We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We present some fundamental aspects of the representation theory of reductive p-adic groups, mapping out some recent developments, including the ABPS conjecture and the description of the structure of the reduced C*-algebras of reductive groups. The exposition proceeds fairly rapidly, but is essentially self-contained.
In this paper we produce infinite families of counterexamples to Jantzen's question posed in 1980 on the existence of Weyl $p$-filtrations for Weyl modules for an algebraic group and Donkin's tilting module conjecture formulated in 1990. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type ${\rm A}_{n}$ or ${\rm B}_{2}$.
Expanding upon the material delivered during the LMS Autumn Algebra School 2020, this volume reflects the fruitful connections between different aspects of representation theory. Each survey article addresses a specific subject from a modern angle, beginning with an exploration of the representation theory of associative algebras, followed by the coverage of important developments in Lie theory in the past two decades, before the final sections introduce the reader to three strikingly different aspects of group theory. Written at a level suitable for graduate students and researchers in related fields, this book provides pure mathematicians with a springboard into the vast and growing literature in each area.
In this paper, we study an analogue of the Bernstein–Gelfand–Gelfand category ${\mathcal {O}}$ for truncated current Lie algebras $\mathfrak {g}_n$ attached to a complex semisimple Lie algebra. This category admits Verma modules and simple modules, each parametrized by the dual space of the truncated currents on a choice of Cartan subalgebra in $\mathfrak {g}$. Our main result describes an inductive procedure for computing composition multiplicities of simples inside Vermas for $\mathfrak {g}_n$, in terms of similar composition multiplicities for ${\mathfrak {l}}_{n-1}$ where ${\mathfrak {l}}$ is a Levi subalgebra. As a consequence, these numbers are expressed as integral linear combinations of Kazhdan–Lusztig polynomials evaluated at 1. This generalizes recent work of the first author, where the case $n=1$ was treated.
We introduce the combinatorial notion of a q-factorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with three decorations: a coloring and a weight map on vertices, and an exponent map on arrows (the exponent map can be seen as a weight map on arrows). Such graphs do not contain oriented cycles and, hence, the set of arrows induces a partial order on the set of vertices. In this first paper on the topic, beside setting the theoretical base of the concept, we establish several criteria for deciding whether or not a tensor product of two simple modules is a highest-$\ell $-weight module and use such criteria to prove, for type A, that a simple module whose q-factorization graph has a totally ordered vertex set is prime.
This detailed yet accessible text provides an essential introduction to the advanced mathematical methods at the core of theoretical physics. The book steadily develops the key concepts required for an understanding of symmetry principles and topological structures, such as group theory, differentiable manifolds, Riemannian geometry, and Lie algebras. Based on a course for senior undergraduate students of physics, it is written in a clear, pedagogical style and would also be valuable to students in other areas of science and engineering. The material has been subject to more than twenty years of feedback from students, ensuring that explanations and examples are lucid and considered, and numerous worked examples and exercises reinforce key concepts and further strengthen readers' understanding. This text unites a wide variety of important topics that are often scattered across different books, and provides a solid platform for more specialized study or research.
The meaning of time reversal can be reconstructed from the structure of time translations. "Instantaneous" time reversal is just its proxy in a state space representation.
There are many representations of time reversal symmetry, including PT, CT, and CPT, but only the standard time reversal operator T is associated with an arrow of time itself.
The classical Gelfand–Naimark theorems provide important insight into the structure of general and of commutative $C^*$-algebras. It is shown that these can be generalized to certain ordered $^*$-algebras. More precisely, for $\sigma $-bounded closed ordered $^*$-algebras, a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand–Naimark representation theorems to classes of $^*$-algebras larger than $C^*$-algebras, and which especially contain $^*$-algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is $\sigma $-bounded, then the order of V is induced by the extremal positive linear functionals on V.
Let $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$, there exists a singular block in the category of $G_1$-modules, which contains precisely $n+1$ irreducible modules. We are interested in the ‘lift’ of this block to the category of $G_1T$-modules. Imposing only mild assumptions on $p$, we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also explicitly compute the Loewy series for the indecomposable projective modules.
We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making systematic use of finite groupoids. This provides a ‘road map’ for the various approaches to the axiomatic representation theory of finite groups, as well as some details that are hard to find in the literature.
This volume contains eight research papers inspired by the 2019 'Equivariant Topology and Derived Algebra' conference, held at the Norwegian University of Science and Technology, Trondheim in honour of Professor J. P. C. Greenlees' 60th birthday. These papers, written by experts in the field, are intended to introduce complex topics from equivariant topology and derived algebra while also presenting novel research. As such this book is suitable for new researchers in the area and provides an excellent reference for established researchers. The inter-connected topics of the volume include: algebraic models for rational equivariant spectra; dualities and fracture theorems in chromatic homotopy theory; duality and stratification in tensor triangulated geometry; Mackey functors, Tambara functors and connections to axiomatic representation theory; homotopy limits and monoidal Bousfield localization of model categories.
Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda [24] conjectured that the formal degree of a square-integrable G-representation
$\pi $
can be expressed in terms of the adjoint
$\gamma $
-factor of the enhanced L-parameter of
$\pi $
. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations.
We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint
$\gamma $
-factors.
This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.
This chapter contains a survey of known results and open problems connected to the combinatorics of (type A) Macdonald polynomials. Macdonald polynomials are symmetric functions in a set of variables which depend on two extra parameters q,t. They include most of the commonly studied bases for the ring of symmetric functions, such as Schur functions and Hall-Littlewood polynomials, as special cases. Macdonald polynomials have geometric interpretations which make them important to algebraic geometry and mathematical physics, and are also fundamental to the study of special functions. Their combinatorial properties are rather mysterious, although a lot of progress has been made on the type A case in the past 20 years, in conjunction with the study of the representation theory of the ring of diagonal coinvariants. This survey shows how to express Macdonald polynomials, and other important objects such as the bigraded Hilbert series of the diagonal coinvariant ring, in terms of popular combinatorial structures including tableaux, Dyck paths, and parking functions.
We prove Breuillard and Green’s theorem that a finite approximate subgroup of a soluble complex linear group G of bounded degree is contained in a union of a few cosets of a nilpotent group of bounded step. We first treat the special case in which G is an upper-triangular group. An important ingredient is Solymosi’s sum-product theorem over the complex numbers, which we state and prove. We introduce some basic representation theory and use it to prove that a soluble complex linear group of bounded degree has a subgroup of bounded index that is conjugate to an upper-triangular group; this is a special case of a result of Mal’cev. We then use this to extend from the upper-triangular case to the general soluble case.
This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.
We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.
In this paper we study certain sheaves of $p$-adically complete rings of differential operators on semistable models of the projective line over the ring of integers in a finite extension $L$ of $\mathbb{Q}_{p}$. The global sections of these sheaves can be identified with (central reductions of) analytic distribution algebras of wide open congruence subgroups. It is shown that the global sections functor furnishes an equivalence between the categories of coherent module sheaves and finitely presented modules over the distribution algebras. Using the work of M. Emerton, we then describe admissible representations of $\text{GL}_{2}(L)$ in terms of sheaves on the projective limit of these formal schemes. As an application, we show that representations coming from certain equivariant line bundles on Drinfeld’s first étale covering of the $p$-adic upper half plane are admissible.