论文标题
$ \ mathrm {sl} _2（\ mathbb {f} _p）$的表示戒指和其自然模块的稳定模块化在特征$ p $中
The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$
论文作者
论文摘要
让$ p $是一个奇怪的素数，让$ k $是特征$ p $的领域。我们提供了$ k \ mathrm {sl} _2（\ mathbb {f} _p）$ modulo Projectives的表示环的实用代数描述。然后，我们研究了一个天然$ k \ mathrm {sl} _2（\ mathbb {f} _p）$ - 模块$ \ nabla^nabla^n n nabla^n v \ mathrm {sym}^l e $的家庭$ k \ mathrm {sl} _2（\ mathbb {f} _p）$ e $ sym {sym}^l e $ sottition $ bes $ p $ p $ p $ p $和0 \ p-p-在这个家庭中，我们将$ e $的模块化整数分类为投影，而模块化的$ e $的模块化成分，它们只有一个不可否认的不可塑性的求和。我们将这些结果概括为类似的分类，这些分类被$ e $的模块化整数被$ k \ mathrm {sl} _2（\ mathbb {f} _p）$ - 表格$ \ nabla^νv$的模块，其中$ v $是一个非投射性的不可能的不合别$ k \ mathrm {sl} _2（\ mathbb {f} _p）$ - 模块和$ |ν| <p $。
Let $p$ be an odd prime and let $k$ be a field of characteristic $p$. We provide a practical algebraic description of the representation ring of $k\mathrm{SL}_2(\mathbb{F}_p)$ modulo projectives. We then investigate a family of modular plethysms of the natural $k\mathrm{SL}_2(\mathbb{F}_p)$-module $E$ of the form $\nabla^ν\mathrm{Sym}^l E$ for a partition $ν$ of size less than $p$ and $0\leq l\leq p-2$. Within this family we classify both the modular plethysms of $E$ which are projective and the modular plethysms of $E$ which have only one non-projective indecomposable summand which is moreover irreducible. We generalise these results to similar classifications where modular plethysms of $E$ are replaced by $k\mathrm{SL}_2(\mathbb{F}_p)$-modules of the form $\nabla^ν V$, where $V$ is a non-projective indecomposable $k\mathrm{SL}_2(\mathbb{F}_p)$-module and $|ν|<p$.
