The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.
Let $\Gamma$ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For $f$ belonging to the reduced group $C^*$ algebra $C_r^*(\Gamma)$, we construct a rearrangement of its individual Fourier terms with a subsequence converging to $f$ in the operator norm. For every $2\le p<\infty$ and $f$...
We establish a uniform asymptotic formula for the smoothly weighted shifted second moment of the Gaussian angular Hecke $L$-functions $L(s,\lambda^k)$. The completed moment is expressed as the sum of four explicit main terms, corresponding to the four functional-equation swaps, with an error of size $O_{\Phi,\epsilon}(...
Han, Jouhet and Zeng established $q$-analogues of the $\gamma$-expansion formulas for Eulerian polynomials of types $A$ and $B$. Combinatorial interpretations of the corresponding coefficients$a_{n,k}(q)$ and $b_{n,k}(q)$, however, remained open. In this paper, we provide combinatorial interpretations for thesecoeffici...
Tai-Feng Ding, Lin-Tong Wang, Sherry H. F. Yan· 0 citations
We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit $\mathrm{L}^2$ norm. The entries before normalization are assumed to have regularly varying tails with tail index $\alpha \in [0,2]$. We prove that, outside the unit disk, the characteristic...
Quentin François, J. Heiny, Xuechun Hu· 0 citations
We study the Galois group $G_f$ of a random polynomial $f$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$. We use a Euclidean height adapted to this relation. Our main result is an asymptotic theorem of van der Waerden--Bhargava type: for fixed $b...
Given a finite dimensional Lie algebra $L$ on a finite field $\mathbb{F}_{p^n}$ of prime power order $p^n$ (with $n$ positive integer and $p$ prime), we consider the number of modular pairs $(A,B)$ in the lattice of all subalgebras $\mathcal{L}(L)$ and introduce the notion of ``subalgebra commutativity degree''of $L$....
Seid Kassaw Muhie, Daniele Ettore Otera, F. Russo· 0 citations
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