On Maximum Norm of Exterior Product and A Conjecture of C.N. Yang

Abstract

Let $V$ be a finite dimensional inner product space over $\mathbb{R}$ with dimension $n$, where $n\in \mathbb{N}$, ${\wedge }^{r}V$ be the exterior algebra of $V$, the problem is to find ${\max }_{\Vert \xi \Vert =1,\Vert \eta \Vert =1}\Vert \xi \wedge \eta \Vert$ where $k,l$ $\in \mathbb{N},$ $\forall \xi \in {\wedge }^{k}V,\eta \in {\wedge }^{l}V.$ This is a problem suggested by the famous Nobel Prize Winner C.N. Yang. He solved this problem for $k\le 2$ in [1], and made the following \textbf{conjecture} in [2] : If $n=2m$, $k=2r$, $l=2s$, then the maximum is achieved when ${\xi }_{max}=\frac{{\omega }^k}{\Vert {\omega }^k\Vert },{\eta }_{max}=\frac{{\omega }^l}{\Vert {\omega }^l\Vert }$, where $\omega ={\Sigma }_{i=1}^m{e}_{2i-1}\wedge e_{2i},$ and $\{e_k{\}}_{k=1}^{2m}$ is an orthonormal basis of V. From a physicist's point of view, this problem is just the dual version of the easier part of the well-known Beauzamy-Bombieri inequality for product of polynomials in many variables, which is discussed in [4]. Here the duality is referred as the well known Bose-Fermi correspondence, where we consider the skew-symmetric algebra(alternative forms) instead of the familiar symmetric algebra(polynomials in many variables) In this paper, for two cases we give estimations of the maximum of exterior products, and the Yang's conjecture is answered partially under some special cases.