论文标题
$ sl_n（\ Mathbb {f} _Q）$的子集的生成标准
A Generation Criterion for Subsets of $SL_n(\mathbb{F}_q)$
论文作者
论文摘要
令$ g_0 $为$ sl_n（\ mathbb {f} _q）$，带有$ q $元素的有限字段上的特殊线性组，或$ psl_n（\ m athbb {f} _q）$，其投影剂，其投射商，让$σ$是$ g_0 $ $ g_0 $ $ g_0 $，namy，name $ ch $ x^{ - 1} \在σ$中。我们找到一套不可约为$ 5 $的$ g_0 $的不可约定表示的$ \ nathcal {r}（g_0）$，以至于$σ$在且仅当$ \ weft \ weft \ weled \lvertς\ right \ right \ rever \ rerver \ ryt \ rerver \ rerver \ r right \ rvert $不是$ eigenvalue of $ { $ρ\ in \ Mathcal {r}（g_0）$。 为了实现此结果，让$ g $为$ gl_n（\ mathbb {f} _q）$或$ pgl_n（\ mathbb {f} _q）$。我们考虑$ \ Mathcal {x}（g）$，这是$ g $的一些不可约的非平地字符，其大小最多为$ 5 $。我们表明，对于不包含$ g_0 $的每个子组$ k \ le g $，限制至少一个字符的$ k $ $ \ nathcal {x}}（g）中的一个字符包含琐碎的字符作为不可约合的求和。然后，我们将字符限制在$ G_0 $中，并使用有关$ G $的Cayley图的标准参数来暗示结果。此外，关于$ g $的对称子集的产生，我们获得的结果略有弱。 我们通过考虑$ s {n} $，$ n $元素上的对称组，并呈现$ \ Mathcal {r}（s_ {n}）$，这是$ s_ {n} $的八组不可修复的非平地表示$ s_ { $ \ left \lvertς\ right \ rvert $不是$ {\ sum_ {σ\ inσ}ρ（σ）} $的特征值，每$ρ\ in \ nathcal {r}（r}（r}（s_ {n}）$，这是$ 12 $ se的$ ired n n of s y mathcal {r}（s_ {n}）$满足这种情况。
Let $G_0$ be a either $SL_n(\mathbb{F}_q)$, the special linear group over the finite field with $q$ elements, or $PSL_n(\mathbb{F}_q)$, its projective quotient, and let $Σ$ be a symmetric subset of $G_0$, namely, if $x \in Σ$ then $x^{-1} \in Σ$. We find a certain set $\mathcal{R}(G_0)$ of irreducible representations of $G_0$ whose size is at most $5$, such that $Σ$ generates $G_0$ if and only if $\left\lvertΣ\right\rvert$ is not an eigenvalue of ${\sum_{σ\in Σ} ρ(σ)}$ for every $ρ\in \mathcal{R}(G_0)$. To achieve this result, let $G$ be either $GL_n(\mathbb{F}_q)$ or $PGL_n(\mathbb{F}_q)$. We consider $\mathcal{X}(G)$, some set of irreducible nontrivial characters of $G$, whose size is at most $5$. We show that for every subgroup $K \le G$ that does not contain $G_0$, the restriction to $K$ of at least one of the characters in $\mathcal{X}(G)$ contains the trivial character as an irreducible summand. We then restrict the characters to $G_0$ and use standard arguments about the Cayley graph of $G$ to imply the result. In addition, we obtain slightly weaker results about the generation of symmetric subsets of $G$. We finish by considering $S_{n}$, the symmetric group on $n$ elements, and presenting $\mathcal{R}(S_{n})$, a set of eight irreducible nontrivial representations of $S_{n}$, such that a symmetric subset $Σ\subseteq S_{n}$ generates $S_{n}$ if and only if $\left\lvertΣ\right\rvert$ is not an eigenvalue of ${\sum_{σ\in Σ} ρ(σ)}$ for every $ρ\in \mathcal{R}(S_{n})$, which is an improvement upon the previously known set of $12$ irreducible nontrivial representations of $S_{n}$ that satisfies this condition.
