Recent zbMATH articles in MSC 20Ghttps://zbmath.org/atom/cc/20G2023-05-31T16:32:50.898670ZUnknown authorWerkzeugAffine Kac-Moody groups and Lie algebras in the language of SGA3https://zbmath.org/1508.140522023-05-31T16:32:50.898670Z"Morita, Jun"https://zbmath.org/authors/?q=ai:morita.jun"Pianzola, Arturo"https://zbmath.org/authors/?q=ai:pianzola.arturo"Shibata, Taiki"https://zbmath.org/authors/?q=ai:shibata.taikiGiven a field \(\mathbb{K}\) of characteristic \(0\), Kac-Moody Lie algebras are defined by generators and relations encoded in a generalized Cartan matrix \(A\). If \(A\) is a Cartan matrix of type \(X\), then the corresponding Kac-Moody Lie algebra is the split simple finite-dimensional \(\mathbb{K}\)-Lie algebra of type \(X\).
In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The authors provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.
Reviewer: Mee Seong Im (Annapolis)Mustafin varieties, moduli spaces and tropical geometryhttps://zbmath.org/1508.140672023-05-31T16:32:50.898670Z"Hahn, Marvin Anas"https://zbmath.org/authors/?q=ai:anas-hahn.marvin"Li, Binglin"https://zbmath.org/authors/?q=ai:li.binglinSummary: Mustafin varieties are flat degenerations of projective spaces, induced by a set of lattices in a vector space over a non-archimedean field. They were introduced by
\textit{G. A. Mustafin} [Math. USSR, Sb. 34, 187--214 (1978; Zbl 0411.14006); translation from Mat. Sb., n. Ser. 105(147), 207--237 (1978)] in the 70s in order to generalise Mumford's groundbreaking work on the unformisation of curves to higher dimension. These varieties have a rich combinatorial structure as can be seen in pioneering work of
\textit{D. Cartwright} et al. [Sel. Math., New Ser. 17, No. 4, 757--793 (2011; Zbl 1248.14013)]. In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied in
[the second author, Int. Math. Res. Not. 2018, No. 13, 4190--4228 (2018; Zbl 1423.14093)]. Applying tropical intersection theory and tropical convex hull computations, we use this method to give a new combinatorial description of the irreducible components of the special fibers of Mustafin varieties. Finally, we outline a first application of our results in limit linear series theory.DG structures on odd categorified quantum \(sl(2)\)https://zbmath.org/1508.170182023-05-31T16:32:50.898670Z"Egilmez, Ilknur"https://zbmath.org/authors/?q=ai:egilmez.ilknur"Lauda, Aaron D."https://zbmath.org/authors/?q=ai:lauda.aaron-dSummary: We equip Ellis and Brundan's version of the odd categorified quantum group for \(sl(2)\) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible decategorifications of the associated dg-2-category. One version gives rise to a categorification of quantum \(sl(2)\) at a fourth root of unity, while the other version produces a subalgebra of quantum \(gl(1|1)\) defined over the integers. Both of these algebras appear in connection with quantum algebraic approaches to the Alexander polynomial.Correction to: ``Dominant and global dimension of blocks of quantised Schur algebras''https://zbmath.org/1508.180112023-05-31T16:32:50.898670Z"Fang, Ming"https://zbmath.org/authors/?q=ai:fang.ming"Hu, Wei"https://zbmath.org/authors/?q=ai:hu.wei"Koenig, Steffen"https://zbmath.org/authors/?q=ai:konig.steffenSeveral typos in the authors' paper [ibid. 300, No. 1, 463--473 (2022; Zbl 1498.18016)] are corrected.Forty years of algebraic groups, algebraic geometry, and representation theory in China. In memory of the centenary year of Xihua Cao's birthhttps://zbmath.org/1508.200022023-05-31T16:32:50.898670ZPublisher's description: Professor Xihua Cao (1920--2005) was a leading scholar at East China Normal University (ECNU) and a famous algebraist in China. His contribution to the Chinese academic circle is particularly the formation of a world-renowned ``ECNU School'' in algebra, covering research areas include algebraic groups, quantum groups, algebraic geometry, Lie algebra, algebraic number theory, representation theory and other hot fields. In January 2020, in order to commemorate Professor Xihua Cao's centenary birthday, East China Normal University held a three-day academic conference. Scholars at home and abroad gave dedications or delivered lectures in the conference. This volume originates from the memorial conference, collecting the dedications of scholars, reminiscences of family members, and 16 academic articles written based on the lectures in the conference, covering a wide range of research hot topics in algebra. The book shows not only scholars' respect and memory for Professor Xihua Cao, but also the research achievements of Chinese scholars at home and abroad.
The articles of this volume will be reviewed individually.On the Malle-Navarro conjecture for 2- and 3-blocks of general linear and unitary groupshttps://zbmath.org/1508.200092023-05-31T16:32:50.898670Z"Brenner, Sofia"https://zbmath.org/authors/?q=ai:brenner.sofiaSummary: The Malle-Navarro conjecture relates central block theoretic invariants in two inequalities. In this article, we prove the conjecture for the 2-blocks and the unipotent 3-blocks of the general linear and unitary groups in non-defining characteristic. Moreover, we show that the conjecture holds for the unipotent 3-blocks of quotients of central subgroups of the special linear and unitary groups.On finite simple images of triangle groupshttps://zbmath.org/1508.200182023-05-31T16:32:50.898670Z"Jambor, Sebastian"https://zbmath.org/authors/?q=ai:jambor.sebastian"Litterick, Alastair"https://zbmath.org/authors/?q=ai:litterick.alastair-j"Marion, Claude"https://zbmath.org/authors/?q=ai:marion.claudeSummary: For a simple algebraic group \(G\) in characteristic \(p\), a triple \((a, b, c)\) of positive integers is said to be rigid for \(G\) if the dimensions of the subvarieties of \(G\) of elements of order dividing \(a\), \(b\), \(c\) sum to \(2 \dim G\). In this paper we complete the proof of a conjecture of the third author, that for a rigid triple \((a, b, c)\) for \(G\) with \(p > 0\), the triangle group \(T_{a,b,c}\) has only finitely many simple images of the form \(G(p^r)\). We also obtain further results on the more general form of the conjecture, where the images \(G(p^r)\) can be arbitrary quasisimple groups of type \(G\).Representation theory of disconnected reductive groupshttps://zbmath.org/1508.200522023-05-31T16:32:50.898670Z"Achar, Pramod N."https://zbmath.org/authors/?q=ai:achar.pramod-n"Hardesty, William D."https://zbmath.org/authors/?q=ai:hardesty.william-d"Riche, Simon"https://zbmath.org/authors/?q=ai:riche.simonSummary: We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic \(0\) and those in characteristic \(p\) (for groups defined over discrete valuation rings of mixed characteristic). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups.Generic character sheaves on reductive groups over a finite ringhttps://zbmath.org/1508.200532023-05-31T16:32:50.898670Z"Chen, Zhe"https://zbmath.org/authors/?q=ai:chen.zheSummary: In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne-Lusztig characters when the parameters are generic. We formulate a conjecture on the simple perversity of these complexes, and we prove it in the level two case (thus extend a result of Lusztig from the function field case). We then discuss the induction and restriction functors, as well as the Frobenius reciprocity, based on the perversity.Geck's conjecture and the generalized Gelfand-Graev representations in bad characteristichttps://zbmath.org/1508.200542023-05-31T16:32:50.898670Z"Dong, Junbin"https://zbmath.org/authors/?q=ai:dong.junbin"Yang, Gao"https://zbmath.org/authors/?q=ai:yang.gaoSummary: For a connected reductive algebraic group \(G\) defined over a finite field \(\mathbb{F}_q\), Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group \(G( \mathbb{F}_q)\) in the case where \(q\) is a power of a good prime for \(G\). This representation has been widely studied and used in various contexts. Recently, Geck proposed a conjecture, characterizing Lusztig's special unipotent classes in terms of weighted Dynkin diagrams. Based on this conjecture, he gave a guideline for extending the definition of GGGRs to the case where \(q\) is a power of a bad prime for \(G\). Here, we will give a proof of Geck's conjecture. Combined with Geck's pioneer work, our proof verifies Geck's conjectural characterization of special unipotent classes, and completes his definition of GGGRs in bad characteristics.On \(p\)-groups with automorphism groups related to the exceptional Chevalley groupshttps://zbmath.org/1508.200552023-05-31T16:32:50.898670Z"Freedman, Saul D."https://zbmath.org/authors/?q=ai:freedman.saul-dSummary: Let \(\hat G\) be the finite simply connected version of an exceptional Chevalley group, and let \(V\) be a nontrivial irreducible module, of minimal dimension, for \(\hat G\) over its field of definition. We explore the overgroup structure of \(\hat G\) in \(\mathrm{GL}(V)\) and the submodule structure of the exterior square (and sometimes the third Lie power) of \(V\). When \(\hat G\) is defined over a field of odd prime order \(p\), this allows us to construct the smallest (with respect to certain properties) \(p\)-groups \(P\) such that the group induced by \(\Aut(P)\) on \(P/\Phi(P)\) is either \(\hat G\) or its normalizer in \(\mathrm{GL}(V).\)Coxeter combinatorics for sum formulas in the representation theory of algebraic groupshttps://zbmath.org/1508.200562023-05-31T16:32:50.898670Z"Gruber, Jonathan"https://zbmath.org/authors/?q=ai:gruber.jonathanThe Jantzen sum formula and the Andersen sum formula are two important results in the representation theory of an arbitrary reductive algebraic group \(G\) in positive characteristic \(p\), giving the characters occurring in particular filtrations of an arbitrary Weyl module of \(G\). While these in principle allow one to calculate composition multiplicities of Weyl modules, in practice this is unwieldy, requiring prohibitively expensive weight calculations. This paper first gives a `recursion formula' which expresses Jantzen sum formula for a given \(p\)-regular weight in terms of a corresponding Jantzen sum formula for a smaller weight (in the Bruhat order). Secondly, the paper gives a `duality formula' which relates the Andersen sum formula and Jantzen sum formula to one another. As a corollary, the author also derives a bound on the length of the Jantzen filtration of a Weyl module with given \(p\)-regular highest weight, in terms of the corresponding Jantzen filtration length for a smaller weight (again in the Bruhat order).
The statements and proofs of the main results are combinatorial in nature. The author makes use of the anti-spherical module for the affine Weyl group to rewrite the sum formulae, and from these these new expressions derives the recursion formula and duality formula. In Section 5, the paper re-interprets these combinatorial results in representation-theoretic terms, which enables the derivation of the bound on Jantzen filtration length. The paper closes with an example application with \(G\) of type \(A\).
Reviewer: Alastair Litterick (Colchester)On the cohomology of integral \(p\)-adic unipotent radicalshttps://zbmath.org/1508.200572023-05-31T16:32:50.898670Z"Ronchetti, NiccolĂ˛"https://zbmath.org/authors/?q=ai:ronchetti.niccoloSummary: Let \(G\) be a reductive split \(p\)-adic group and let \( \mathrm U\) be the unipotent radical of a Borel subgroup. We study the cohomology with trivial \(\mathbb Z_p\)-coefficients of the profinite nilpotent group \(N = \mathrm U (\mathcal O_F)\) and its Lie algebra \(\mathfrak n\) by extending a classical result of Kostant to our integral \(p\)-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.Gluing compact matrix quantum groupshttps://zbmath.org/1508.200582023-05-31T16:32:50.898670Z"Gromada, Daniel"https://zbmath.org/authors/?q=ai:gromada.danielSummary: We study glued tensor and free products of compact matrix quantum groups with cyclic groups -- so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing.Correction to: ``Gluing compact matrix quantum groups''https://zbmath.org/1508.200592023-05-31T16:32:50.898670Z"Gromada, Daniel"https://zbmath.org/authors/?q=ai:gromada.danielCorrection to the author's paper [ibid. 25, No. 1, 53--88 (2022; Zbl 1508.20058)].Characters of Renner monoids and their Hecke algebrashttps://zbmath.org/1508.201222023-05-31T16:32:50.898670Z"Hardt, Andrew"https://zbmath.org/authors/?q=ai:hardt.andrew"Marx-Kuo, Jared"https://zbmath.org/authors/?q=ai:marx-kuo.jared"McDonald, Vaughan"https://zbmath.org/authors/?q=ai:mcdonald.vaughan"O'Brien, John M."https://zbmath.org/authors/?q=ai:obrien.john-m"Vetter, Alexander"https://zbmath.org/authors/?q=ai:vetter.alexanderSummary: This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type \(A\) Renner monoid) was computed earlier by \textit{M. Dieng} et al. [J. Algebr. Comb. 17, No. 2, 99--123 (2003; Zbl 1019.05065)], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type \(C\) and use them to obtain an explicit description of the character table for the type \(C\) Renner monoid Hecke algebra.Maximal entries of elements in certain matrix monoidshttps://zbmath.org/1508.201252023-05-31T16:32:50.898670Z"Han, Sandie"https://zbmath.org/authors/?q=ai:han.sandie"Masuda, Ariane M."https://zbmath.org/authors/?q=ai:masuda.ariane-m"Singh, Satyanand"https://zbmath.org/authors/?q=ai:singh.satyanand"Thiel, Johann"https://zbmath.org/authors/?q=ai:thiel.johann-aSummary: Let \(L_u = \begin{bmatrix} 1 & 0 \\ u & 1 \end{bmatrix}\) and \(R_v = \begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}\) be matrices in \(\mathrm{SL}_2(\mathbb{Z})\) with \(u, v \geq 1\). Since the monoid generated by \(L_u\) and \(R_v\) is free, we can associate a depth to each element based on its product representation. In the cases where \(u = v = 2\) and \(u = v = 3\), Bromberg, Shpilrain, and Vdovina found a depth \(n\) matrix containing the maximal entry for each \(n \geq 1\). By using ideas from our previous work on positive linear fractional transformation \((u, v)\)-Calkin-Wilf trees and a polynomial partial ordering, we extend their results for any \(u, v \geq 1\) and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of Cayley hash functions based on \(L_u\) and \(R_v\).Corrigendum to: ``Effective mixing and counting in trees''https://zbmath.org/1508.370452023-05-31T16:32:50.898670Z"Kwon, Sanghoon"https://zbmath.org/authors/?q=ai:kwon.sanghoonCorrigendum to the author's paper [ibid. 38, No. 1, 257--283 (2018; Zbl 1498.37057)].Gauge field theory in natural geometric language. A revisitation of mathematical notions of quantum physicshttps://zbmath.org/1508.810052023-05-31T16:32:50.898670Z"Canarutto, Daniel"https://zbmath.org/authors/?q=ai:canarutto.danielPublisher's description: Gauge Field theory in Natural Geometric Language addresses the need to clarify basic mathematical concepts at the crossroad between gravitation and quantum physics. Selected mathematical and theoretical topics are exposed within a brief, integrated approach that exploits standard and non-standard notions, as well as recent advances, in a natural geometric language in which the role of structure groups can be regarded as secondary even in the treatment of the gauge fields themselves.
In proposing an original bridge between physics and mathematics, this text will appeal not only to mathematicians who wish to understand some of the basic ideas involved in quantum particle physics, but also to physicists who are not satisfied with the usual mathematical presentations of their field.Linear codes with arbitrary dimensional hull and their applications to EAQECCshttps://zbmath.org/1508.941032023-05-31T16:32:50.898670Z"Sok, Lin"https://zbmath.org/authors/?q=ai:sok.lin"Qian, Gang"https://zbmath.org/authors/?q=ai:qian.gangSummary: The Euclidean hull dimension of a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting code (EAQECC) if the Euclidean construction is applied. In this paper, we study the Euclidean hull of a linear code by means of orthogonal matrices. We provide some methods to construct linear codes over \(\mathbb{F}_{p^m}\) with hull of arbitrary dimensions. With existence of self-dual bases of \(\mathbb{F}_{p^m}\) over \(\mathbb{F}_p\), we determine a Gray map from \(\mathbb{F}_{p^m}\) to \(\mathbb{F}_p^m\), and from a given linear code over \(\mathbb{F}_{p^m}\) with one-dimensional hull, we construct, using such a Gray map, a linear code over \(\mathbb{F}_p\) with \(m\)-dimensional hull for all \(m\) when \(p\) is even and for all \(m\) odd when \(p\) is odd. Comparisons with classical constructions are made, and some good EAQECCs over \(\mathbb{F}_q\), \(q = 2, 3, 4, 5, 9, 13, 17, 49\) are presented.