Cores of Partitions in Rectangles

Weights for symmetric and general linear groups

p ∗-Theory and modular representation theory

Symmetric local algebras and small blocks of finite groups

Working with tensor triangulated categories, we prove two theorems relative to supports and then discuss their incarnations in algebraic geometry and in modular representation theory. First, we show that an indecomposable object has connected support. Then we consider the filtrations by dimension or codimension of the support and prove that the associated subquotients decompose as sums of local terms.

Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebro-geometric problems, such as the derived equivalence conjecture and description of T. Bridgeland’s space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a t-structure on the derived category of the resolution. The intriguing fact that the same t-structure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory.

Representations of finite association schemes

The modular representation theory of finite groups has its origins in the work of Richard Brauer. In this survey article, we first discuss the work being done on some outstanding conjectures in the theory. We then describe work done in the eighties and nineties on modular representations in non-defining characteristic of finite groups of Lie type. In the second part of the article, we discuss some recent developments in the theory for symmetric groups and Hecke algebras, where remarkable connections with Lie theory and graded representation theory have been made.Modularrepresentations

Three dimensional (3-D) laser scanning has recently proved to be an ideal technique for quantifying aggregate particle shape parameters. This is because 3-D laser scanners measure fine details and capture free-form shapes. This paper presents some results of an on-going study, which utilises 3-D laser scanning technology to quantify aggregate shape parameters. Specific correlations and shape parameters have been considered. The results have shown that meaningful correlations exist between 3-D sphericity and aspect ratio as well as between specific surface area and the particle diameter as defined in this paper. The results of the current study are consistent with generally accepted relationships between shape parameters and particle size, further validating the application of the 3-D laser scanning technology

The main aim of the present paper is to study the robustness of the developed sequential probability ratio test (SPRT) for testing the hypothesis about scale parameter of gamma distribution with known shape parameter and exponential distribution with location parameter. The robustness of the SPRT for scale parameter of gamma distribution is studied when the shape parameter has undergone a change. The similar study is conducted for the scale parameter of exponential distribution when the location parameter has undergone a change. The expressions for operating characteristic and average sample number functions are derived. It is found in both the cases that the SPRT is robust only when there is a slight variation in the shape and location parameter in the respective distributions.

Modular Representation Theory: New Trends and Methods

We show that the restricted Lie algebra structure on Hochschild cohomology is\ninvariant under stable equivalences of Morita type between self-injective\nalgebras. Thereby we obtain a number of positive characteristic stable\ninvariants, such as the $p$-toral rank of $\\mathrm{HH}^1(A,A)$. We also prove a\nmore general result concerning Iwanaga-Gorenstein algebras, using a more\ngeneral notion of stable equivalences of Morita type. Several applications are\ngiven to commutative algebra and modular representation theory. These results\nare proven by first establishing the stable invariance of the\n$B_\\infty$-structure of the Hochschild cochain complex. In the appendix we\nexplain how the $p$-power operation on Hochschild cohomology can be seen as an\nartifact of this $B_\\infty$-structure. In particular, we establish\nwell-definedness of the $p$-power operation, following some -- originally\ntopological -- methods due to May, Cohen and Turchin, using the language of\noperads.\n

In this note, we give a brief overview of the telescope conjecture and the chromatic splitting conjecture in stable homotopy theory. In particular, we provide a proof of the folklore result that Ravenel’s telescope conjecture for all heights combined is equivalent to the generalized telescope conjecture for the stable homotopy category, and explain some similarities with modular representation theory.

The equivariant cohomology ring of a G-space X defines a homogeneous affine variety. Quillen [Q] and W. Y. Hsiang [Hs] have determined the relation between such varieties and the family of isotropy subgroups as well as their fixed point sets when dim X < oo. In modular representation theory, J. Carlson [Cj] introduced cohomological support varieties and rank varieties (the latter depending on the group algebra) and explored their relationship. We define rank and support varieties for G-spaces and G-chain complexes and apply them to cohomological problems in transformation groups. As a corollary, a useful criterion for ZG-projectivity of the reduced total homology of certain G-spaces is obtained, which improves the projectivity criteria of Rim [R], Chouinard [Ch], and Dade [D].

A characterization of blocks with vertices

We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of square-free periods, whose scarcity in the case of the partition function was investigated recently. We complement our obstructions on maximal Ramanujan-type congruences with several existence statements. Our results are based on a framework that leverages classical results on integral models of modular curves via modular representation theory, and applies to congruences of all weakly holomorphic modular forms. Steinberg representations govern all maximal Ramanujan-type congruences for integral weights. We discern the scope of our framework in the case of half-integral weights through example calculations.