Leonard对24个碱基之间的跃迁映射
Transition maps between the 24 bases for a Leonard pair
摘要：
设$V$表示具有有限正维数的向量空间。我们考虑一对线性变换$A : V \to V$和$A^* : V \to V$满足以下（i）和（ii）：（i）存在一个基$V$，关于该基，表示$A$的矩阵是不可约三对角的，表示$A^*$的矩阵是对角的. (ii)存在$V$的基，关于该基，表示$A^*$的矩阵是不可约三对角的，并且表示$A$的矩阵是对角的。我们称这样的对为$V$上的Leonard对。在以前的论文中，我们描述了24个特殊基地$V$。这些基的一个特征是，对于它们中的每一个，表示$A$和$A^*$的矩阵是（i）对角和不可约三对角的或（ii）不可约三对角和对角的或（iii）下双对角和上双对角的或（iv）上双对角和下双对角的。对于24个有序基对中的每一个，都存在从$V$到$V$的唯一线性变换，它将第一个基发送到第二个基;我们称之为转换映射。在本文中，我们发现每个转换映射明确作为一个多项式在$A,A^*$。
英文摘要：
Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In an earlier paper we described 24 special bases for $V$. One feature of these bases is that with respect to each of them the matrices that represent $A$ and $A^*$ are (i) diagonal and irreducible tridiagonal or (ii) irreducible tridiagonal and diagonal or (iii) lower bidiagonal and upper bidiagonal or (iv) upper bidiagonal and lower bidiagonal. For each ordered pair of bases among the 24, there exists a unique linear transformation from $V$ to $V$ that sends the first basis to the second basis; we call this the transition map. In this paper we find each transition map explicitly as a polynomial in $A,A^*$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
