Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Even degree characters in principal blocks

We characterise finite groups such that for an odd prime p all the irreducible characters in its principal p-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by p unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\ne 7$ or if $M_{22}$ is not a composition factor of a group G, then the condition above is equivalent to $G/{\mathbf{O}}_{p^{\prime }}(G)$ having odd order.     

Classification of metabelian 2-groups G with Gab ≃ (2,2n),n ≥ 2, and rank d(G′)=2; Applications to real quadratic number fields

We characterize all finite metabelian 2-groups G whose abelianizations $G^{\mathrm{ab}}$ are of type $(2,2^n)$, with $n\ge 2$, and for which their commutator subgroups $G^{\prime }$ have $\text{rank}=2$. This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group ${\mathrm{Cl}}_2(k)?(2,2^n)$, $n\ge 2$, such that the rank of ${\mathrm{Cl}}_2(k^1)=2$ where $k^1$ is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.     

Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

We prove that any weak* continuous semigroup ${(T_t)}_{t?0}$ of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

An improvement to the Hilton-Zhao vertex-splitting conjecture

For a simple graph G, denote by n, $\mathrm{\Delta }(G)$, and ${\chi }^{\prime }(G)$ its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ and ${\chi }^{\prime }(H)<{\chi }^{\prime }(G)$ for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let $G^{?}$ be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that $G^{?}$ must be edge-chromatic critical if $\mathrm{\Delta }(G)>n/3$, and they verified this when $\mathrm{\Delta }(G)\ge \frac{n}{2}(\sqrt{7}?1)\approx 0.82n$. In this paper, we prove it for $\mathrm{\Delta }(G)\ge 0.75n$.     

A note on the Hambleton-Taylor-Williams conjecture

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in the spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group G and noetherian ring R. The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,{\mathbb{F}}_3)$ is also a counterexample to the conjectured HTW-decomposition. Nevertheless, we prove that for any finite group G the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition. We also show that the HTW-decomposition predicts correct torsion for $G_1(\mathbb{Z}G)$ for any finite group G. Furthermore, we prove that for any degree other than $n=1$ the conjecture gives a correct prediction for the rank of $G_n(\mathbb{Z}G)$.     

Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions

In this paper we study the family of embeddings ${\mathrm{\Phi }}_t$ of a compact ${\mathrm{RCD}}^{?}(K,N)$ space $(X,\mathsf{d},\mathfrak{m})$ into $L^2(X,\mathfrak{m})$ via eigenmaps. Extending part of the classical results [10], [11] known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaled pull-back metrics ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ in $L^2(X,\mathfrak{m})$ induced by ${\mathrm{\Phi }}_t$. Moreover we discuss the behavior of ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.     

On Hr(curl,Ω) estimates for a Maxwell type system in convex domains

In bounded convex domains, the regularity of a vector field u with its $\mathrm{div}\mathbf{u}$, $\mathrm{curl}\mathbf{u}$ in $L^r$ space and the tangential component or the normal component of u over the boundary in $L^r$ space, is established for $1<r<\infty$. As an application, we derive an $H^r(\mathrm{curl},\mathrm{\Omega })$ estimate for solutions to a Maxwell type system with an inhomogeneous boundary condition in convex domains. In contrast to the well-posed region of r in the space $H^r(\mathrm{curl},\mathrm{\Omega })$ for the Maxwell type system in Lipschitz domains given by Kar and Sini (2016) [16], we extend the well-posed region to be optimal.     

On coloring numbers of graph powers

The weak $r$-coloring numbers ${\mathrm{wcol}}_r\left(G\right)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $\mathrm{col}\left(G\right)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let $G^p$ denote the $p$th power of $G$. We show that, all integers $p>0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta \left(G\right)\le \Delta$ satisfy $\mathrm{col}\left(G^p\right)\in O\left(p\cdot {\mathrm{wcol}}_{\lceil p∕2\rceil }\left(G\right){(\Delta -1)}^{\lfloor p∕2\rfloor }\right)$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2<\mathrm{mad}\left(G\right)\le 2k$, then $\mathrm{col}\left(G^2\right)\le (2k-1)\Delta \left(G\right)+2k+1$.