokonuma type algebra for torus normalisers in $\ell$-compact groups which may be of independent interest.

Height zero characters in principal blocks

We show that the principal p-block of a finite group of order divisible by p has at least 2p−1 height-zero characters. Along the way, we describe the p-local structure of finite groups whose principal p-blocks have at most five height-zero ordinary irreducible characters.

Principal Blocks for Different Primes, II

If G is a finite group, we have proposed three new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper,we give strong support to the validity of Conjectures B and C.

Toward a granular molecular-anatomic map of the blood vasculature – single-cell RNA sequencing makes the leap

Single-cell RNA sequencing (scRNAseq) marks the birth of a new era in physiology and medicine. Within foreseeable future, we will know exactly what genes are expressed - and at what levels - in all the different cell types and subtypes that make up our bodies. We will also learn how a particular cell state, whether it occurs during development, tissue repair, or disease, reflects precise changes in gene expression. While profoundly impacting all areas of life science, scRNAseq may lead to a particular leap in vascular biology research. Blood vessels pervade and fulfill essential functions in all organs, but the functions differ. Innumerable organ-specific vascular adaptations and specializations are required. These, in turn, are dictated by differential gene expression by the two principal cellular building blocks of blood vessels: endothelial cells and mural cells. An organotypic vasculature is essential for functions as diverse as thinking, gas exchange, urine excretion, and xenobiotic detoxification in the brain, lung, kidney, and liver, respectively. In addition to the organotypicity, vascular cells also differ along the vascular arterio-venous axis, referred to as zonation, differences that are essential for the regulation of blood pressure and flow. Moreover, gene expression-based molecular changes dictate states of cellular activity, necessary for angiogenesis, vascular permeability, and immune cell trafficking, i.e. functions necessary for development, inflammation, and repair. These different levels of cellular heterogeneity create a nearly infinite phenotypic diversity among vascular cells. In this review, I summarize and exemplify what scRNAseq has brought to the picture in just a few years and point out where it will take us.