The depth of a subgroup H of a finite group G is a positive integer defined with respect to the inclusion of the corresponding complex group algebras $\mathbb {C}H \subseteq \mathbb {C}G$. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

Let $\lambda (G)$ be the maximum number of subgroups in an irredundant cover of the finite group G. We establish bounds on the order, exponent and derived length of the group in terms of this invariant.

In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here, we verify some of these conjectures for fusion systems on an extraspecial group of order $p^3$, which contain among them the Ruiz–Viruel exotic fusion systems at the prime $7$. As a byproduct, we verify Robinson’s ordinary weight conjecture for principal p-blocks of almost simple groups G realizing such (nonconstrained) fusion systems.

For a group G and $m\ge 1$, $G^m$ denotes the subgroup generated by the elements $g^m$, where g runs through G. The subgroups not of the form $G^m$ are called nonpower subgroups. We extend the classification of groups with few nonpower subgroups from groups with at most nine nonpower subgroups to groups with at most 13 nonpower subgroups.

Let K be a genus one two-bridge knot. Let p be a prime number and let ${\mathbb {Z}}_{p}$ denote the ring of p-adic integers. In the spirit of arithmetic topology, we observe that if $p\neq 2$ and p divides (or $p=2$ and $2^3$ divides) the size of the 1st homology group of some odd-th cyclic branched cover of the knot K, then its group $\pi _1(S^3-K)$ admits a liminal $\mathrm { SL}_2{\mathbb {Z}}_p$-character. In addition, we discuss the existence of liminal $\mathrm {SL}_2{\mathbb {Z}}_{p}$-representations and give a remark on a general two-bridge knot. In the course of the argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols.

Given a group G and an automorphism $\varphi $ of G, two elements $x,y\in G$ are said to be $\varphi $-conjugate if $x=gy\varphi (g)^{-1}$ for some $g\in G$. The number $R(\varphi )$ of equivalence classes with respect to this relation is called the Reidemeister number of $\varphi $ and the set $\{R(\varphi ) \mid \varphi \in \text {Aut}(G)\}$ is called the Reidemeister spectrum of G. We determine the Reidemeister spectrum of ZM-groups, extending some results of Senden [‘The Reidemeister spectrum of split metacyclic groups’, Preprint, 2022, arXiv:2109.12892].

Let ${\mathbb {F}_q}$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod p representations of ${G_q} = \text {GL}_2({{\mathbb {F}_q}})$ to ${G_p} = \text {GL}_2({\mathbb {F}_p})$. We first study the restrictions of principal series which are obtained by induction from a Borel subgroup ${B_q}$. We then analyze the restrictions of inductions from an anisotropic torus ${T_q}$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of f. The proofs depend on writing down explicit orbit decompositions of ${G_p} \backslash {G_q} / H,$ where $H = {B_q}$ or ${T_q}$ using the fact that ${G_q}/H$ is an explicit orbit in a certain projective line, along with Mackey theory.

Let G be a finite group and p be a prime number. An element g of G is called an $\mathrm {SM}^*$-vanishing element of G if there exists a strongly monolithic character $\chi $ of G satisfying $Z(\chi )=\ker \chi $ and $\chi (g)=0$. In this paper, we present some results on the relationship between the orders of $\mathrm {SM}^*$-vanishing elements of G and the structure of G.

We give a presentation of the torus-equivariant (small) quantum K-ring of flag manifolds of type C as an explicit quotient of a Laurent polynomial ring; our presentation can be thought of as a quantization of the classical Borel presentation of the ordinary K-ring of flag manifolds. Also, we give an explicit Laurent polynomial representative for each special Schubert class in our Borel-type presentation of the quantum K-ring.

For a prime p, let $\mathcal {N}_p(G)$ denote the intersection of the normalisers of all non-p-nilpotent subgroups of a finite group G and set $\mathcal {N}_p(G)=G$ if G itself is p-nilpotent. We give some properties of $\mathcal {N}_p(G)$ and investigate the influence of $\mathcal {N}_p(G)$ on G.

In this article, we provide a specific characterization of invariants of classical Lie superalgebras from the super-analog of the Schur–Weyl duality in a unified way. We establish $\mathfrak {g}$-invariants of the tensor algebra $T(\mathfrak {g})$, the supersymmetric algebra $S(\mathfrak {g})$, and the universal enveloping algebra $\mathrm {U}(\mathfrak {g})$ of a classical Lie superalgebra $\mathfrak {g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak {g})^{\mathfrak {g}}$ of the projection from $T(\mathfrak {g})$ to $\mathrm {U}(\mathfrak {g})$ is surjective, which enables us to determine the generators of the center $\mathcal {Z}(\mathfrak {g})$ except for $\mathfrak {g}=\mathfrak {osp}_{2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal {Z}(\mathfrak {p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams.

Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$, where $a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.

Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial t-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte T-designs. We explore when a T-design is also a $T'$-design, where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.

Commutator blueprints can be seen as blueprints for constructing RGD systems over $\mathbb {F}_2$ with prescribed commutation relations. In this paper, we construct several families of Weyl-invariant commutator blueprints, mostly of universal type. Also applying another result of the author, we obtain new examples of exotic RGD systems of universal type over $\mathbb {F}_2$. In particular, we generalize Tits’ construction of uncountably many trivalent Moufang twin trees to higher rank, we obtain an example of an RGD system of rank $3$ such that the nilpotency degree of the groups $U_w$ is unbounded, and we construct a commutator blueprint of type $(4, 4, 4)$ that is used to answer a question of Tits from the late $1980$s about twin buildings.

This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R. The focus is on lattices, namely, finitely generated G-modules that are projective as R-modules, and on the full subcategory of all G-modules projective over R generated by the lattices. The stable category of such G-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney–Tate stratifications of Beilinson–Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne’s modification of the dual group and a modified form of Vinberg’s monoid over the integers.

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that its equivariant volume, given by the evaluation of its equivariant $H^*$-series at $1$, is the permutation character of decorated ordered set partitions under the natural action of $S_n$. This verifies a conjecture of Stapledon for the hypersimplex. To prove this result, we give a formula for the coefficients of the $H^*$-polynomial. Additionally, for the $(2,n)$-hypersimplex, we use this formula to show that trivial character need not appear as a direct summand of a coefficient of the $H^*$-polynomial, which gives a family of counterexamples to a different conjecture of Stapledon.

The supersingular locus of the $\mathrm {GU}(1,n-1)$ Shimura variety at a ramified prime p is stratified by Coxeter varieties attached to finite symplectic groups. In this article, we compute the $\ell $-adic cohomology of the Zariski closure of any such stratum. These are known as closed Bruhat–Tits strata. We prove that the cohomology groups of odd degree vanish, and those of even degree are explicitly determined as representations of the symplectic group with a Frobenius action. Each closed Bruhat–Tits stratum is linearly stratified by Coxeter varieties attached to smaller symplectic groups. Thanks to results of Lusztig who computed the cohomology of Coxeter varieties for classical groups, we make use of the spectral sequence associated with this stratification and describe explicitly all the terms at infinity. We point out that the closed Bruhat–Tits strata have isolated singularities when the dimension is greater than 1. Our analysis requires discussing the smoothness of the blow-up at the singular points, as well as comparing the ordinary $\ell $-adic cohomology with intersection cohomology. A by-product of our computations is that these two cohomologies actually coincide, so that surprisingly the presence of singularities does not interfere with the cohomology.

We study new properties of generalised Harish-Chandra theory aiming at explaining the inductive local-global conditions for finite groups of Lie type in nondefining characteristic. In particular, we consider a parametrisation of generalised Harish-Chandra series that is compatible with Clifford theory and with the action of automorphisms on irreducible characters and we reduce it to the verification of certain requirements on stabilisers and extendibility of characters. This parametrisation is used by the author in a separate paper to obtain new conjectures for finite reductive groups that can be seen as geometric realisations of the local-global counting conjectures and their inductive conditions. As a by-product, we extend the parametrisation of generalised Harish-Chandra series given by Broué–Malle–Michel to the nonunipotent case by assuming maximal extendibility.

Let $W_{\mathrm {aff}}$ be an extended affine Weyl group, $\mathbf {H}$ be the corresponding affine Hecke algebra over the ring $\mathbb {C}[\mathbf {q}^{\frac {1}{2}}, \mathbf {q}^{-\frac {1}{2}}]$, and J be Lusztig’s asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}$. Viewing J as a subalgebra of the $(\mathbf {q}^{-\frac {1}{2}})$-adic completion of $\mathbf {H}$ via Lusztig’s map $\phi $, we use Harish-Chandra’s Plancherel formula for p-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf {q}$, with denominator depending only on the two-sided cell containing w, and dividing a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known.

Along the way, we show that upon specializing $\mathbf {q}=q>1$, the map from J to the Harish-Chandra Schwartz algebra is injective. As an application of injectivity, we give a novel criterion for an Iwahori-spherical representation to have fixed vectors under a larger parahoric subgroup in terms of its Kazhdan-Lusztig parameter.