Recent zbMATH articles in MSC 20Chttps://zbmath.org/atom/cc/20C2024-03-13T18:33:02.981707ZUnknown authorWerkzeugSpin Kostka polynomials and vertex operatorshttps://zbmath.org/1528.050712024-03-13T18:33:02.981707Z"Jing, Naihuan"https://zbmath.org/authors/?q=ai:jing.naihuan"Liu, Ning"https://zbmath.org/authors/?q=ai:liu.ningSummary: An algebraic iterative formula for the spin Kostka-Foulkes polynomial \(K^-_{\xi\mu}(t)\) is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur \(Q\)-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of \textit{K. Aokage} [Math. J. Okayama Univ. 63, 133--151 (2021; Zbl 1508.20016)] on the expansion of the Schur \(P\)-function in terms of Schur functions. Tables of \(K^-_{\xi\mu}(t)\) for \(|\xi|\leq6\) are listed.Moments of Kloosterman sums, supercharacters, and elliptic curveshttps://zbmath.org/1528.110712024-03-13T18:33:02.981707Z"Sayed, Fahim"https://zbmath.org/authors/?q=ai:sayed.fahim"Kalita, Gautam"https://zbmath.org/authors/?q=ai:kalita.gautamAuthors' abstract: In this paper, we use a supercharacter theory for \(\mathbb{F}_p^2\) induced by the action of a subgroup of \(\mathrm{GL}_2\left(\mathbb{F}_p\right)\) to express the fifth and seventh power moments of Kloosterman sums in terms of traces of Frobenius endomorphism for certain families of elliptic curves.
Reviewer: Alexey Ustinov (Khabarovsk)Rings of invariants for three dimensional modular representationshttps://zbmath.org/1528.130032024-03-13T18:33:02.981707Z"Herzog, Jürgen"https://zbmath.org/authors/?q=ai:herzog.jurgen"Trivedi, Vijaylaxmi"https://zbmath.org/authors/?q=ai:trivedi.vijaylaxmiLet \(p>3\) be a prime number, let \(\mathbb{F}\) be a field of characteristic \(p,\) let \(G\) be a finite abelian group and let \(V\) be an \(n\)-dimensional representation of \(G\) over \(\mathbb{F}.\) Let \(\mathbb{F}[V]^G\) be the ring of invariants. The aim of the paper is to study the ring of invariants in the case \(G=(\mathbb{Z}/p\mathbb{Z})^r,\) for a 3-dimensional generic representation. The ring of invariants are computed and it is shown that these rings are complete intersection rings with embedding dimension \(\lceil \frac{r}{2} \rceil +3.\)
Reviewer: Cristodor-Paul Ionescu (Bucureşti)Wedderburn decomposition of a semisimple group algebra \(\mathbb{F}_qG\) from a subalgebra of factor group of \(G\)https://zbmath.org/1528.160242024-03-13T18:33:02.981707Z"Mittal, Gaurav"https://zbmath.org/authors/?q=ai:mittal.gaurav"Sharma, R. K."https://zbmath.org/authors/?q=ai:sharma.rajendra-kumarLet \(\mathbb F_q\) be the field of \(q=p^k\) elements, and let \( G\) be a finite group of order coprime with \(p\). In the paper under review, it is shown that the Wedderburn decomposition of a semisimple group algebra \(\mathbb F_q G\) is determined by the Wedderburn decomposition of the group algebra \(\mathbb F_q(G/H)\) for a normal subgroup \(H\) of \(G\) of order \(2\), provided that all the following conditions hold:
\begin{itemize}
\item[1.] \(H\cap G'=1\).
\item[2.] The number of conjugacy classes of \(G\) equals twice the number of conjugacy classes of \(G/H\).
\item[3.] The center of each one of the Wedderburn components of \(\mathbb F_p G\) is \(\mathbb F_p\).
\end{itemize}
As an application, for each \(p>5\), the Wedderburn decomposition of the group algebra of the non-metabelian group \(A_5\rtimes C_4\) over \(\mathbb F_q\) is described.
Reviewer: Diego García-Lucas (Murcia)Quantum affine wreath algebrashttps://zbmath.org/1528.170112024-03-13T18:33:02.981707Z"Rosso, Daniele"https://zbmath.org/authors/?q=ai:rosso.daniele"Savage, Alistair"https://zbmath.org/authors/?q=ai:savage.alistairSummary: To each symmetric algebra we associate a family of algebras that we call \textit{quantum affine wreath algebras}. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type \(A\) and as quantum deformations of affine wreath algebras. We study the structure theory of these new algebras and their natural cyclotomic quotients.Character sheaves for graded Lie algebras: stable gradingshttps://zbmath.org/1528.170222024-03-13T18:33:02.981707Z"Vilonen, Kari"https://zbmath.org/authors/?q=ai:vilonen.kari"Xue, Ting"https://zbmath.org/authors/?q=ai:xue.tingThe authors initiate the study of character sheaves for general \(\mathbb{Z}/m\mathbb{Z}\)-graded Lie algebras, with the goal to determine the cuspidal character sheaves, i.e., the character sheaves which cannot be obtained (as a direct summand) by parabolic induction from smaller graded Lie algebras. That is, the authors construct full support character sheaves for stably graded Lie algebras. Conjecturally, these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description via analyzing the Fourier transform of the nearby cycle sheaves.
Reviewer: Mee Seong Im (Annapolis)Derived traces of Soergel categorieshttps://zbmath.org/1528.180132024-03-13T18:33:02.981707Z"Gorsky, Eugene"https://zbmath.org/authors/?q=ai:gorsky.eugene"Hogancamp, Matthew"https://zbmath.org/authors/?q=ai:hogancamp.matthew"Wedrich, Paul"https://zbmath.org/authors/?q=ai:wedrich.paulIn this paper the authors study two kinds of categorical traces on monoidal dg categories with applications to categories of Soergel bimodules. More precisely, they (i) compute explicitly the derived vertical trace (the usual Hochschild homology) of the category of Soergel bimodules in arbitrary types, and (ii) introduce the notion of derived horizontal trace of a monoidal dg category and compute it for Soergel bimodules in type \(A\). As an application, the authors define a derived annular Khovanov-Rozansky link invariant as the derived horizontal class of a Rouquier complex, which has an action of full twist insertion. This this way a categorification of the HOMFLY-PT skein module of the solid torus is obtained.
Reviewer: Pedro Vaz (Louvain-la-Neuve)Evaluation birepresentations of affine type \(A\) Soergel bimoduleshttps://zbmath.org/1528.180212024-03-13T18:33:02.981707Z"Mackaay, Marco"https://zbmath.org/authors/?q=ai:mackaay.marco"Miemietz, Vanessa"https://zbmath.org/authors/?q=ai:miemietz.vanessa"Vaz, Pedro"https://zbmath.org/authors/?q=ai:vaz.pedroFinitary birepresentation theory of finite type Soergel bimodules in characteristic zero has been a topic intensive study [\textit{T. Kildetoft} et al., Trans. Am. Math. Soc. 371, No. 8, 5551--5590 (2019; Zbl 1409.18005); \textit{M. Mackaay} and \textit{V. Mazorchuk}, J. Pure Appl. Algebra 221, No. 3, 565--587 (2017; Zbl 1404.18013); \textit{M. Mackaay} et al., Proc. Lond. Math. Soc. (3) 126, No. 5, 1585--1655 (2023; Zbl 1522.18033); \textit{M. Mackaaij} and \textit{D. Tubbenhauer}, Can. J. Math. 71, No. 6, 1523--1566 (2019; Zbl 1512.20016); \textit{J. Zimmermann}, J. Pure Appl. Algebra 221, No. 3, 666--690 (2017; Zbl 1360.18008)]. This paper initiates the study of a class of finitary and triangulated birepresentations of affine type \(A\) Soergel bimodules. In type \(A\), as is well-known, there are evaluation maps from the affine Hecke algebra to the finite type Hecke algebra, which are homomorphisms of algebras so that any representation of the latter algebra can be pulled back to a representation of the former algebra through such a map. These so-called \textit{evaluation representations} form an important and well-studied class of finite-dimensional representations of affine type \ Hecke algebras [\textit{V. Chari} and \textit{A. Pressley}, Pac. J. Math. 174, No. 2, 295--326 (1996; Zbl 0881.17011); \textit{J. Du} and \textit{Q. Fu}, Algebr. Represent. Theory 19, No. 2, 355--376 (2016; Zbl 1403.20059); \textit{B. Leclerc} et al., Prog. Math. 210, 115--153 (2003; Zbl 1085.17010)]. It has been conjectured that these evaluation maps can be categorified by monoidal \textit{evaluation functors} from affine type \(A\) Soergel bimodules to the homotopy category of bounded complexes in finite type \(A\) Soergel bimodules. This paper defines such functors, using them to categorify the aforementioned evaluation representations in \ the form of triangulated birepresentations, obtained by pulling back the triangulated birepresentations induced by finitary birepresentations of finite type \(A\) Soergel bimodules through these functors. Moreover, in case the original finitary birepresentation is simple transitive, it is shown that the evaluation birepresentation admits a \textit{finitary cover}, i.e., a finitary birepresentation together with an essentially surjective and epimorphic morphism of additive birepresentations from that cover to the evaluation birepresentation, which categorifies the well-known fact that the corresponding evaluation representations are quotients of certain cell representations defined by \textit{J. J. Graham} and \textit{G. I. Lehrer} [Enseign. Math. (2) 44, No. 3--4, 173--218 (1998; Zbl 0964.20002)].
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] recalls the basics of extended and non-extended affine Hecke algebras of affine type \(A\), the evaluation maps, the Grothendieck-Lehrer cell modules and the evaluation representations.
\item[\S 3] briefly recalls Soergel calculus in finite and affine type \(A\), the latter both in non-extended and the extended version.
\item[\S 4] recalls some basic results on Rouquier complexes in finite type \(A\), focusing on a special type of Rouquier complex, which is fundamental for the definition of the evaluation functors in the next section. A mixed diagrammatic calculus for morphisms between products of Bott-Samelson bimodules and these special Rouquier complexes, all in finite type \(A\), is developed.
\item[\S 5] defines the evaluation functors by assigning a bounded complex of finite type \(A\) Soergel bimodules to each extended affine type \(A\) Bott-Samelson bimodule and a map between such complexes to each generating extended affine type \(A\) Soergel calculus diagram. The main result of this section is that this assignment is well-defined up to homotopy equivalence.
\item[\S 6] introduces the notion of a triangulated birepresentation of an additive bicategory, defining evaluation birepresentations of Soergel bimodules in extened affine type \(A\). It is then established that each evaluation birepresentation has a finitary cover. The simplest non-trivial evaluation birepresentations, which are the ones induced by cell birepresentaion of finite type \(A\) with subregular apex, are investigated. It is shown that these admit a simple transitive finitary cover whose underlying algebra is a signed version of the zigzag algebra of affine type \(A\).
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Representations of finite groups. Abstracts from the workshop held April 16--21, 2023https://zbmath.org/1528.200012024-03-13T18:33:02.981707ZSummary: The workshop \textit{Representations of Finite Groups} was organised by Olivier Dudas (Marseille), Meinolf Geck (Stuttgart), Radha Kessar (Manchester), and Gabriel Navarro (Valencia). It covered a wide variety of aspects of the representation theory of finite groups and related topics, and showcased several recent breakthrough results.Explicit universal axioms for Kaplansky groupshttps://zbmath.org/1528.200052024-03-13T18:33:02.981707Z"Bakulin, Sergey"https://zbmath.org/authors/?q=ai:bakulin.sergey"Miasnikov, Alexei"https://zbmath.org/authors/?q=ai:myasnikov.alexei-gSummary: In this paper, we give an explicit set of universal axioms in the pure group theory language for the class of groups whose group algebras over an arbitrary field have no zero divisors. This answer a question posed by \textit{B. Fine} et al. [Groups Complex. Cryptol. 10, No. 1, 43--52 (2018; Zbl 1507.20019)].
For the entire collection see [Zbl 1435.20002].Affine wreath product algebrashttps://zbmath.org/1528.200062024-03-13T18:33:02.981707Z"Savage, Alistair"https://zbmath.org/authors/?q=ai:savage.alistairSummary: We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke-Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new results, including proofs of two open conjectures of Kleshchev and Muth.On the basic representation of the double affine Hecke algebra at critical levelhttps://zbmath.org/1528.200072024-03-13T18:33:02.981707Z"van Diejen, J. F."https://zbmath.org/authors/?q=ai:van-diejen.jan-felipe"Emsiz, E."https://zbmath.org/authors/?q=ai:emsiz.erdal"Zurrián, I. N."https://zbmath.org/authors/?q=ai:zurrian.ignacio-nahuelThe aim of the paper under review is to construct a monomorphism of the double affine Hecke algebra at critical level \(\mathcal{H}\) associated to any irreducible reduced affine root system \(R\) with a reduced gradient root system. To do this, the authors adapt the corresponding results of \textit{I. Cherednik} [Double affine Hecke algebras. Cambridge: Cambridge University Press (2005; Zbl 1087.20003)] and \textit{I. G. Macdonald} [Affine Hecke algebras and orthogonal polynomials. Cambridge: Cambridge University Press (2003: Zbl 1024.33001] for general Macdonald parameter values \(q\) (not equal to a root of unity) to the critical level \(q=1\).
Reviewer: Enrico Jabara (Venezia)A characterization of nested groups in terms of conjugacy classeshttps://zbmath.org/1528.200082024-03-13T18:33:02.981707Z"Burkett, Shawn T."https://zbmath.org/authors/?q=ai:burkett.shawn-t"Lewis, Mark L."https://zbmath.org/authors/?q=ai:lewis.mark-lSummary: A group is nested if the centers of the irreducible characters form a chain. In this paper, we will show that there is a set of subgroups associated with the conjugacy classes of group so that a group is nested if and only if these subgroups form a chain.On the irreducible characters of Suzuki \(p\)-groupshttps://zbmath.org/1528.200092024-03-13T18:33:02.981707Z"Di, Wendi"https://zbmath.org/authors/?q=ai:di.wendi"Feng, Tao"https://zbmath.org/authors/?q=ai:feng.tao"He, Zhiwen"https://zbmath.org/authors/?q=ai:he.zhiwen\textit{G. Higman} [Ill. J. Math. 7, 79--96 (1963; Zbl 0112.02107)] determined all groups \(G\) of order a power of \(2\) which possess an automorphism that permutes their involutions cyclically. With some modifications this definition also applies when \(p\) is an odd prime number, so it makes sense to consider Suzuki \(p\)-groups for a general prime \(p\).
In the paper under review, the authors completely determine the irreducible characters of the four families of Suzuki \(p\)-groups.
Reviewer: Enrico Jabara (Venezia)Non-solvable groups whose character degree graph has a cut-vertex. IIIhttps://zbmath.org/1528.200102024-03-13T18:33:02.981707Z"Dolfi, Silvio"https://zbmath.org/authors/?q=ai:dolfi.silvio"Pacifici, Emanuele"https://zbmath.org/authors/?q=ai:pacifici.emanuele"Sanus, Lucia"https://zbmath.org/authors/?q=ai:sanus.luciaFor a finite group \(G\), the character degree graph of \(G\) is a simple, undirected graph whose vertices correspond to the prime divisors of the numbers in the set \(cd(G)\), representing the irreducible complex characters of \(G\), and two distinct vertices \(p\) and \(q\) are adjacent if and only if \(pq\) divides some number in \(cd(G)\). The graph contains a cut-vertex if the removal of a vertex increases the number of connected components in the graph.
The paper under review is the third and final part of a series on non-solvable groups whose character degree graph contains a cut-vertex.
In the first paper [Vietnam J. Math. 51, No. 3, 731--753 (2023; Zbl 1522.20032)], the authors establish that these groups possess a unique non-solvable composition factor \(S\), which is a non-abelian simple group. In the second paper [Ann. Mat. Pura Appl. (4) 202, No. 4, 1753--1780 (2023; Zbl 1517.20010)], they explore all isomorphism types for \(S\), except the case \(S = \mathrm{PSL}_2(2^a)\) for some integer \(a \geq 2\). The remaining case is addressed in the paper under review, completing the classification of groups with the cut-vertex property.
Reviewer: Amin Saeidi (Polokwane)Finite groups with special codegreeshttps://zbmath.org/1528.200112024-03-13T18:33:02.981707Z"Yang, Dongfang"https://zbmath.org/authors/?q=ai:yang.dongfang"Gao, Cong"https://zbmath.org/authors/?q=ai:gao.cong"Lv, Heng"https://zbmath.org/authors/?q=ai:lv.hengSummary: In this paper, it is proved that the finite group \(G\) is solvable if \(\text{cod}(\chi ) \le p_{\chi }\cdot \chi (1)\) for any nonlinear irreducible character \(\chi\) of \(G\), where \(p_{\chi }\) is the largest prime divisor of \(|G:\ker \chi |\).Blob algebra approach to modular representation theoryhttps://zbmath.org/1528.200122024-03-13T18:33:02.981707Z"Libedinsky, Nicolas"https://zbmath.org/authors/?q=ai:libedinsky.nicolas"Plaza, David"https://zbmath.org/authors/?q=ai:plaza.davidSummary: Two decades ago, \textit{P. P. Martin} and \textit{D. Woodcock} [LMS J. Comput. Math. 6, 249--296 (2003; Zbl 1080.20004)] made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type \(\widetilde{A}_1\). In this paper, we take that observation far beyond its original scope. We conjecture that for \(\widetilde{A}_n\) there is an equivalence of categories between the characteristic \(p\) diagrammatic Hecke category and a `blob category' that we introduce (using certain quotients of KLR algebras called \textit{generalized blob algebras}). Using alcove geometry we prove the `graded degree' part of this equivalence for all \(n\) and all prime numbers \(p\). If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic \(p\) give the \(p\)-Kazhdan-Lusztig polynomials in type \(\widetilde{A}_n\). We prove this for \(\widetilde{A}_1\), the only case where the \(p\)-Kazhdan-Lusztig polynomials are known.
This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures.\(\mathrm{Sym}(n)\)- and \(\mathrm{Alt}(n)\)-modules with an additive dimensionhttps://zbmath.org/1528.200132024-03-13T18:33:02.981707Z"Corredor, Luis Jaime"https://zbmath.org/authors/?q=ai:corredor.luis-jaime"Deloro, Adrien"https://zbmath.org/authors/?q=ai:deloro.adrien"Wiscons, Joshua"https://zbmath.org/authors/?q=ai:wiscons.joshuaThe paper studies \(\mathrm{Sym}(n)\)- and \(\mathrm{Alt}(n)\)-modules that inherit a natural notion of \textit{dimension} introduced by the authors, who ``fully identify the faithful such modules of least dimension''. More precisely, the authors define a \textit{modular universe} \(\mathcal U\) as a subcategory of the one of abelian groups that satisfies some natural closure properties (among which taking inverses, Cartesian products, quotients, images and kernels, sums). Examples of modular universes, given by the authors, include the category of all abelian groups and the categories of abelian groups that are interpretable in some first-order theory, with the interpretable group morphisms as arrows. Given a modular universe~\(\mathcal U\), the authors call an \textit{additive dimension on \(\mathcal U\)} any function \(\dim: \mathrm{Ob}(\mathcal U)\rightarrow\mathbb N\) such that the equality
\[
\dim V=\dim\ker f+\dim\mathrm{im} f
\]
holds for each arrow \(f: V\rightarrow W\) of \(\mathcal U\) (the reviewer notes that with the authors' setting, this property is equivalent with asking that dim be preserved by one-to-one arrows of~\(\mathcal U\) and satisfy \(\dim A/B=\dim A-\dim B\) for all objects \(B\leqslant A\) of \(\mathcal U\)). Examples of additive dimensions, given by the authors, include the vector-space linear dimension, the (finite) Morley rank and the o-minimal dimension (the reviewer also notes that in the category of T-affine spaces -- which is not a modular universe -- as considered in [\textit{C. Milliet}, J. Algebra 569, 143--168 (2021; Zbl 1468.16031)], the Zariski dimension is additive by Theorem 5.11). If \(V\) is an object from such a pair \((\mathcal U,\dim)\), then \(V\) is said \textit{dimension-connected} if \(\dim W<\dim V\) (or equivalently \(\dim(V/W)>0\)) holds for every proper subobject \(W<V\) of \(\mathcal U\). The authors point out that the objects of a modular universe that carries an additive dimension need not satisfy a descending chain condition; still, they satisfy a Wagnerian descending chain condition ``up to \((\dim 0)\)-index'' (see Theorem 1.9 of [\textit{F. Wagner}, Lect. Notes Log. 20, 440--467 (2005; Zbl 1081.03032)]). Finally, in the paper, an abelian group \(V\) is said to have \textit{characteristic \(p\)} for \(p\) prime, if it is of exponent~\(p\). It is said to have \textit{characteristic \(0\)} if it divisible.
Now let \(G\) be a fixed group and \(\mathcal U\) a fixed modular universe with an additive dimension. The authors call any \(V\in\mathrm{Ob}(\mathcal U)\) on which \(G\) acts by arrows of \(\mathcal U\) a \textit{\(G\)-module}; a dimension-connected \(G\)-module is called \textit{dc-irreducible} (as a \(G\)-module) if it contains no proper non-trivial dimension-connected \(G\)-submodule in \(\mathrm{Ob}(\mathcal U)\). The main result of the paper can be summarised as follows:
{Theorem.} Let \(V\) be a faithful, dc-irreducible \(\mathrm{Sym}(n)\)- or \(\mathrm{Alt}(n)\)-module of dimension \(d\) and characteristic \(q\) with \(d<n\) and \(n\geqslant 7\); in the \(\mathrm{Alt}(n)\)-module case, \(q=2\), further assume \(n\geqslant 10\). Then there is a dimension \(1\), dimension-connected subobject \(L\leqslant V\) in \(\mathcal U\) such that \(V\) can be explicitly reconstructed from \(L\), namely as a quotient \(\operatorname{std}(n,L)/H\) of the \textit{standard module} \(\operatorname{std}(n,L)\) over \(L\), ``up to tensoring with the signature''.
In the theorem, when \(q\) is prime, \(H\) is simply the submodule of elements fixed by \(\mathrm{Sym}(n)\); when \(q\) is zero, \(H\) has dimension~\(0\). The authors tell that the theorem ``generalizes a century-old theorem by \textit{L. E. Dickson} [Trans. Am. Math. Soc. 9, 121--148 (1908; JFM 39.0198.02)] and its more recent companions by \textit{A. Wagner} [Math. Z. 151, 127--137 (1976; Zbl 0321.20008); Math. Z. 154, 103--114 (1977; Zbl 0336.20008)]''; the reviewer would have enjoyed a recall of these results. The impressive proof of the theorem is about 20 pages long and its articulation in 3 main steps is very well explained by the authors.
``The paper is dedicated to those labouring for peace and freedom of speech.''
Writing this review was facilitated by the slides of a September 2021 talk given at Institut Camille Jordan by the third author and available at \url{http://math.univ-lyon1.fr/homes-www/logicum/RankedGroups/Slides/Wiscons.pdf}.
Reviewer: Cédric Milliet (Lyon)On the generalised Springer correspondence for groups of type \(E_8\)https://zbmath.org/1528.200142024-03-13T18:33:02.981707Z"Hetz, Jonas"https://zbmath.org/authors/?q=ai:hetz.jonasLet \(\mathbf{G}\) be a connected reductive algebraic group over an algebraic closure \(k\) of the finite field \(\mathbb{F}_{p}\). Let \(\mathbf{W}\) be the Weyl group of \(\mathbf{G}\) and \(\mathcal{N}\mathbf{G}\) be the set of all pairs \((\mathcal{O},\mathcal{E})\) where \(\mathcal{O} \subseteq \mathbf{G}\) is a unipotent conjugacy class and \(\mathcal{E}\) is an irreducible local system on \(\mathcal{O}\) which is equivariant for the conjugation action of \(\mathbf{G}\). The Springer correspondence (defined in [\textit{T. A. Springer}, Invent. Math. 36, 173--207 (1976; Zbl 0374.20054)]) for \(p\) not too small (for arbitrary \(p\) see [\textit{G. Lusztig}, Adv. Math. 42, 169--178 (1981; Zbl 0473.20029)]) defines an injective map \(\iota_{\mathbf{G}}: \mathrm{Irr}(\mathbf{G})\rightarrow \mathcal{N}_{\mathbf{G}}\) which plays a crucial role, for example, in the determination of the values of the Deligne-Lusztig Green functions of [\textit{P. Deligne} and \textit{G. Lusztig}, Ann. Math. (2) 103, 103--161 (1976; Zbl 0336.20029)]. The map \(\iota_{\mathbf{G}}\) is not surjective in general. In order to understand the missing pairs in \(\mathcal{N}\mathbf{G}\), \textit{G. Lusztig} [Invent. Math. 75, 205--272 (1984; Zbl 0547.20032)] developed a generalisation of Springer's correspondence.
In the paper under review, the author proves a conjecture proposed by \textit{G. Lusztig} ,[Proc. Symp. Pure Math. 101, 219--253 (2019; Zbl 1481.20179)]. He completes the determination of the generalised Springer correspondence for connected reductive algebraic groups, by proving the conjecture on the last open cases which occur for groups of type \(\mathsf{E}_{8}\).
Reviewer: Enrico Jabara (Venezia)Axiomatic representation theory of finite groups by way of groupoidshttps://zbmath.org/1528.200152024-03-13T18:33:02.981707Z"Dell'Ambrogio, Ivo"https://zbmath.org/authors/?q=ai:dellambrogio.ivoSummary: 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.
For the entire collection see [Zbl 1476.55006].Hodge theory of Soergel bimodules [after Soergel and Elias-Williamson]https://zbmath.org/1528.200602024-03-13T18:33:02.981707Z"Riche, Simon"https://zbmath.org/authors/?q=ai:riche.simonFor the entire collection see [Zbl 1436.00053].Modular representations and reflection subgroupshttps://zbmath.org/1528.200622024-03-13T18:33:02.981707Z"Williamson, Geordie"https://zbmath.org/authors/?q=ai:williamson.geordieSummary: The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the ``philosophy of deformations'' both as a conceptual and computational tool, and suggest possible connections to Lusztig's ``philosophy of generations''. On the geometric side one can understand deformations in terms of localisation in equivariant cohomology. Recently \textit{D. Treumann} [Math. Ann. 375, No. 1--2, 595--628 (2019; Zbl 1440.20001)] and \textit{S. Leslie} and \textit{G. Lonergan} [J. Reine Angew. Math. 777, 49--87 (2021; Zbl 1478.14033)] have added Smith theory, which provides a useful tool when considering mod \(p\) coefficients. In this context, we make contact with some remarkable work of \textit{A. Hazi} [``Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups'', Preprint, \url{arXiv:1708.07072}]. Using recent work of Abe on Soergel bimodules, we are able to reprove and generalise some of Hazi's results [loc. cit.]. Our aim is to convince the reader that the work of Hazi [loc. cit.] and Leslie-Lonergan [loc. cit.] can usefully be viewed as some kind of localisation to ``good'' reflection subgroups. These are notes for my lectures at the 2019 Current Developments in Mathematics at Harvard.
For the entire collection see [Zbl 1475.00108].Splendor of Deligne-Lusztig varieties [after Deligne-Lusztig, Broué, Rickard, Bonnafé-Dat-Rouquier]https://zbmath.org/1528.200832024-03-13T18:33:02.981707Z"Dudas, Olivier"https://zbmath.org/authors/?q=ai:dudas.olivierFor the entire collection see [Zbl 1436.00053].On the second largest eigenvalue of some Cayley graphs of the symmetric grouphttps://zbmath.org/1528.200862024-03-13T18:33:02.981707Z"Siemons, Johannes"https://zbmath.org/authors/?q=ai:siemons.johannes"Zalesski, Alexandre"https://zbmath.org/authors/?q=ai:zalesski.alexandre-eGiven a group \(G\) and a subset \(H \subset G\) such that \(e \not \in H\) and \(H=H^{-1}\), one can form the Cayley graph of this data: it is the graph with set of vertices \(G\) and where \(g_1,g_2\) are adjacent whenever \(g_2 \in Hg_1\). Assuming moreover that \(H\) is included in no proper subgroup of \(G\) will imply the connectedness of this graph.
The paper is concerned with the eigenvalues of this graph (defined as eigenvalues of the adjacency matrix). More specifically, the largest eigenvalue is always \(|H|\), so the authors study the second largest eigenvalue, in some cases where \(G\) is the symmetric or alternating group on \(\{1,\dots,n\}\).
First, the authors consider the case where \(H\) is the set \(C(n,k)\) of all \(k\)-cycles for \(k=n\) or \(k=n-1\). They give an explicit formula for the second eigenvalue. This follows from the theory of characters. Then, they consider a more complicated subset \(H\). For \(r,k\) such that \(1 \leq r < k \leq n\), they define \(C(n,k;r)\) as the set of \(k\)-cycles whose support contain \(\{1,\dots,r\}\). In that case, they show that some explicit number is an eigenvalue and conjecture that it is the second eigenvalue. This conjecture is proved when \(k=r+1\). This result follows from the fact that the element \(\sum_{h \in H} h\) in the group algebra of \(G\) has this eigenvalue for a specific \(G\)-module (in fact the \(n\)-dimensional permutation \(G\)-module).
Reviewer: Pierre-Emmanuel Chaput (Vandœuvre-lès-Nancy)Multiplication formulas and isomorphism theorem of \({\imath}\)Schur superalgebrashttps://zbmath.org/1528.200912024-03-13T18:33:02.981707Z"Chen, Jian"https://zbmath.org/authors/?q=ai:chen.jian.4"Luo, Li"https://zbmath.org/authors/?q=ai:luo.li|luo.li.1The main player of the paper is the \(\iota\)Schur superalgebra. Let us first precise what the paper means by ``\(\iota\)Schur'' and by ``superalgebra''.
First, recall the definition of the \(q\)-Schur algebra in the most classical case (type \(A\)). Let \(W=\mathfrak S_n\) be the permutation group. Let \(H(W)\) be the associated Hecke algebra. The \(q\)-Schur algebra (of type \(A\)) is defined as the endomorphism algebra of some \(H(W)\)-module \(M\), where \(M=\bigoplus_{\lambda}M_\lambda\), the sum is taken by the parabolic subgroups \(W_\lambda\subset W\), the module \(M_\lambda\) is obtained by induction from the \(1\)-dimensional \(H(W_\lambda)\)-module. Here, there are two possible choices of the \(1\)-dimensional module: the \(q\)-version of the trivial representation (all standard generators act by \(q\)) and the sign representation (all standard generators act by \(-1\)). These two choices give us isomorphic \(q\)-Schur algebras.
Now, the ``\(\iota\)Schur superalgebra'' contains two modifications of this definition. ``\(\iota\)Schur'' means that we start from the group \(W\) of type \(B/C\) instead of taking \(W=\mathfrak S_n\) (type \(A\)). ``Superalgebra'' means that we consider some parabolic subgroups with special decompositions \(W_\lambda=W_{\lambda^{(0)}}\times W_{\lambda^{(1)}}\) and in the definition of the \(H(W)\)-module \(M_\lambda\) we induce from the \(1\)-dimensional representation which is \(q\)-trivial for \(H(W_{\lambda^{(0)}})\) and sign for \(H(W_{\lambda^{(1)}})\).
The paper proves the following results.
\begin{itemize}
\item It proves that the \(H(W)\)-module \(M\) in the definition of the \(\iota\)Schur superalgebra is isomorphic to the \(q\)-tensor superspace.
\item The paper obtains some multiplication formulas for some special generators of the \(\iota\)Schur superalgebra. These formulas are used to construct a monomial basis and a canonical basis.
\item The paper constructs, under some light assumption on \(q\) (this assumption is always true for generic \(q\)) that the \(\iota\)Schur superalgebra is isomorphic to a direct sum of tensor products of \(q\)-Schur superalgebras (of type \(A\)). As a consequence, the paper gives a semisimplicity criteria for the \(\iota\)Schur superalgebra (under the same assumption on \(q\)).
\item The paper also considers a slightly different version of the \(\iota\)Schur superalgebra. In the definition of this version, we take less parabolic subgroups \(W_\lambda\) than in the previous construction. For this alternative version, the paper gives similar results (bases, isomorphism, semisimplicity criteria).
\end{itemize}
Reviewer: Ruslan Maksimau (Paris)Signature cocycles on the mapping class group and symplectic groupshttps://zbmath.org/1528.200972024-03-13T18:33:02.981707Z"Benson, Dave"https://zbmath.org/authors/?q=ai:benson.david-john"Campagnolo, Caterina"https://zbmath.org/authors/?q=ai:campagnolo.caterina"Ranicki, Andrew"https://zbmath.org/authors/?q=ai:ranicki.andrew-a"Rovi, Carmen"https://zbmath.org/authors/?q=ai:rovi.carmenSummary: \textit{W. Meyer} [Math. Ann. 201, 239--264 (1973; Zbl 0241.55019)] constructed a cocycle in \(H^2(\mathsf{Sp}(2g, \mathbb{Z}); \mathbb{Z})\) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer \(N\). Using these results, we are able to give a complete answer for \(N = 2, 4, \text{ and } 8\), and based on a theorem of \textit{P. Deligne} [C. R. Acad. Sci., Paris, Sér. A 287, 203--208 (1978; Zbl 0416.20042)], we show that this is the best we can hope for using this method.Nearby cycle sheaves for symmetric pairshttps://zbmath.org/1528.220112024-03-13T18:33:02.981707Z"Grinberg, Mikhail"https://zbmath.org/authors/?q=ai:grinberg.mikhail"Vilonen, Kari"https://zbmath.org/authors/?q=ai:vilonen.kari"Xue, Ting"https://zbmath.org/authors/?q=ai:xue.tingThe authors present a nearby cycle sheaf construction in the development of the theory of character sheaves in the context of symmetric spaces. They allow equivariant local systems as coefficients, and this construction, and its variant, produce all character sheaves up to parabolic induction in the setting of classical symmetric spaces.
They consider a connected complex reductive group \(G\) and an involution \(\theta: G \to G\), giving rise to a symmetric pair \((G, K)\), with \(K = G^\theta\). They have a decomposition \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) into \(+1\) and \(-1\) eigenspaces of the involution \(\theta\). Writing \(\mathcal{N}\) for the nilpotent cone in \(\mathfrak{g}\), let \(\mathcal{N}_{\mathfrak{p}} = \mathcal{N} \cap \mathfrak{p}\). Let \(\mathfrak{a} \subset \mathfrak{p}\) be a Cartan subspace, which is a maximal abelian subspace of \(\mathfrak{p}\) consisting of semisimple elements, and let \(W_{\mathfrak{a}} = N_K (\mathfrak{a}) / Z_K (\mathfrak{a})\) be the little Weyl group. Write \(\mathfrak{p}^{\text{rs}}\) for the set of regular semisimple elements of \(\mathfrak{p}\) and let \(\mathfrak{a}^{\text{rs}} = \mathfrak{a} \cap \mathfrak{p}^{\text{rs}}\). This gives the adjoint quotient map: \(f: \mathfrak{p} \to \mathfrak{p} /\!\!/ K \cong \mathfrak{a} / W_{\mathfrak{a}}\). Note that \(f^{-1} (0) = \mathcal{N}_{\mathfrak{p}}\). The fiber of this map at a point \(\bar{a}_0 \in \mathfrak{a}^{\text{rs}} / W_{\mathfrak{a}}\) is a regular semisimple \(K\)-orbit \(X_{\bar{a}_0}\), with equivariant fundamental group \(\pi_1^K (X_{\bar{a}_0}) = I := Z_K (\mathfrak{a}) / Z_K (\mathfrak{a})^0\).
The authors construct a nearby cycle sheaf \(P_\chi \in \text{Perv}_K (\mathcal{N}_{\mathfrak{p}})\), a \(K\)-equivariant perverse sheaf on the nilpotent cone \(\mathcal{N}_{\mathfrak{p}}\), for each character \(\chi \in \hat{I}\). They study the topological Fourier transform of \(P_\chi\), and show that this Fourier transform is an IC-extension of a \(K\)-equivariant local system on \(\mathfrak{p}^{\text{rs}}\) under a suitable identification of \(\mathfrak{p}\) and \(\mathfrak{p}^*\). The local systems on \(\mathfrak{p}^{\text{rs}}\) arising in this way are described in Theorem 3.6 (page 21), where Hecke algebras with parameters \(\pm 1\), attached to certain Coxeter subgroups of \(W_\mathfrak{a}\), enter the description. The construction of these Hecke algebras can be viewed as an instance of endoscopy. This nearby cycle sheaf construction can be regarded as a replacement for the Grothendieck--Springer resolution in classical Springer theory.
Reviewer: Mee Seong Im (Annapolis)On the attribute of uniform convergence of Fourier series of the Vilenkin system in the case of unbounded \(p_k\)https://zbmath.org/1528.420372024-03-13T18:33:02.981707Z"Voronov, S. M."https://zbmath.org/authors/?q=ai:voronov.sergei-mirzoevichSummary: Series with respect to a system of characters of a zero-dimensional compact commutative group are considered. Generalization of the test of convergence of Fourier series of the Vilenkin system in the case of unbounded quasimonotone \(p_k\) for functions having a generalized bounded \(\Phi \)-fluctuation, which was earlier obtained in the case of bounded sequences \({p_k}\), is proved.