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.     

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 $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Construction and enumeration of left dihedral codes satisfying certain duality properties

Let ${\mathbb{F}}_q$ be the finite field of q elements and let $D_{2n}=〈x,y|x^n=1,y^2=1,yxy=x^{n?1}〉$ be the dihedral group of 2n elements. Left ideals of the group algebra ${\mathbb{F}}_q[D_{2n}]$ are known as left dihedral codes over ${\mathbb{F}}_q$ of length 2n, and abbreviated as left $D_{2n}$-codes. Let $\gcd (n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over ${\mathbb{F}}_q$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over ${\mathbb{F}}_q$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over ${\mathbb{F}}_q$, and present several numerical examples to illustrative our applications.     

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.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.     

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 $G_n(\mathbb{Z}G)$.     

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.     

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 ${\mathbb{F}}_p$.