The Magic Square and Symmetric Compositions II

On Magic Finite Projective Space

This paper studies a generalization of magic squares to finite projective space $\mathbb{P}^n(q)$. We classify at all functions from $\mathbb{P}^n(q)$ into a finite field where the sum along any $r$-flat is $0$. In doing so we show connections to elementary number theory and the modular representation theory of $\operatorname{GL}(n,q)$.

A Kind of Magic

We introduce the extended Freudenthal-Rosenfeld-Tits magic square based on six algebras: the reals $\mathbb{R}$, complexes $\mathbb{C}$, ternions $\mathbb{T}$, quaternions $\mathbb{H}$, sextonions $\mathbb{S}$ and octonions $\mathbb{O}$. The ternionic and sextonionic rows/columns of the magic square yield non-reductive Lie algebras, including $\mathfrak{e}_{7\scriptscriptstyle{\frac{1}{2}}}$. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives th…

Semi-magic matrices for dihedral groups

After reviewing the group structure and representation theory for the dihedral group $D_{2n},$ we consider an intertwining operator $Φ_ρ$ from the group algebra $\mathbb{C}[D_{2n}]$ into a corresponding space of semi-magic matrices. From this intertwining operator, one obtains the generating function for enumerating the associated semi-magic squares with fixed line sum and an algebra extending the circulant matrices. While this work complements the approach to $D_{2n}$ through permutation polytopes, we use only methods from representation theory.

幻方、对称群与默比乌斯随机性

Magic squares, the symmetric group and Möbius randomness

本文利用特征映射给出了CUE系综长期系数矩的新组合证明，扩展至对称幂迹的矩，并提出了关于Liouville或Möbius函数扭曲特征和矩的猜想，在函数域中验证，且给出了矩阵积分的新公式。

Gradings of Lie algebras, magical spin geometries and matrix factorizations

We describe a remarkable rank fourtenn matrix factorization of the octic Spin(14)-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular Z-grading of the exceptional Lie algebra $\mathfrak{e}_8$. Intriguingly, the whole story can be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the s…