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.

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.

For a finite group G, let $\operatorname { {AD}}(G)$ denote the Fourier norm of the antidiagonal in $G\times G$. The author showed recently in [‘An explicit minorant for the amenability constant of the Fourier algebra’, Int. Math. Res. Not. IMRN 2023 (2023), 19390–19430] that $\operatorname { {AD}}(G)$ coincides with the amenability constant of the Fourier algebra of G and is equal to the normalized sum of the cubes of the character degrees of G. Motivated by a gap result for amenability constants from Johnson [‘Non-amenability of the Fourier algebra of a compact group’, J. Lond. Math. Soc. (2) 50 (1994), 361–374], we determine exactly which numbers in the interval $[1,2]$ arise as values of $\operatorname { {AD}}(G)$. As a by-product, we show that the set of values of $\operatorname { {AD}}(G)$ does not contain all its limit points. Some other calculations or bounds for $\operatorname { {AD}}(G)$ are given for familiar classes of finite groups. We also indicate a connection between $\operatorname { {AD}}(G)$ and the commuting probability of G, and use this to show that every finite group G satisfying $\operatorname { {AD}}(G)< {61}/{15}$ must be solvable; here, the value ${61}/{15}$ is the best possible.

Let W be a group endowed with a finite set S of generators. A representation $(V,\rho )$ of W is called a reflection representation of $(W,S)$ if $\rho (s)$ is a (generalized) reflection on V for each generator $s \in S$. In this article, we prove that for any irreducible reflection representation V, all the exterior powers $\bigwedge ^d V$, $d = 0, 1, \dots , \dim V$, are irreducible W-modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic W-modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups.

The minimal faithful permutation degree $\mu (G)$ of a finite group G is the least integer n such that G is isomorphic to a subgroup of the symmetric group $S_n$. If G has a normal subgroup N such that $\mu (G/N)> \mu (G)$, then G is exceptional. We prove that the proportion of exceptional groups of order $p^6$ for primes $p \geq 5$ is asymptotically zero. We identify $(11p+107)/2$ such groups and conjecture that there are no others.

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 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.

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.

Let G be a finite group and let $\chi $ be an irreducible character of G. The number $|G:\mathrm {ker}\chi |/\chi (1)$ is called the codegree of the character $\chi $. We provide several relations between the structure of G and the codegrees of the characters in a given subset of $\mathrm {Irr}(G)$, where $\mathrm {Irr}(G)$ is the set of all complex irreducible characters of G. For example, we show that if the codegrees of all strongly monolithic characters of G are odd, then G is solvable, analogous to the well-known fact that if all irreducible character degrees of a finite group are odd, then that group is solvable.

In this article, we study rational matrix representations of VZ p-groups (p is any prime). Using our findings on VZ p-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p-groups of order $\leq p^4$. Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p-groups and all p-groups of order $\leq p^4$, ensuring simplicity in the process.

Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G, then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian.

Let G be a finite group. We investigate the structure of finite groups whose irreducible character codegrees are consecutive integers.

Let $\Gamma $ be a finite group, let $\theta $ be an involution of $\Gamma $ and let $\rho $ be an irreducible complex representation of $\Gamma $. We bound ${\operatorname {dim}} \rho ^{\Gamma ^{\theta }}$ in terms of the smallest dimension of a faithful $\mathbb {F}_p$-representation of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$, where p is any odd prime and $\operatorname {\mathrm {Rad}}_p(\Gamma )$ is the maximal normal p-subgroup of $\Gamma $.

This implies, in particular, that if $\mathbf {G}$ is a group scheme over $\mathbb {Z}$ and $\theta $ is an involution of $\mathbf {G}$, then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf {G}(\mathbb {Z}_p)/ \mathbf {G} ^{\theta }(\mathbb {Z}_p) \right)$ is bounded, uniformly in p.

Let G be a finite group and $\mathrm {Irr}(G)$ the set of all irreducible complex characters of G. Define the codegree of $\chi \in \mathrm {Irr}(G)$ as $\mathrm {cod}(\chi ):={|G:\mathrm {ker}(\chi ) |}/{\chi (1)}$ and let $\mathrm {cod}(G):=\{\mathrm {cod}(\chi ) \mid \chi \in \mathrm {Irr}(G)\}$ be the codegree set of G. Let $\mathrm {A}_n$ be an alternating group of degree $n \ge 5$. We show that $\mathrm {A}_n$ is determined up to isomorphism by $\operatorname {cod}(\mathrm {A}_n)$.

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.

In 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.

Assume that G is a finite group, N is a nontrivial normal subgroup of G and p is an odd prime. Let $\mathrm{Irr}_p(G)=\{\chi \in \mathrm{Irr}(G) : \chi (1)=1~\mathrm{or}~ p \mid \chi (1)\}$ and $\mathrm{Irr}_p(G|N)=\{\chi \in \mathrm{Irr}_p(G) : N \not \leq \mathrm{ker}\,\chi \}$. The average character degree of irreducible characters of $\mathrm{Irr}_p(G)$ and the average character degree of irreducible characters of $\mathrm{Irr}_p(G|N)$ are denoted by $\mathrm{acd}_p(G)$ and $\mathrm{acd}_p(G|N)$, respectively. We show that if $\mathrm{Irr}_p(G|N) \neq \emptyset $ and $\mathrm{acd}_p(G|N) < \mathrm{acd}_p(\mathrm{PSL}_2(p))$, then G is p-solvable and $O^{p'}(G)$ is solvable. We find examples that make this bound best possible. Moreover, we see that if $\mathrm{Irr}_p(G|N) = \emptyset $, then N is p-solvable and $P \cap N$ and $PN/N$ are abelian for every $P \in \mathrm{Syl}_p(G)$.

We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G, then the number of conjugacy classes of G is at least $Dp/\log_2p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$.

Let $p \;:\; Y \to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\;{\mathbb{C}})$ of the cover as a $G$-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand and topological properties of homology classes in $H_1(Y;\;{\mathbb{C}})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\;{\mathbb{C}})$.

The homology class of a lift of a primitive element in $\pi _1(X)$ spans an induced subrepresentation in $H_1(Y;\;{\mathbb{C}})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \leq H_1(Y;\;{\mathbb{C}})$—the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $\pi _1(X)$. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \neq \ker\!(p_*)$.