Jespers and Sun conjectured in [27] that if a finite group G has the property ND, i.e. for every nilpotent element n in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in \mathbb{Z}G$, then at most one of the simple components of the group algebra $\mathbb{Q} G$ has reduced degree bigger than 1. With the exception of one very special series of groups we are able to answer their conjecture, showing that it is true—up to exactly one exception. To do so, we first classify groups with the so-called SN property which was introduced by Liu and Passman in their investigation of the Multiplicative Jordan Decomposition for integral group rings.

The conjecture of Jespers and Sun can also be formulated in terms of a group q(G) made from the group generated by the unipotent units, which is trivial if and only if the ND property holds for the group ring. We answer two more open questions about q(G) and notice that this notion allows to interpret the studied properties in the general context of linear semisimple algebraic groups. Here we show that q(G) is finite for lattices of big rank but can contain elements of infinite order in small rank cases.

We then study further two properties which appeared naturally in these investigations. A first which shows that property ND has a representation theoretical interpretation, while the other can be regarded as indicating that it might be hard to decide ND. Among others we show these two notions are equivalent for groups with SN.

A finite group is said to be n-cyclic if it contains n cyclic subgroups. For a finite group G, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group G and is denoted by $\textit {cdeg} (G)$. In this paper, we classify all $12$-cyclic groups. We also prove that the set of cyclicity degrees for all the finite groups is dense in $[0,1]$, which solves a problem posed by Tărnăuceanu and Tóth [‘Cyclicity degrees of finite groups’, Acta Math. Hungar. 145(2) (2015), 489–504].

We propose and present evidence for a conjectural global-local phenomenon concerning the p-rationality of height-zero characters. Specifically, if $\chi $ is a height-zero character of a finite group G and D is a defect group of the p-block of G containing $\chi $, then the p-rationality of $\chi $ can be captured inside the normalizer ${\mathbf {N}}_G(D)$.

In 1954, B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018, G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.

In this article, we study the Johnson homomorphisms of basis-conjugating automorphism groups of free groups. We construct obstructions for the surjectivity of the Johnson homomorphisms. By using it, we determine their cokernels of degree up to four and give further observations for degree greater than four. As applications, we give the affirmative answer to the Andreadakis problem for the basis-conjugating automorphism groups of free groups at degree four. Moreover, we calculate twisted first cohomology groups of the braid-permutation automorphism groups of free groups.

The trigonometric double affine Hecke algebra $\mathbf {H}_c$ for an irreducible root system depends on a family of complex parameters c. Given two families of parameters c and $c'$ which differ by integers, we construct the translation functor from $\mathbf {H}_{c}\text{-}{\mathrm{Mod}}$ to $\mathbf {H}_{c'}\text{-}{\mathrm{Mod}}$ and prove that it induces equivalence of derived categories. This is a trigonometric counterpart of a theorem of Losev on the derived equivalences for rational Cherednik algebras.

We strengthen two results of Moretó. We prove that the index of the Fitting subgroup is bounded in terms of the degrees of the irreducible monomial Brauer characters of the finite solvable group G and it is also bounded in terms of the average degree of the irreducible Brauer characters of G that lie over a linear character of the Fitting subgroup.

We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the $\ell $-adic local systems on affine spaces in characteristic $p> 0$ whose trace functions are these exponential sums. The exponential sums here are much more general than we previously were able to consider. As a byproduct, we determine the number of irreducible components of maximal dimension in certain intersections of Fermat surfaces. We also show that in any family of such local systems, say parameterized by an affine space S, there is a dense open set of S over which the geometric monodromy group of the corresponding local system is a fixed known group.

A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. These rings generalize the GKM construction for equivariant cohomology rings of flag, regular semisimple Hessenberg and permutohedral varieties. These cohomology rings carry two actions of the symmetric group $S_n$ whose graded characters are both of general interest in algebraic combinatorics. In this paper, we generalize the graded $S_n$-representations from the cohomologies of the above varieties to splines on Cayley graphs of $S_n$ and then (1) give explicit module and ring generators for whenever the $S_n$-generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one $S_n$-representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets.

A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture [15], which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open.

A Steiner coset partition of G, with respect to distinct subgroups $H_1,\dots ,H_r$, is a coset partition of G that contains exactly one coset of each $H_i$. In the quest of a more structural version of the Herzog–Schönheim Conjecture, it was shown that there is no Steiner coset partition of G with respect to any $r\geq 2$ subgroups $H_i$ that mutually commute [1]. In this article, we show that this result holds for $r=4$ mutually commuting subgroups provided that G does not have $C_2\times C_2\times C_2$ as a quotient, where $C_2$ is the cyclic group of order $2$. We further give an explicit construction of Steiner coset partitions of the n-fold direct product $G^*=C_p\times \ldots \times C_p$ for p prime and $n\geq 3$. This construction lifts to every group extension of $G^*$.

If all of the entries of a large $S_n$ character table are covered up and you are allowed to uncover one entry at a time, then how can you quickly identify all of the indexing characters and conjugacy classes? We present a fast algorithmic solution that works even when n is so large that almost none of the entries of the character table can be computed. The fraction of the character table that needs to be uncovered is $O( n^2 \exp({-}2\pi\sqrt{n/6}))$, and for many of these entries we are only interested in whether the entry is zero.

Continuing our work on group-theoretic generalisations of the prime Ax–Katz Theorem, we give a lower bound on the p-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\dots,f_r)$ of r maps $f_j\,{:}\,A\rightarrow B_j$ between arbitrary finite commutative groups A and $B_j$ in terms of the invariant factors of $A, B_1,B_2, \cdots,B_r$ and the functional degrees of the maps $f_1,f_2, \dots,f_r$.

In their celebrated paper [CLR10], Caputo, Liggett and Richthammer proved Aldous’ conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality — a certain inequality of operators in the group ring $\mathbb{R}\left[{\mathrm{Sym}}_{n}\right]$ of the symmetric group. Here we generalise the Octopus Inequality and apply it to generalising the Caputo–Liggett–Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

Let G be a finite group and p be a prime. We prove that if G has three codegrees, then G is an M-group. We prove for some prime p that if the degree of every nonlinear irreducible Brauer character of G is a prime, then for every normal subgroup N of G, either $G/N$ or N is an $M_p$-group.

In their 1988 paper ‘Gluing of perverse sheaves and discrete series representations’, D. Kazhdan and G. Laumon constructed an abelian category $\mathcal{A}$ associated to a reductive group G over a finite field with the aim of using it to construct discrete series representations of the finite Chevalley group $G(\mathbb{F}_q)$. The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproved in 2001 by R. Bezrukavnikov and A. Polishchuk, who found a counterexample in the case $G = SL_3$. Polishchuk then made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension, and suggested that this conjecture could lead toward a proof that Kazhdan and Laumon’s construction is well defined. He proved this conjecture in Types $A_1, A_2, A_3$, and $B_2$. In the present paper, we prove Polishchuk’s conjecture for all types, and prove that Kazhdan and Laumon’s construction is indeed well defined, giving a new geometric construction of discrete series representations of $G(\mathbb{F}_q)$.

We describe several exotic fusion systems related to the sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow 3-subgroups of the Conway group $\textrm{Co}_1$ and the Thompson group $\textrm{F}_3$, and a Sylow 5-subgroup of the Monster M, as well as a particular maximal subgroup of the latter two p-groups. This work is supported by computations in MAGMA.

We describe algebraically, diagrammatically, and in terms of weight vectors, the restriction of tensor powers of the standard representation of quantum $\mathfrak {sl}_2$ to a coideal subalgebra. We realize the category as a module category over the monoidal category of type $\pm 1$ representations in terms of string diagrams and via generators and relations. The idempotents projecting onto the quantized eigenspaces are described as type $B/D$ analogues of Jones–Wenzl projectors. As an application, we introduce and give recursive formulas for analogues of $\Theta$-networks.

We introduce an explicit family of representations of the double affine Hecke algebra $\mathbb {H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial representations provide concrete realisations of a natural family of cyclic Y-parabolically induced $\mathbb {H}$-representations. We recover Cherednik’s well-known polynomial representation as a special case.

The quasi-polynomial representation gives rise to a family of commuting operators acting on spaces of quasi-polynomials. These generalize the Cherednik operators, which are fundamental in the study of Macdonald polynomials. We provide a detailed study of their joint eigenfunctions, which may be regarded as quasi-polynomial, multi-parametric generalisations of nonsymmetric Macdonald polynomials. We also introduce generalizations of symmetric Macdonald polynomials, which are invariant under a multi-parametric generalization of the standard Weyl group action.

We connect our results to the representation theory of metaplectic covers of reductive groups over non-archimedean local fields. We introduce root system generalizations of the metaplectic polynomials from our previous work by taking a suitable restriction and reparametrization of the quasi-polynomial generalizations of Macdonald polynomials. We show that metaplectic Iwahori-Whittaker functions can be recovered by taking the Whittaker limit of these metaplectic polynomials.

We introduce a new algebra $\mathcal {U}=\dot {\mathrm {\mathbf{U}}}_{0,N}(L\mathfrak {sl}_n)$ called the shifted $0$-affine algebra, which emerges naturally from studying coherent sheaves on n-step partial flag varieties through natural correspondences. This algebra $\mathcal {U}$ has a similar presentation to the shifted quantum affine algebra defined by Finkelberg-Tsymbaliuk. Then, we construct a categorical $\mathcal {U}$-action on a certain 2-category arising from derived categories of coherent sheaves on n-step partial flag varieties. As an application, we construct a categorical action of the affine $0$-Hecke algebra on the bounded derived category of coherent sheaves on the full flag variety.

For a group G, a subgroup $U \leqslant G$ and a group A such that $\mathrm {Inn}(G) \leqslant A \leqslant \mathrm {Aut}(G)$, we say that U is an A-covering group of G if $G = \bigcup _{a\in A}U^a$. A theorem of Jordan (1872), implies that if G is a finite group, $A = \mathrm {Inn}(G)$ and U is an A-covering group of G, then $U = G$. Motivated by a question concerning Kronecker classes of field extensions, Neumann and Praeger (1990) conjectured that, more generally, there is an integer function f such that if G is a finite group and U is an A-covering subgroup of G, then $|G:U| \leqslant f(|A:\mathrm {Inn}(G)|)$. A key piece of evidence for this conjecture is a theorem of Praeger [‘Kronecker classes of fields and covering subgroups of finite groups’, J. Aust. Math. Soc. 57 (1994), 17–34], which asserts that there is a two-variable integer function g such that if G is a finite group and U is an A-covering subgroup of G, then $|G:U|\leqslant g(|A:\mathrm {Inn}(G)|,c)$, where c is the number of A-chief factors of G. Unfortunately, the proof of this theorem contains an error. In this paper, using a different argument, we give a correct proof of the theorem.