CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

Let $n\ge 2$. A collection of points $y_1,y_2,\dots ,y_m$ on the unit sphere ${\Sigma }_{n-2}$ in ${ℝ}^{n-1}$ is called a spherical $t$-design for some integer $t\ge 1$ if the average value of every polynomial $f\left(X_1,X_2,\dots ,X_{n-1}\right)$ with real coefficients of degree $\le t$ on the unit sphere ${\Sigma }_{n-2}$ in ${ℝ}^{n-1}$ equals $\frac{1}{m}{\sum }_{k=1}^{m}f\left(y_k\right)$. A full-rank lattice in ${ℝ}^{n-1}$ is called strongly eutactic if its set of normalized minimal vectors form a spherical $2$-design. Given a finite Abelian group $G=\left\{g_0:=0,g_1,\dots ,g_{n-1}\right\}$ of order $n$, one can form a lattice $L_G:=\left\{\left(x_0,x_1,\dots x_{n-1}\right)\in {\mathbb{Z}}^n:x_0+x_1+\dots +x_{n-1}=0and{\sum }_{j=1}^n{x}_j{g}_j=0\right\}$ of rank $n-1$. This talk will be about recent joint work with Albrecht Böttcher, Lenny Fukshansky and Stephan Garcia in which we showed that $L_G$ is strongly eutactic iff $n$ is odd or $G={\left(\mathbb{Z}∕2\mathbb{Z}\right)}^k$ for some $k\ge 1$.