标题： 雅可比-特鲁迪行列式在有限域上 显示英文标题

标题： Jacobi-Trudi determinants over finite fields

摘要： 在本文中，我们致力于回答以下问题：给定从整数上的对称函数环到有限域 $\mathbb{F}_q$的均匀随机代数同态，Schur 函数 $s_\lambda$映射到零的概率是多少？ 我们证明这个概率至少为 $1/q$，并且渐近为 $1/q$。 此外，我们给出了所有可以达到概率 $1/q$的形状的完整分类。 另外，我们识别出某些形状族，其中对应的 Schur 函数被发送到零是独立事件，并我们研究了 Schur 函数在 $\mathbb{F}_q$中被映射到非零值的概率。 显示英文摘要

摘要： In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_\lambda$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.