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 $G_n(\mathbb{Z}G)$.     

Realization of PBW-deformations of type An quantum groups via multiple Ore extensions

The notion of multiple Ore extension is introduced as a natural generalization of Ore extensions and double Ore extensions. For a PBW-deformation ${\mathfrak{B}}_q(\mathfrak{sl}(n+1,\mathbb{C}))$ of type ${\mathbb{A}}_n$ quantum group, we explicitly obtain the commutation relations of its root vectors, then show that it can be realized via a series of multiple Ore extensions, which we call a ladder Ore extension of type $(1,2,?,n)$. Moreover, we analyze the quantum algebras ${\mathfrak{B}}_q(\mathfrak{g})$ with $\mathfrak{g}$ of type ${\mathbb{D}}_4$, ${\mathbb{B}}_2$ and ${\mathbb{G}}_2$ and give some examples and counterexamples that can be realized by a ladder Ore extension.     

Invariant polynomials on truncated multicurrent algebras

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field $\mathbb{F}$ of characteristic zero, and I is a finite-codimensional ideal of $\mathbb{F}[t_1,\dots ,t_{?}]$ generated by monomials. In particular, when $\mathfrak{g}$ is semisimple and $\mathbb{F}$ is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$. As an application of our main result, we show that the center of the universal enveloping algebra of $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$ acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.     

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.     

On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of ${\mathbb{F}}_p{\mathbb{F}}_p[\theta ]$-linear codes, where $p\in \{2;3\}$ and $\theta \ne {\theta }^2=0$ that provides LCD codes over ${\mathbb{F}}_p$.     

Non-integrated defect relation for meromorphic mappings on Kähler manifolds intersecting non-subdegenerate families of hyperplanes

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball ${\mathbb{B}}^m(R_0)$ in ${\mathbb{C}}^m$ ($0<R_0\le +\infty$). Our purpose of this article is to establish a non-integrated defect relation with truncated level for a meromorphic mapping on M intersecting a family of hyperplanes in ${\mathbb{P}}^n(\mathbb{C})$ which is non-subdegenerate with respect to the mapping.     

The Orlicz inequality for multilinear forms

The Orlicz $({?}_2,{?}_1)$-mixed inequality states that${\left(\sum _{j_1=1}^n{\left(\sum _{j_2=1}^n|A(e_{j_1},e_{j_2})|\right)}^2\right)}^{\frac{1}{2}}\le \sqrt{2}6A6$ for all bilinear forms $A:{\mathbb{K}}^n\times {\mathbb{K}}^n\to \mathbb{K}$ and all positive integers n, where ${\mathbb{K}}^n$ denotes ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ endowed with the supremum norm. In this paper we extend this inequality to multilinear forms, with ${\mathbb{K}}^n$ endowed with ${?}_p$ norms for all $p\in [1,\infty ]$.