Let ${\mathscr {G}} $ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb {C}$, excluding the absolutely special case of $A^{(2)}_{2\ell }$. Using the methods and results of Zhu, we prove a duality theorem for general ${\mathscr {G}} $: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for ${\mathscr {G}} $. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of ${\mathscr {G}} $. In particular, this confirms a conjecture of Haines and Richarz.

We give a notion of boundary pair $(\mathcal{B}_-,\mathcal{B}_+)$ for measured groupoids which generalizes the one introduced by Bader and Furman [BF14] for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=\Gamma \ltimes X$ obtained by a probability measure preserving action $\Gamma \curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$, where $(B_-,B_+)$ is a boundary pair for $\Gamma$. For any measured groupoid $(\mathcal{G},\nu )$, we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\nu$ provide other examples of our definition. Following Bader and Furman [BF], we define algebraic representability for an ergodic groupoid $(\mathcal{G},\nu )$. In this way, given any measurable representation $\rho \,:\,\mathcal{G} \rightarrow H$ into the $\kappa$-points of an algebraic $\kappa$-group $\mathbf{H}$, we obtain $\rho$-equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$, where $L_\pm =\mathbf{L}_\pm (\kappa )$ for some $\kappa$-subgroups $\mathbf{L}_\pm \lt \mathbf{H}$. In the particular case when $\kappa =\mathbb{R}$ and $\rho$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.

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.

Let W be a simply laced Weyl group of finite type and rank n. If W has type $E_7$, $E_8$ or $D_n$ for n even, then the root system of W has subsystems of type $nA_1$. This gives rise to an irreducible Macdonald representation of W spanned by n-roots, which are products of n orthogonal roots in the symmetric algebra of the reflection representation. We prove that in these cases, the set of all maximal sets of orthogonal positive roots has the structure of a quasiparabolic set in the sense of Rains–Vazirani. The quasiparabolic structure can be described in terms of certain quadruples of orthogonal positive roots which we call crossings, nestings and alignments. This leads to nonnesting and noncrossing bases for the Macdonald representation, as well as some highly structured partially ordered sets. We use the $8$-roots in type $E_8$ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the $E_8$ root system.

We prove that Thompson’s group T and, more generally, all the Higman–Thompson groups $T_n$ have quadratic Dehn function.

We investigate semigroups S which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on S. We call such an S a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups, including periodic, stable, inverse and several well-known types of simple semigroups, the only DSC members are groups. However, it turns out that there exist nongroup DSC semigroups, which we obtain by utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple.

A subgroup X of a group G is said to be transitively normal if X is normal in any subgroup Y of G such that $X\leq Y$ and X is subnormal in Y. We investigate the structure of generalised soluble groups with dense transitively normal subgroups, that is, groups in which every nonempty open interval in their subgroup lattice contains a transitively normal subgroup. In particular, it will be proved that nonperiodic generalised soluble groups with dense transitively normal subgroups are abelian.

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 compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups. This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians and to compute a Morava K-theory analogue of the J-invariant in terms of the ordinary (Chow) J-invariant.

The Hanna Neumann conjecture (HNC) for a free group G predicts that $\overline{\chi}(U\cap V)\leqslant \overline{\chi} (U)\overline{\chi}(V)$ for all finitely generated subgroups U and V, where $\overline{\chi}(H) = \max\{-\chi(H),0\}$ denotes the reduced Euler characteristic of H. A strengthened version of the HNC was proved independently by Friedman and Mineyev in 2011. Recently, Antolín and Jaikin-Zapirain introduced the $L^2$-Hall property and showed that if G is a hyperbolic limit group that satisfies this property, then G satisfies the HNC. Antolín and Jaikin-Zapirain established the $L^2$-Hall property for free and surface groups, which Brown and Kharlampovich extended to all limit groups. In this paper, we prove the $L^2$-Hall property for graphs of free groups with cyclic edge groups that are hyperbolic relative to virtually abelian subgroups. We also give another proof of the $L^2$-Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.

We study random walks on metric spaces with contracting isometries. In this first article of the series, we establish sharp deviation inequalities by adapting Gouëzel’s pivotal time construction. As an application, we establish the exponential bounds for deviation from below, central limit theorem, law of the iterated logarithms, and the geodesic tracking of random walks on mapping class groups and CAT(0) spaces.

Let g be an element of a group G. For a positive integer n, let $R_n(g)$ be the subgroup generated by all commutators $[\ldots [[g,x],x],\ldots ,x]$ over $x\in G$, where x is repeated n times. Similarly, $L_n(g)$ is defined as the subgroup generated by all commutators $[\ldots [[x,g],g],\ldots ,g]$, where $x\in G$ and g is repeated n times. In the literature, there are several results showing that certain properties of groups with small subgroups $R_n(g)$ or $L_n(g)$ are close to those of Engel groups. The present article deals with orderable groups in which, for some $n\geq 1$, the subgroups $R_n(g)$ are polycyclic. Let $h\geq 0$, $n>0$ be integers and G be an orderable group in which $R_n(g)$ is polycyclic with Hirsch length at most h for every $g\in G$. It is proved that there are $(h,n)$-bounded numbers $h^*$ and $c^*$ such that G has a finitely generated normal nilpotent subgroup N with $h(N)\leq h^*$ and $G/N$ nilpotent of class at most $c^*$. The analogue of this theorem for $L_n(g)$ was established in 2018 by Shumyatsky [‘Orderable groups with Engel-like conditions’, J. Algebra 499 (2018), 313–320].

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 $G^*$.

Let $G \leqslant \mathrm {Sym}(\Omega )$ be a finite transitive permutation group and recall that an element in G is a derangement if it has no fixed points on $\Omega $. Let $\Delta (G)$ be the set of derangements in G and define $\delta (G) = |\Delta (G)|/|G|$ and $\Delta (G)^2 = \{ xy \,:\, x,y \in \Delta (G)\}$. In recent years, there has been a focus on studying derangements in simple groups, leading to several remarkable results. For example, by combining a theorem of Fulman and Guralnick with recent work by Larsen, Shalev and Tiep, it follows that $\delta (G) \geqslant 0.016$ and $G = \Delta (G)^2$ for all sufficiently large simple transitive groups G. In this paper, we extend these results in several directions. For example, we prove that $\delta (G) \geqslant 89/325$ and $G = \Delta (G)^2$ for all finite simple primitive groups with soluble point stabilisers, without any order assumptions, and we show that the given lower bound on $\delta (G)$ is best possible. We also prove that every finite simple transitive group can be generated by two conjugate derangements, and we present several new results on derangements in arbitrary primitive permutation groups.

It is a theorem due to F. Haglund and D. Wise that reflection groups (aka Coxeter groups) virtually embed into right-angled reflection groups (aka right-angled Coxeter groups). In this article, we generalize this observation to rotation groups, which can be thought of as a common generalization of Coxeter groups and graph products of groups. More precisely, we prove that rotation groups (aka periagroups) virtually embed into right-angled rotation groups (aka graph products of groups).

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 $f_1,f_2, \dots,f_r$.