In this paper we introduce a novel family of Markov chains on the simple representations of SL₂ \(\left ({\mathbb {F}_p}\right )\) in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL₂ \(\left ({\mathbb {F}_p}\right )\), emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL₂ \(\left ({\mathbb {F}_p}\right )\)-representations.

在本文中，我们通过定义固定的简单模块并选择张量来定义SL ₂ \（\ left（{left（{\ mathbb {F} _p} \ right）\）不可分解的非投影求和。我们证明了这些链是可逆的，并找到了它们连接的组件和它们的固定分布。我们在链的性质和SL ₂ \（\ left（{left（{\ mathbb {F} _p} \ right）\）的表示理论之间绘制联系，强调张量积的对称性。我们还提供了简单SL ₂ \（\ left（{\ mathbb {F} _p} \ right）\） -表示形式的张量积分解的基本证明。