We geometrize the mod p Satake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric mod p Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod p Hecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a mod p Local Langlands Correspondence.

A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is given. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set by Tsang at Ischia Group Theory 2024 Conference is provided.

Let G be an almost simple group with socle $G_0$. In this paper we prove that whenever $G/G_0$ is abelian, then there exists an abelian subgroup A of G such that $G=AG_0$. We propose a few applications of this structural property of almost simple groups.

In this article, $\mathcal{F}_{S}(G)$ denotes the fusion category of G on a Sylow p-subgroup S of G where p denotes a prime. A subgroup K of G has normal complement in G if there is a normal subgroup T of G satisfying that G = KT and $T \cap K = 1$. We investigate the supersolvability of $\mathcal{F}_{S}(G)$ under the assumption that some subgroups of S are normal in G or have normal complement in G.

For finite nilpotent groups $J$ and $N$, suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$-subgroups of $J$ that mirrors the primary decomposition of $H^1(J,N)$ for abelian $N$. We then show that if $N \rtimes J$ acts on some non-empty set $\Omega$, where the action of $N$ is transitive and for each prime $p$ a Sylow $p$-subgroup of $J$ fixes an element of $\Omega$, then $J$ fixes an element of $\Omega$.

We define oriented Temperley–Lieb algebras for Hermitian symmetric spaces. This allows us to explain the existence of closed combinatorial formulae for the Kazhdan–Lusztig polynomials for these spaces.

We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of $0$-Hecke algebras. We apply this framework in type B to give representation–theoretic interpretations of a number of noteworthy families of type-B quasisymmetric functions. Next, we construct modules of the type-B $0$-Hecke algebra corresponding to type-B analogs of Schur functions and introduce a type-B analog of Schur Q-functions; we prove that these shifted domino functions expand positively in the type-B peak functions. We define a type-B analog of the $0$-Hecke–Clifford algebra, and we use this to provide representation–theoretic interpretations for both the type-B peak functions and the shifted domino functions. We consider the modules of this algebra induced from type-B $0$-Hecke modules constructed via ascent-compatibility and prove a general formula, in terms of type-B peak functions, for the type-B quasisymmetric characteristics of the restrictions of these modules.

We define an involution on the elliptic space of tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence for the conjecture, including proofs for ${\mathsf {SL}}_n$ and ${\mathsf {PGL}}_n$.

Let $\mathbf {G}$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk $ and ${\mathbf B}$ be a Borel subgroup of ${\mathbf G}$. In this paper, we completely determine the composition factors of the permutation module $\mathbb {F}[{\mathbf G}/{\mathbf B}]$ for any field $\mathbb {F}$.

We introduce the concept of almost $\mathcal {P}$-numbers where $\mathcal {P}$ is a class of groups. We survey the existing results in the literature for almost cyclic numbers, and give characterisations for almost abelian and almost nilpotent numbers proving these two concepts are equivalent.

Let G be a finite group and r be a prime divisor of the order of G. An irreducible character of G is said to be quasi r-Steinberg if it is non-zero on every r-regular element of G. A quasi r-Steinberg character of degree $\displaystyle |Syl_r(G)|$ is said to be weak r-Steinberg if it vanishes on the r-singular elements of $G.$ In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group $GL(n,q).$ Then we characterize the quasi r-Steinberg characters of $GL(2,q)$ and $GL(3,q).$ Finally, we obtain a classification of the weak r-Steinberg characters of $GL(n,q).$

We obtain an adaptation of Dade’s Conjecture and Späth’s Character Triple Conjecture to unipotent characters of simple, simply connected finite reductive groups of type $\mathbf {A}$, $\mathbf {B}$ and $\mathbf {C}$. In particular, this gives a precise formula for counting the number of unipotent characters of each defect d in any Brauer $\ell $-block B in terms of local invariants associated to e-local structures. This provides a geometric version of the local-global principle in representation theory of finite groups. A key ingredient in our proof is the construction of certain parametrisations of unipotent generalised Harish-Chandra series that are compatible with isomorphisms of character triples.

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in characteristic $p\geq 7$. We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via G-Hilbert schemes. In the appendix, we discuss several approaches towards the notion of conjugacy classes for finite group schemes: This is an ingredient in McKay correspondences, but also of independent interest.

We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group G, where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in G, to a sum over values of irreducible characters of G weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.

Assuming Stanley’s P-partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets P with underlying set $\{1, 2, \ldots , |P|\}$ such that the P-partition generating function is symmetric and the set of linear extensions of P, denoted $\Sigma _L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak {S}_{|P|}$. We describe the permutations in $\Sigma _L(P)$ in terms of reading words of standard Young tableaux when P is a regular Schur labeled skew shape poset, and classify $\Sigma _L(P)$’s up to descent-preserving isomorphism as P ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf {M}_P$ associated with regular Schur labeled skew shape posets P up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the finite posets P whose linear extensions form a dual plactic-closed subset of $\mathfrak {S}_{|P|}$. Using this characterization, we construct distinguished filtrations of $\mathsf {M}_P$ with respect to the Schur basis when P is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf {M}_P$ are also discussed.

In this paper, the authors introduce a new notion called the quantum wreath product, which is the algebra $B \wr _Q \mathcal {H}(d)$ produced from a given algebra B, a positive integer d and a choice $Q=(R,S,\rho ,\sigma )$ of parameters. Important examples that arise from our construction include many variants of the Hecke algebras, such as the Ariki–Koike algebras, the affine Hecke algebras and their degenerate version, Wan–Wang’s wreath Hecke algebras, Rosso–Savage’s (affine) Frobenius Hecke algebras, Kleshchev–Muth’s affine zigzag algebras and the Hu algebra that quantizes the wreath product $\Sigma _m \wr \Sigma _2$ between symmetric groups.

In the first part of the paper, the authors develop a structure theory for the quantum wreath products. Necessary and sufficient conditions for these algebras to afford a basis of suitable size are obtained. Furthermore, a Schur–Weyl duality is established via a splitting lemma and mild assumptions on the base algebra B. Our uniform approach encompasses many known results which were proved in a case by case manner. The second part of the paper involves the problem of constructing natural subalgebras of Hecke algebras that arise from wreath products. Moreover, a bar-invariant basis of the Hu algebra via an explicit formula for its extra generator is also described.

We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of an open problem by Stanley from 2000 and an open problem by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is ${\textsf {NP}}$-hard, and computing them is ${{{\textsf {#P}}}}$-hard under parsimonious many-one reductions. Our proof also provides an explicit isomorphism of the corresponding highest weight vector spaces.

For any prime p and S a p-group isomorphic to a Sylow p-subgroup of a rank $2$ simple group of Lie type in characteristic p, we determine all saturated fusion systems supported on S up to isomorphism.

Let $X=GC$ be a group, where C is a cyclic group and G is either a generalized quaternion group or a dihedral group such that $C\cap G=1$. In this paper, X is characterized and, moreover, a complete classification for $X$ is given, provided that G is a generalized quaternion group and C is core-free.