Let $F_n$ be the free group of rank $n\geq2$ with basis $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ and $e_{i,j}$ be the automorphisms of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $e_{i,j}(x_i)=x_jx_i$, respectively, and fixing the remaining free generators. Write $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$ and $A_n^+=\langle d_{i,j},e_{i,j}\mid 1\leq j<i\leq n\rangle$. We prove that $D_n$ admits a presentation on the generators $d_{r+1,r}$, $1\leq r\leq n-1$, with three families of relations given by commutators of weights two, three, and four, respectively. We extend this to a presentation of $A_n^+$ on the generators $d_{r+1,r}$ and $e_{r+1,r}$, $1\leq r\leq n-1$. These generating sets have minimum cardinality. Moreover, the presentation of $A_n^+$ may be chosen so that every defining relator is a single commutator.
Let $F_n$ be a free group of rank $n\geq3$, freely generated by $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $d_{i,j}(x_k)=x_k$ for $k\neq i$, and let $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$. We determine the lower central series of $D_n$. We first prove that, for $1\leq r<i\leq n$, $d_{i,r}\in \gamma_{i-r}(D_n)\setminus\gamma_{i-r+1}(D_n)$. For each $i=2,\ldots,n$, this calculation leads to a filtration $\{W_{i,c}\}_{c\geq1}$ of $U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$. For every $c\geq1$, we obtain an explicit iterated semidirect-product decomposition of $\gamma_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\cap\gamma_c(D_n)=W_{i,c}$ for $i=2,\ldots,n$. The construction also gives an explicit basis for each quotient $\gamma_c(D_n)/\gamma_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.
Let $S=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with ${\rm deg} x_1=\cdots ={\rm deg} x_n = 1$ and ${\bf a} = (a_1,\ldots,a_n) \in {\mathbb Z}_{>0}^n$. Given a squarefree monomial $u=x_{i_1} \cdots x_{i_d}$ of $S$ with $1 \leq i_1<\cdots<i_d \leq n$, we set $u^{[{\bf a}]}:=x_{i_1}^{a_{i_1}}\cdots x_{i_d}^{a_{i_d}}$. Let $I$ be a squarefree monomial ideal of $S$ and $G(I)$ its unique minimal set of monomial generators. We introduce the monomial ideal $I^{[{\bf a}]}$ with $G(I^{[{\bf a}]})=\{u^{[{\bf a}]} : u \in G(I)\}$. In the present paper, componentwise linearity of a squarefree monomial ideal $I$ and that of $I^{[{\bf a}]}$ is studied.
For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm1},x_1,\ldots,x_n]$ and $\mathfrak{g}_n=\bigoplus_{i=1}^n A_nd_i$, where $d_i=t_i\frac{\partial}{\partial t_i} +\frac{\partial}{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple $\mathfrak{gl}_n$-module. We then prove a canonical algebra isomorphism $A_n\#U(\mathfrak{g}_n)\cong D_n\otimes U(\mathfrak{m}_{\mathbf{1},\mathbf{0}}\Delta)$, and use it to show that every simple cuspidal $\mathfrak{g}_n$-module is isomorphic to a simple quotient of some $T(A_n(\lambda),V)$, where $V$ is a finite-dimensional simple $\mathfrak{gl}_n$-module.
Let $q$ be a primitive $n$-th root of unity, $n>1$, and let $f$ be a nonzero polynomial such that $n\nmid(j+1)$ for every $j\in\supp(f)$. Set $e=\gcd(n,\{j+1:j\in\supp(f)\})$. We show that $\Oz(B_q(f))\cong\mu_e\times\mu_e$: the defining relations are homogeneous for a $\mathbb{Z}/e\times\mathbb{Z}/e$ grading, the center sits in degree zero, and the ozone group is the character group of that grading. The determination of the ozone group only requires the central elements $u^n$, $v^n$, and $\Omega$. The regular normal elements modulo the center form the same group, generated by $u^{n/e}$ and $v^{n/e}$, so every normal element is central exactly when $e=1$. For $f=t^2$ we recover a computation of Chan, Gaddis, Won and Zhang, and for $e>1$ we obtain an infinite family of Calabi--Yau algebras with nontrivial ozone group.
Let $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$, and define the Fibonacci zeta function by $$ Z_F(s)=\sum_{n\ge1}F_n^{-s},\qquad \operatorname{Re}s>0. $$ We determine the exact right edge of the closure of the real parts of its zeros in the half-plane of absolute convergence. If $\sigma_F$ is the unique solution of $$ Z_F(\sigma_F)=4+2\,144^{-\sigma_F}, $$ then $$ \sigma_F=0.743163398726901648\ldots, $$ $Z_F(s)\neq0$ for $\operatorname{Re}s\ge\sigma_F$, while $$ \overline{\{\operatorname{Re}\rho:Z_F(\rho)=0,\ \operatorname{Re}\rho>0\}}=[0,\sigma_F]. $$ The edge is sharp in an almost-periodic sense: zeros occur with relatively dense ordinates near every admissible vertical line. We prove growing-dimensional phase locking near the edge and a Diophantine zero-free cusp, and describe the associated Jessen function and smooth mean vertical zero density. For every partial sum with $N\ge12$ we determine the corresponding exact closure edge $\sigma_N$, prove $\sigma_N\nearrow\sigma_F$, and obtain an exponential asymptotic for $\sigma_F-\sigma_N$. We also derive a finite-core theorem for positive integral Lucas zeta functions, with the Pell zeta function as an explicit example. Finally, using the known meromorphic continuation, we construct a natural $q$-Pochhammer completion that is entire of exact order $2$ and type $\log\varphi/4$.
Let $k=\mathbb{F}_q$, $E=\mathbb{F}_{q^n}$ and $\mathrm{Tr}=\mathrm{Tr}_{E/k}$. For $r\ge 2$, $a\in k^{\times}$ and $x\in E^{\times}$, let $\mathrm{N}(E,r,x,a)$ be the number of $r$-tuples $(x_1,\cdots,x_r)$ in $(E^{\times})^r$ satisfying $x_1\cdots x_r=x$ and $\mathrm{Tr}(x_1+\cdots+x_r)=a$. We prove $\left|\mathrm{N}(E,r,x,a)-\left((q^n-1)^{r-1}+(-1)^r\right)/q\right|\le (r^n-1) q^{\frac{(r-1)n-1}{2}}$. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra $B=\prod\limits_{i=1}^s M_{d_i}(\mathbb{F}_{q^{n_i}})$ over $k$ and a regular element $x\in B^{\times}$, the same method combined with Zelingher's formula leads to analogous square-root estimates.
Xue-Jun Guo, Chen Lin, Chenhao Tang· 0 citations
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