$q$-Racah型Leonard三元组
Leonard triples of $q$-Racah type
摘要：
设$\mathbb F$表示域，$V$表示$\mathbb F$上具有有限正维数的向量空间。选择一个非零的$q \in \mathbb F$，使得$q^4 \not=1$，并让$A,B,C$表示$V$上具有$q$-Racah类型的Leonard三元组。我们发现，在${\rm End}(V)$中存在可逆的$W, W', W'' $，使得（i）$A$与$W$和$W^{-1}BW-C$互换;（ii）$B$与$W'$和$(W')^{-1}CW'-A$互换;（iii）$C$与$W''$和$(W'')^{-1}AW''-B$互换。此外，$W,W', W''$中的每一个直到乘以$\mathbb F$中的非零标量都是唯一的。我们证明了三个元素$W'W, W''W', WW''$相互交换，并且它们的乘积是恒等式的标量倍。得到了一些相关的结果。
英文摘要：
Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. Pick a nonzero $q \in \mathbb F$ such that $q^4 \not=1$, and let $A,B,C$ denote a Leonard triple on $V$ that has $q$-Racah type. We show that there exist invertible $W, W', W'' $ in ${\rm End}(V)$ such that (i) $A$ commutes with $W$ and $W^{-1}BW-C$; (ii) $B$ commutes with $W'$ and $(W')^{-1}CW'-A$; (iii) $C$ commutes with $W''$ and $(W'')^{-1}AW''-B$. Moreover each of $W,W', W''$ is unique up to multiplication by a nonzero scalar in $\mathbb F$. We show that the three elements $W'W, W''W', WW''$ mutually commute, and their product is a scalar multiple of the identity. A number of related results are obtained.
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