Related papers: Character triples and relative defect zero charact…

The fundamental bijection is a bijection $\theta:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations…

Let $F$ denote a number field and let $\mathfrak{q}\subset O_F$ traverse a sequence of prime ideals with norm $N(\mathfrak{q}) \to \infty$ and for each $\mathfrak{q}$, let $\chi \in \widehat{F^{\times}\setminus \mathbb{A}^\times}$ be a…

Let $G$ be a finite group, and $\pi$ be a set of primes. The $\pi$-core $\mathbf{O}_\pi(G)$ is the unique maximal normal $\pi$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved…

This is a contribution to the study of $\operatorname {Irr}(G)$ as an $\operatorname {Aut}(G)$-set for $G$ a finite quasi-simple group. Focusing on the last open case of groups of Lie type $\mathrm D$ and $^2\mathrm D$, a crucial property…

In 2021, Navarro and Tiep proposed a conjecture on character fields of finite quasi-simple groups. We develop some theory on sums of roots of unity and use this theory to prove the conjecture for some infinite families of finite…

Let $G$ be a finite group and $cd(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is Mathieu group such that $cd(G)…

We derive several identities that feature irreducible characters of the general linear, the symplectic, the orthogonal, and the special orthogonal groups. All the identities feature characters that are indexed by shapes that are "nearly"…

For $G={\rm GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(\chi,g)$ in ${\rm Irr}(G)\times G$ with $\chi(g)\neq 0$ satisfies $P_{n,q}\to 0$ as $n\to\infty$.

Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees of the irreducible complex characters of $G$, we consider the {\it character degree graph} of $G$: this is the (simple, undirected) graph whose vertices are the…

If $G$ is a finite classical group, linear or unitary in any characteristic, and orthogonal in odd characteristic, we give an approximate formula for $\chi(g)$ in which the error term is much smaller than the estimate, when $g\in G$ is an…

The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial…

We introduce the notion of quasi-BNS invariants, where we replace homomorphism to $\mathbb R$ by homogenous quasimorphisms to $\mathbb R$ in the theory of Bieri-Neumann-Strebel invariants. We prove that the quasi-BNS invariant $Q\Sigma(G)$…

For a compact Lie group G we define a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundles over non-compact Kahler manifolds. The new cohomology is infinite-dimensional, but as a representation of G…

Let $G$ be a reductive group over a field $k$ of characteristic $\neq 2$, let ${\mathfrak g}=\Lie(G)$, let $\theta$ be an involutive automorphism of $G$ and let ${\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}$ be the associated symmetric…

Suppose $\Gamma$ is a finite simple graph. If $D$ is a dominating set of $\Gamma$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $\Gamma$, then we say that $D$ is an odd dominating set for $\Gamma$. For a…

Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $\Theta (c n^t d^n)$, where $d$…

Let $G$ be a finite group and $N<G$ a normal subgroup with $G/N$ abelian. We show how the conjugacy classes of $G$ in a given coset $qN$ relate to the irreducible characters of $G$ that are not identically $0$ on $qN$. We describe several…

This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the…

Let $R$ be an algebraically closed field and $\ell$ be its characteristic. Let $G$ be a locally profinite group having a compact open subgroup of invertible pro-order in $R$. Take $N$ a closed subgroup of $G$ exhausted by compact subgroups…

We focus on derangement characters of $GL(n,q)$ which depend solely on the dimension of the space of fixed vectors. This family includes Thoma characters which become asymptotically irreducible as $n\to\infty$. We find explicit…