论文标题
Dixon和Pressman的猜想的图理论方法
A graph-theoretic approach to a conjecture of Dixon and Pressman
论文作者
论文摘要
给定$ n \ times n $矩阵，$ a_1，\ dots，a_k $，考虑线性操作员$ l（a_1，\ dots，a_k）\，\ colon \; \ operatorName {m} _n \ to \ operatotorName {m} _n $由\ [l（a_1，\ dots，a_k）给出（a_ {k+1}）= \ sum_ {= \ sum_ {σ\ a_ {σ（1）} a_ {σ（2）} \ cdots a_ {σ（k+1）}。 \] Amitsur-Levitzki定理断言$ l（a_1，\ ldots，a_k）$对于每$ k \ geq 2n-1 $的$ 0 $。 Dixon和Pressman猜想，如果$ k $是$ 2 $至2 $ 2n-2 $的偶数数字，则$ l（a_1，\ ldots，a_k）$的内核为$ k $ for $ a_1，\ ldots，\ ldots，\ ldots，a_k \ in \ in \ perperateRonname {m} _n（m} _n（mather \ nration）我们使用图理论技术证明了这种猜想。
Given $n \times n$ matrices, $A_1, \dots, A_k$, consider the linear operator $L(A_1,\dots,A_k) \, \colon \; \operatorname{M}_n \to \operatorname{M}_n$ given by \[ L(A_1,\dots,A_k)(A_{k+1})= \sum_{σ\in S_{k+1}} \operatorname{sign}(σ) A_{σ(1)}A_{σ(2)} \cdots A_{σ(k+1)}. \] The Amitsur-Levitzki theorem asserts that $L(A_1, \ldots, A_k)$ is identically $0$ for every $k \geq 2n-1$. Dixon and Pressman conjectured that if $k$ is an even number between $2$ and $2n - 2$, then the kernel of $L(A_1, \ldots, A_k)$ is of dimension $k$ for $A_1, \ldots, A_k\in \operatorname{M}_n(\mathbb{R})$ in general position. We prove this conjecture using graph-theoretic techniques.
